cp-algo/math/poly.hpp

• View this file on GitHub
• Last update: 2025-12-30 00:54:24+01:00
• Include: #include "cp-algo/math/poly.hpp"
• Link: https://codeforces.com/blog/entry/73947?#comment-581173

Depends on

• cp-algo/math/affine.hpp
• cp-algo/math/combinatorics.hpp
• cp-algo/math/common.hpp
• cp-algo/math/cvector.hpp
• cp-algo/math/fft.hpp
• cp-algo/math/poly/impl/div.hpp
• cp-algo/math/poly/impl/euclid.hpp
• cp-algo/number_theory/discrete_sqrt.hpp
• cp-algo/number_theory/modint.hpp
• cp-algo/random/rng.hpp
• cp-algo/util/big_alloc.hpp
• cp-algo/util/checkpoint.hpp
• cp-algo/util/complex.hpp
• cp-algo/util/simd.hpp

Required by

• cp-algo/linalg/frobenius.hpp

Verified with

• Characteristic Polynomial (verify/linalg/characteristic.test.cpp)
• Pow of Matrix (Frobenius) (verify/linalg/pow_fast.test.cpp)
• Chromatic Polynomial (verify/math/chromatic_poly.test.cpp)
• Bell Number (verify/poly/bell.test.cpp)
• Division of Polynomials (verify/poly/div.test.cpp)
• Exp of Power Series (verify/poly/exp.test.cpp)
• Find Linear Recurrence (verify/poly/find_linrec.test.cpp)
• Polynomial Interpolation (verify/poly/inter.test.cpp)
• Polynomial Interpolation (Geometric Sequence) (verify/poly/inter_geo.test.cpp)
• Inv of Power Series (verify/poly/inv.test.cpp)
• Inv of Polynomials (verify/poly/inv_mod.test.cpp)
• Consecutive Terms of Linear Recursion (verify/poly/linrec_consecutive.test.cpp)
• Kth term of Linearly Recurrent Sequence (verify/poly/linrec_single.test.cpp)
• Log of Power Series (verify/poly/log.test.cpp)
• Multipoint Evaluation (verify/poly/multieval.test.cpp)
• Multipoint Evaluation (Geometric Sequence) (verify/poly/multieval_geo.test.cpp)
• Conversion to Newton Basis (verify/poly/newton.test.cpp)
• Pow of Power Series (verify/poly/pow.test.cpp)
• Product of Polynomial Sequence (verify/poly/prod_sequence.test.cpp)
• Polynomial Root Finding (verify/poly/roots.test.cpp)
• Shift of Sampling Points of Polynomial (verify/poly/sampling.test.cpp)
• Sqrt of Power Series (verify/poly/sqrt.test.cpp)
• $\sum\limits_{i=0}^\infty r^i i^d$ (verify/poly/sum_exp_poly.test.cpp)
• Polynomial Taylor Shift (verify/poly/taylor.test.cpp)

Code

#ifndef CP_ALGO_MATH_POLY_HPP
#define CP_ALGO_MATH_POLY_HPP
#include "poly/impl/euclid.hpp"
#include "poly/impl/div.hpp"
#include "combinatorics.hpp"
#include "../number_theory/discrete_sqrt.hpp"
#include "fft.hpp"
#include <functional>
#include <algorithm>
#include <iostream>
#include <optional>
#include <utility>
#include <vector>
#include <list>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
    template<typename T>
    struct poly_t {
        using Vector = big_vector<T>;
        using base = T;
        Vector a;
        
        poly_t& normalize() {
            while(deg() >= 0 && lead() == base(0)) {
                a.pop_back();
            }
            return *this;
        }
        
        poly_t(){}
        poly_t(T a0): a{a0} {normalize();}
        poly_t(Vector const& t): a(t) {normalize();}
        poly_t(Vector &&t): a(std::move(t)) {normalize();}
        
        poly_t& negate_inplace() {
            std::ranges::transform(a, begin(a), std::negate{});
            return *this;
        }
        poly_t operator -() const {
            return poly_t(*this).negate_inplace();
        }
        poly_t& operator += (poly_t const& t) {
            a.resize(std::max(size(a), size(t.a)));
            std::ranges::transform(a, t.a, begin(a), std::plus{});
            return normalize();
        }
        poly_t& operator -= (poly_t const& t) {
            a.resize(std::max(size(a), size(t.a)));
            std::ranges::transform(a, t.a, begin(a), std::minus{});
            return normalize();
        }
        poly_t operator + (poly_t const& t) const {return poly_t(*this) += t;}
        poly_t operator - (poly_t const& t) const {return poly_t(*this) -= t;}
        
        poly_t& mod_xk_inplace(size_t k) {
            a.resize(std::min(size(a), k));
            return normalize();
        }
        poly_t& mul_xk_inplace(size_t k) {
            a.insert(begin(a), k, T(0));
            return normalize();
        }
        poly_t& div_xk_inplace(int64_t k) {
            if(k < 0) {
                return mul_xk_inplace(-k);
            }
            a.erase(begin(a), begin(a) + std::min<size_t>(k, size(a)));
            return normalize();
        }
        poly_t &substr_inplace(size_t l, size_t k) {
            return mod_xk_inplace(l + k).div_xk_inplace(l);
        }
        poly_t mod_xk(size_t k) const {return poly_t(*this).mod_xk_inplace(k);}
        poly_t mul_xk(size_t k) const {return poly_t(*this).mul_xk_inplace(k);}
        poly_t div_xk(int64_t k) const {return poly_t(*this).div_xk_inplace(k);}
        poly_t substr(size_t l, size_t k) const {return poly_t(*this).substr_inplace(l, k);}
        
        poly_t& operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
        poly_t operator * (const poly_t &t) const {return poly_t(*this) *= t;}

        poly_t& operator /= (const poly_t &t) {return *this = divmod(t)[0];}
        poly_t& operator %= (const poly_t &t) {return *this = divmod(t)[1];}
        poly_t operator / (poly_t const& t) const {return poly_t(*this) /= t;}
        poly_t operator % (poly_t const& t) const {return poly_t(*this) %= t;}

        poly_t& operator *= (T const& x) {
            for(auto &it: a) {
                it *= x;
            }
            return normalize();
        }
        poly_t& operator /= (T const& x) {return *this *= x.inv();}
        poly_t operator * (T const& x) const {return poly_t(*this) *= x;}
        poly_t operator / (T const& x) const {return poly_t(*this) /= x;}
        
        poly_t& reverse(size_t n) {
            a.resize(n);
            std::ranges::reverse(a);
            return normalize();
        }
        poly_t& reverse() {return reverse(size(a));}
        poly_t reversed(size_t n) const {return poly_t(*this).reverse(n);}
        poly_t reversed() const {return poly_t(*this).reverse();}
        
        std::array<poly_t, 2> divmod(poly_t const& b) const {
            return poly::impl::divmod(*this, b);
        }
        
        // reduces A/B to A'/B' such that
        // deg B' < deg A / 2
        static std::pair<std::list<poly_t>, linfrac<poly_t>> half_gcd(auto &&A, auto &&B) {
            return poly::impl::half_gcd(A, B);
        }
        // reduces A / B to gcd(A, B) / 0
        static std::pair<std::list<poly_t>, linfrac<poly_t>> full_gcd(auto &&A, auto &&B) {
            return poly::impl::full_gcd(A, B);
        }
        static poly_t gcd(poly_t &&A, poly_t &&B) {
            full_gcd(A, B);
            return A;
        }
        
        // Returns a (non-monic) characteristic polynomial
        // of the minimum linear recurrence for the sequence
        poly_t min_rec(size_t d) const {
            return poly::impl::min_rec(*this, d);
        }
        
        // calculate inv to *this modulo t
        std::optional<poly_t> inv_mod(poly_t const& t) const {
            return poly::impl::inv_mod(*this, t);
        };
        
        poly_t negx() const { // A(x) -> A(-x)
            auto res = *this;
            for(int i = 1; i <= deg(); i += 2) {
                res.a[i] = -res[i];
            }
            return res;
        }
        
        void print(int n) const {
            for(int i = 0; i < n; i++) {
                std::cout << (*this)[i] << ' ';
            }
            std::cout << "\n";
        }
        
        void print() const {
            print(deg() + 1);
        }
        
        T eval(T x) const { // evaluates in single point x
            T res(0);
            for(int i = deg(); i >= 0; i--) {
                res *= x;
                res += a[i];
            }
            return res;
        }
        
        T lead() const { // leading coefficient
            assert(!is_zero());
            return a.back();
        }
        
        int deg() const { // degree, -1 for P(x) = 0
            return (int)a.size() - 1;
        }
        
        bool is_zero() const {
            return a.empty();
        }
        
        T operator [](int idx) const {
            return idx < 0 || idx > deg() ? T(0) : a[idx];
        }
        
        T& coef(size_t idx) { // mutable reference at coefficient
            return a[idx];
        }
        
        bool operator == (const poly_t &t) const {return a == t.a;}
        bool operator != (const poly_t &t) const {return a != t.a;}
        
        poly_t& deriv_inplace(int k = 1) {
            if(deg() + 1 < k) {
                return *this = poly_t{};
            }
            for(int i = k; i <= deg(); i++) {
                a[i - k] = fact<T>(i) * rfact<T>(i - k) * a[i];
            }
            a.resize(deg() + 1 - k);
            return *this;
        }
        poly_t deriv(int k = 1) const { // calculate derivative
            return poly_t(*this).deriv_inplace(k);
        }

        poly_t& integr_inplace() {
            a.push_back(0);
            for(int i = deg() - 1; i >= 0; i--) {
                a[i + 1] = a[i] * small_inv<T>(i + 1);
            }
            a[0] = 0;
            return *this;
        }
        poly_t integr() const { // calculate integral with C = 0
            Vector res(deg() + 2);
            for(int i = 0; i <= deg(); i++) {
                res[i + 1] = a[i] * small_inv<T>(i + 1);
            }
            return res;
        }
        
        size_t trailing_xk() const { // Let p(x) = x^k * t(x), return k
            if(is_zero()) {
                return -1;
            }
            int res = 0;
            while(a[res] == T(0)) {
                res++;
            }
            return res;
        }
        
        // calculate log p(x) mod x^n
        poly_t& log_inplace(size_t n) {
            assert(a[0] == T(1));
            mod_xk_inplace(n);
            return (inv_inplace(n) *= mod_xk_inplace(n).deriv()).mod_xk_inplace(n - 1).integr_inplace();
        }
        poly_t log(size_t n) const {
            return poly_t(*this).log_inplace(n);
        }
        
        poly_t& mul_truncate(poly_t const& t, size_t k) {
            fft::mul_truncate(a, t.a, k);
            return normalize();
        }

        poly_t& exp_inplace(size_t n) {
            if(is_zero()) {
                return *this = T(1);
            }
            assert(a[0] == T(0));
            a[0] = 1;
            size_t a = 1;
            while(a < n) {
                poly_t C = log(2 * a).div_xk_inplace(a) - substr(a, 2 * a);
                *this -= C.mul_truncate(*this, a).mul_xk_inplace(a);
                a *= 2;
            }
            return mod_xk_inplace(n);
        }

        poly_t exp(size_t n) const { // calculate exp p(x) mod x^n
            return poly_t(*this).exp_inplace(n);
        }
        
        poly_t pow_bin(int64_t k, size_t n) const { // O(n log n log k)
            if(k == 0) {
                return poly_t(1).mod_xk(n);
            } else {
                auto t = pow(k / 2, n);
                t = (t * t).mod_xk(n);
                return (k % 2 ? *this * t : t).mod_xk(n);
            }
        }

        poly_t circular_closure(size_t m) const {
            if(deg() == -1) {
                return *this;
            }
            auto t = *this;
            for(size_t i = t.deg(); i >= m; i--) {
                t.a[i - m] += t.a[i];
            }
            t.a.resize(std::min(t.a.size(), m));
            return t;
        }

        static poly_t mul_circular(poly_t const& a, poly_t const& b, size_t m) {
            return (a.circular_closure(m) * b.circular_closure(m)).circular_closure(m);
        }

        poly_t powmod_circular(int64_t k, size_t m) const {
            if(k == 0) {
                return poly_t(1);
            } else {
                auto t = powmod_circular(k / 2, m);
                t = mul_circular(t, t, m);
                if(k % 2) {
                    t = mul_circular(t, *this, m);
                }
                return t;
            }
        }
        
        poly_t powmod(int64_t k, poly_t const& md) const {
            return poly::impl::powmod(*this, k, md);
        }
        
        // O(d * n) with the derivative trick from
        // https://codeforces.com/blog/entry/73947?#comment-581173
        poly_t pow_dn(int64_t k, size_t n) const {
            if(n == 0) {
                return poly_t(T(0));
            }
            assert((*this)[0] != T(0));
            Vector Q(n);
            Q[0] = bpow(a[0], k);
            auto a0inv = a[0].inv();
            for(int i = 1; i < (int)n; i++) {
                for(int j = 1; j <= std::min(deg(), i); j++) {
                    Q[i] += a[j] * Q[i - j] * (T(k) * T(j) - T(i - j));
                }
                Q[i] *= small_inv<T>(i) * a0inv;
            }
            return Q;
        }
        
        // calculate p^k(n) mod x^n in O(n log n)
        // might be quite slow due to high constant
        poly_t pow(int64_t k, size_t n) const {
            if(is_zero()) {
                return k ? *this : poly_t(1);
            }
            size_t i = trailing_xk();
            if(i > 0) {
                return k >= int64_t(n + i - 1) / (int64_t)i ? poly_t(T(0)) : div_xk(i).pow(k, n - i * k).mul_xk(i * k);
            }
            if(std::min(deg(), (int)n) <= magic) {
                return pow_dn(k, n);
            }
            if(k <= magic) {
                return pow_bin(k, n);
            }
            T j = a[i];
            poly_t t = *this / j;
            return bpow(j, k) * (t.log(n) * T(k)).exp(n).mod_xk(n);
        }
        
        // returns std::nullopt if undefined
        std::optional<poly_t> sqrt(size_t n) const {
            if(is_zero()) {
                return *this;
            }
            size_t i = trailing_xk();
            if(i % 2) {
                return std::nullopt;
            } else if(i > 0) {
                auto ans = div_xk(i).sqrt(n - i / 2);
                return ans ? ans->mul_xk(i / 2) : ans;
            }
            auto st = math::sqrt((*this)[0]);
            if(st) {
                poly_t ans = *st;
                size_t a = 1;
                while(a < n) {
                    a *= 2;
                    ans -= (ans - mod_xk(a) * ans.inv(a)).mod_xk(a) / 2;
                }
                return ans.mod_xk(n);
            }
            return std::nullopt;
        }
        
        poly_t mulx(T a) const { // component-wise multiplication with a^k
            T cur = 1;
            poly_t res(*this);
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= cur;
                cur *= a;
            }
            return res;
        }

        poly_t mulx_sq(T a) const { // component-wise multiplication with a^{k choose 2}
            T cur = 1, total = 1;
            poly_t res(*this);
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= total;
                cur *= a;
                total *= cur;
            }
            return res;
        }

        // be mindful of maxn, as the function
        // requires multiplying polynomials of size deg() and n+deg()!
        poly_t chirpz(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
            if(is_zero()) {
                return Vector(n, 0);
            }
            if(z == T(0)) {
                Vector ans(n, (*this)[0]);
                if(n > 0) {
                    ans[0] = accumulate(begin(a), end(a), T(0));
                }
                return ans;
            }
            auto A = mulx_sq(z.inv());
            auto B = ones(n+deg()).mulx_sq(z);
            return semicorr(B, A).mod_xk(n).mulx_sq(z.inv());
        }

        // res[i] = prod_{1 <= j <= i} 1/(1 - z^j)
        static auto _1mzk_prod_inv(T z, int n) {
            Vector res(n, 1), zk(n);
            zk[0] = 1;
            for(int i = 1; i < n; i++) {
                zk[i] = zk[i - 1] * z;
                res[i] = res[i - 1] * (T(1) - zk[i]);
            }
            res.back() = res.back().inv();
            for(int i = n - 2; i >= 0; i--) {
                res[i] = (T(1) - zk[i+1]) * res[i+1];
            }
            return res;
        }
        
        // prod_{0 <= j < n} (1 - z^j x)
        static auto _1mzkx_prod(T z, int n) {
            if(n == 1) {
                return poly_t(Vector{1, -1});
            } else {
                auto t = _1mzkx_prod(z, n / 2);
                t *= t.mulx(bpow(z, n / 2));
                if(n % 2) {
                    t *= poly_t(Vector{1, -bpow(z, n - 1)});
                }
                return t;
            }
        }

        poly_t chirpz_inverse(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
            if(is_zero()) {
                return {};
            }
            if(z == T(0)) {
                if(n == 1) {
                    return *this;
                } else {
                    return Vector{(*this)[1], (*this)[0] - (*this)[1]};
                }
            }
            Vector y(n);
            for(int i = 0; i < n; i++) {
                y[i] = (*this)[i];
            }
            auto prods_pos = _1mzk_prod_inv(z, n);
            auto prods_neg = _1mzk_prod_inv(z.inv(), n);

            T zn = bpow(z, n-1).inv();
            T znk = 1;
            for(int i = 0; i < n; i++) {
                y[i] *= znk * prods_neg[i] * prods_pos[(n - 1) - i];
                znk *= zn;
            }

            poly_t p_over_q = poly_t(y).chirpz(z, n);
            poly_t q = _1mzkx_prod(z, n);

            return (p_over_q * q).mod_xk_inplace(n).reverse(n);
        }

        static poly_t build(big_vector<poly_t> &res, int v, auto L, auto R) { // builds evaluation tree for (x-a1)(x-a2)...(x-an)
            if(R - L == 1) {
                return res[v] = Vector{-*L, 1};
            } else {
                auto M = L + (R - L) / 2;
                return res[v] = build(res, 2 * v, L, M) * build(res, 2 * v + 1, M, R);
            }
        }

        poly_t to_newton(big_vector<poly_t> &tree, int v, auto l, auto r) {
            if(r - l == 1) {
                return *this;
            } else {
                auto m = l + (r - l) / 2;
                auto A = (*this % tree[2 * v]).to_newton(tree, 2 * v, l, m);
                auto B = (*this / tree[2 * v]).to_newton(tree, 2 * v + 1, m, r);
                return A + B.mul_xk(m - l);
            }
        }

        poly_t to_newton(Vector p) {
            if(is_zero()) {
                return *this;
            }
            size_t n = p.size();
            big_vector<poly_t> tree(4 * n);
            build(tree, 1, begin(p), end(p));
            return to_newton(tree, 1, begin(p), end(p));
        }

        Vector eval(big_vector<poly_t> &tree, int v, auto l, auto r) { // auxiliary evaluation function
            if(r - l == 1) {
                return {eval(*l)};
            } else {
                auto m = l + (r - l) / 2;
                auto A = (*this % tree[2 * v]).eval(tree, 2 * v, l, m);
                auto B = (*this % tree[2 * v + 1]).eval(tree, 2 * v + 1, m, r);
                A.insert(end(A), begin(B), end(B));
                return A;
            }
        }
        
        Vector eval(Vector x) { // evaluate polynomial in (x1, ..., xn)
            size_t n = x.size();
            if(is_zero()) {
                return Vector(n, T(0));
            }
            big_vector<poly_t> tree(4 * n);
            build(tree, 1, begin(x), end(x));
            return eval(tree, 1, begin(x), end(x));
        }
        
        poly_t inter(big_vector<poly_t> &tree, int v, auto ly, auto ry) { // auxiliary interpolation function
            if(ry - ly == 1) {
                return {*ly / a[0]};
            } else {
                auto my = ly + (ry - ly) / 2;
                auto A = (*this % tree[2 * v]).inter(tree, 2 * v, ly, my);
                auto B = (*this % tree[2 * v + 1]).inter(tree, 2 * v + 1, my, ry);
                return A * tree[2 * v + 1] + B * tree[2 * v];
            }
        }
        
        static auto inter(Vector x, Vector y) { // interpolates minimum polynomial from (xi, yi) pairs
            size_t n = x.size();
            big_vector<poly_t> tree(4 * n);
            return build(tree, 1, begin(x), end(x)).deriv().inter(tree, 1, begin(y), end(y));
        }

        static auto resultant(poly_t a, poly_t b) { // computes resultant of a and b
            if(b.is_zero()) {
                return 0;
            } else if(b.deg() == 0) {
                return bpow(b.lead(), a.deg());
            } else {
                int pw = a.deg();
                a %= b;
                pw -= a.deg();
                auto mul = bpow(b.lead(), pw) * T((b.deg() & a.deg() & 1) ? -1 : 1);
                auto ans = resultant(b, a);
                return ans * mul;
            }
        }
                
        static poly_t xk(size_t n) { // P(x) = x^n
            return poly_t(T(1)).mul_xk(n);
        }
        
        static poly_t ones(size_t n) { // P(x) = 1 + x + ... + x^{n-1} 
            return Vector(n, 1);
        }
        
        static poly_t expx(size_t n) { // P(x) = e^x (mod x^n)
            return ones(n).borel();
        }

        static poly_t log1px(size_t n) { // P(x) = log(1+x) (mod x^n)
            Vector coeffs(n, 0);
            for(size_t i = 1; i < n; i++) {
                coeffs[i] = (i & 1 ? T(i).inv() : -T(i).inv());
            }
            return coeffs;
        }

        static poly_t log1mx(size_t n) { // P(x) = log(1-x) (mod x^n)
            return -ones(n).integr();
        }
        
        // [x^k] (a corr b) = sum_{i} a{(k-m)+i}*bi
        static poly_t corr(poly_t const& a, poly_t const& b) { // cross-correlation
            return a * b.reversed();
        }

        // [x^k] (a semicorr b) = sum_i a{i+k} * b{i}
        static poly_t semicorr(poly_t const& a, poly_t const& b) {
            return corr(a, b).div_xk(b.deg());
        }
        
        poly_t invborel() const { // ak *= k!
            auto res = *this;
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= fact<T>(i);
            }
            return res;
        }
        
        poly_t borel() const { // ak /= k!
            auto res = *this;
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= rfact<T>(i);
            }
            return res;
        }
        
        poly_t shift(T a) const { // P(x + a)
            return semicorr(invborel(), expx(deg() + 1).mulx(a)).borel();
        }
        
        poly_t x2() { // P(x) -> P(x^2)
            Vector res(2 * a.size());
            for(size_t i = 0; i < a.size(); i++) {
                res[2 * i] = a[i];
            }
            return res;
        }
        
        // Return {P0, P1}, where P(x) = P0(x) + xP1(x)
        std::array<poly_t, 2> bisect(size_t n) const {
            n = std::min(n, size(a));
            Vector res[2];
            for(size_t i = 0; i < n; i++) {
                res[i % 2].push_back(a[i]);
            }
            return {res[0], res[1]};
        }
        std::array<poly_t, 2> bisect() const {
            return bisect(size(a));
        }
        
        // Find [x^k] P / Q
        static T kth_rec_inplace(poly_t &P, poly_t &Q, int64_t k) {
            while(k > Q.deg()) {
                size_t n = Q.a.size();
                auto [Q0, Q1] = Q.bisect();
                auto [P0, P1] = P.bisect();
                
                size_t N = fft::com_size((n + 1) / 2, (n + 1) / 2);
                
                auto Q0f = fft::dft<T>(Q0.a, N);
                auto Q1f = fft::dft<T>(Q1.a, N);
                auto P0f = fft::dft<T>(P0.a, N);
                auto P1f = fft::dft<T>(P1.a, N);
                
                Q = poly_t(Q0f * Q0f) -= poly_t(Q1f * Q1f).mul_xk_inplace(1);
                if(k % 2) {
                    P = poly_t(Q0f *= P1f) -= poly_t(Q1f *= P0f);
                } else {
                    P = poly_t(Q0f *= P0f) -= poly_t(Q1f *= P1f).mul_xk_inplace(1);
                }
                k /= 2;
            }
            return (P *= Q.inv_inplace(Q.deg() + 1))[(int)k];
        }
        static T kth_rec(poly_t const& P, poly_t const& Q, int64_t k) {
            return kth_rec_inplace(poly_t(P), poly_t(Q), k);
        }

        // inverse series mod x^n
        poly_t& inv_inplace(size_t n) {
            return poly::impl::inv_inplace(*this, n);
        }
        poly_t inv(size_t n) const {
            return poly_t(*this).inv_inplace(n);
        }
        // [x^k]..[x^{k+n-1}] of inv()
        // supports negative k if k+n >= 0
        poly_t& inv_inplace(int64_t k, size_t n) {
            return poly::impl::inv_inplace(*this, k, n);
        }
        poly_t inv(int64_t k, size_t n) const {
            return poly_t(*this).inv_inplace(k, n);
        }
        
        // compute A(B(x)) mod x^n in O(n^2)
        static poly_t compose(poly_t A, poly_t B, int n) {
            int q = std::sqrt(n);
            big_vector<poly_t> Bk(q);
            auto Bq = B.pow(q, n);
            Bk[0] = poly_t(T(1));
            for(int i = 1; i < q; i++) {
                Bk[i] = (Bk[i - 1] * B).mod_xk(n);
            }
            poly_t Bqk(1);
            poly_t ans;
            for(int i = 0; i <= n / q; i++) {
                poly_t cur;
                for(int j = 0; j < q; j++) {
                    cur += Bk[j] * A[i * q + j];
                }
                ans += (Bqk * cur).mod_xk(n);
                Bqk = (Bqk * Bq).mod_xk(n);
            }
            return ans;
        }
        
        // compute A(B(x)) mod x^n in O(sqrt(pqn log^3 n))
        // preferrable when p = deg A and q = deg B
        // are much less than n
        static poly_t compose_large(poly_t A, poly_t B, int n) {
            if(B[0] != T(0)) {
                return compose_large(A.shift(B[0]), B - B[0], n);
            }
            
            int q = std::sqrt(n);
            auto [B0, B1] = std::make_pair(B.mod_xk(q), B.div_xk(q));
            
            B0 = B0.div_xk(1);
            big_vector<poly_t> pw(A.deg() + 1);
            auto getpow = [&](int k) {
                return pw[k].is_zero() ? pw[k] = B0.pow(k, n - k) : pw[k];
            };
            
            std::function<poly_t(poly_t const&, int, int)> compose_dac = [&getpow, &compose_dac](poly_t const& f, int m, int N) {
                if(f.deg() <= 0) {
                    return f;
                }
                int k = m / 2;
                auto [f0, f1] = std::make_pair(f.mod_xk(k), f.div_xk(k));
                auto [A, B] = std::make_pair(compose_dac(f0, k, N), compose_dac(f1, m - k, N - k));
                return (A + (B.mod_xk(N - k) * getpow(k).mod_xk(N - k)).mul_xk(k)).mod_xk(N);
            };
            
            int r = n / q;
            auto Ar = A.deriv(r);
            auto AB0 = compose_dac(Ar, Ar.deg() + 1, n);
            
            auto Bd = B0.mul_xk(1).deriv();
            
            poly_t ans = T(0);
            
            big_vector<poly_t> B1p(r + 1);
            B1p[0] = poly_t(T(1));
            for(int i = 1; i <= r; i++) {
                B1p[i] = (B1p[i - 1] * B1.mod_xk(n - i * q)).mod_xk(n - i * q);
            }
            while(r >= 0) {
                ans += (AB0.mod_xk(n - r * q) * rfact<T>(r) * B1p[r]).mul_xk(r * q).mod_xk(n);
                r--;
                if(r >= 0) {
                    AB0 = ((AB0 * Bd).integr() + A[r] * fact<T>(r)).mod_xk(n);
                }
            }
            
            return ans;
        }
    };
    template<typename base>
    static auto operator * (const auto& a, const poly_t<base>& b) {
        return b * a;
    }
};
#pragma GCC pop_options
#endif // CP_ALGO_MATH_POLY_HPP

#line 1 "cp-algo/math/poly.hpp"


#line 1 "cp-algo/math/poly/impl/euclid.hpp"


#line 1 "cp-algo/math/affine.hpp"


#include <optional>
#include <utility>
#include <cassert>
#include <tuple>
namespace cp_algo::math {
    // a * x + b
    template<typename base>
    struct lin {
        base a = 1, b = 0;
        std::optional<base> c;
        lin() {}
        lin(base b): a(0), b(b) {}
        lin(base a, base b): a(a), b(b) {}
        lin(base a, base b, base _c): a(a), b(b), c(_c) {}

        // polynomial product modulo x^2 - c
        lin operator * (const lin& t) {
            assert(c && t.c && *c == *t.c);
            return {a * t.b + b * t.a, b * t.b + a * t.a * (*c), *c};
        }

        // a * (t.a * x + t.b) + b
        lin apply(lin const& t) const {
            return {a * t.a, a * t.b + b};
        }

        void prepend(lin const& t) {
            *this = t.apply(*this);
        }

        base eval(base x) const {
            return a * x + b;
        }
    };

    // (ax+b) / (cx+d)
    template<typename base>
    struct linfrac {
        base a, b, c, d;
        linfrac(): a(1), b(0), c(0), d(1) {} // x, identity for composition
        linfrac(base a): a(a), b(1), c(1), d(0) {} // a + 1/x, for continued fractions
        linfrac(base a, base b, base c, base d): a(a), b(b), c(c), d(d) {}

        // composition of two linfracs
        linfrac operator * (linfrac t) const {
            return t.prepend(linfrac(*this));
        }

        linfrac operator-() const {
            return {-a, -b, -c, -d};
        }

        linfrac adj() const {
            return {d, -b, -c, a};
        }
        
        linfrac& prepend(linfrac const& t) {
            t.apply(a, c);
            t.apply(b, d);
            return *this;
        }

        // apply linfrac to A/B
        void apply(base &A, base &B) const {
            std::tie(A, B) = std::pair{a * A + b * B, c * A + d * B};
        }
    };
}

#line 1 "cp-algo/math/fft.hpp"


#line 1 "cp-algo/number_theory/modint.hpp"


#line 1 "cp-algo/math/common.hpp"


#include <functional>
#include <cstdint>
#line 6 "cp-algo/math/common.hpp"
#include <bit>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
    const int maxn = CP_ALGO_MAXN;
#else
    const int maxn = 1 << 19;
#endif
    const int magic = 64; // threshold for sizes to run the naive algo

    auto bpow(auto const& x, auto n, auto const& one, auto op) {
        if (n == 0) {
            return one;
        }
        auto ans = x;
        for(int j = std::bit_width<uint64_t>(n) - 2; ~j; j--) {
            ans = op(ans, ans);
            if((n >> j) & 1) {
                ans = op(ans, x);
            }
        }
        return ans;
    }
    auto bpow(auto x, auto n, auto ans) {
        return bpow(x, n, ans, std::multiplies{});
    }
    template<typename T>
    T bpow(T const& x, auto n) {
        return bpow(x, n, T(1));
    }
    inline constexpr auto inv2(auto x) {
        assert(x % 2);
        std::make_unsigned_t<decltype(x)> y = 1;
        while(y * x != 1) {
            y *= 2 - x * y;
        }
        return y;
    }
}

#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {

    template<typename modint, typename _Int>
    struct modint_base {
        using Int = _Int;
        using UInt = std::make_unsigned_t<Int>;
        static constexpr size_t bits = sizeof(Int) * 8;
        using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
        using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
        constexpr static Int mod() {
            return modint::mod();
        }
        constexpr static Int remod() {
            return modint::remod();
        }
        constexpr static UInt2 modmod() {
            return UInt2(mod()) * mod();
        }
        constexpr modint_base() = default;
        constexpr modint_base(Int2 rr) {
            to_modint().setr(UInt((rr + modmod()) % mod()));
        }
        modint inv() const {
            return bpow(to_modint(), mod() - 2);
        }
        modint operator - () const {
            modint neg;
            neg.r = std::min(-r, remod() - r);
            return neg;
        }
        modint& operator /= (const modint &t) {
            return to_modint() *= t.inv();
        }
        modint& operator *= (const modint &t) {
            r = UInt(UInt2(r) * t.r % mod());
            return to_modint();
        }
        modint& operator += (const modint &t) {
            r += t.r; r = std::min(r, r - remod());
            return to_modint();
        }
        modint& operator -= (const modint &t) {
            r -= t.r; r = std::min(r, r + remod());
            return to_modint();
        }
        modint operator + (const modint &t) const {return modint(to_modint()) += t;}
        modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
        modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
        modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
        // Why <=> doesn't work?..
        auto operator == (const modint &t) const {return to_modint().getr() == t.getr();}
        auto operator != (const modint &t) const {return to_modint().getr() != t.getr();}
        auto operator <= (const modint &t) const {return to_modint().getr() <= t.getr();}
        auto operator >= (const modint &t) const {return to_modint().getr() >= t.getr();}
        auto operator < (const modint &t) const {return to_modint().getr() < t.getr();}
        auto operator > (const modint &t) const {return to_modint().getr() > t.getr();}
        Int rem() const {
            UInt R = to_modint().getr();
            return R - (R > (UInt)mod() / 2) * mod();
        }
        constexpr void setr(UInt rr) {
            r = rr;
        }
        constexpr UInt getr() const {
            return r;
        }

        // Only use these if you really know what you're doing!
        static uint64_t modmod8() {return uint64_t(8 * modmod());}
        void add_unsafe(UInt t) {r += t;}
        void pseudonormalize() {r = std::min(r, r - modmod8());}
        modint const& normalize() {
            if(r >= (UInt)mod()) {
                r %= mod();
            }
            return to_modint();
        }
        void setr_direct(UInt rr) {r = rr;}
        UInt getr_direct() const {return r;}
    protected:
        UInt r;
    private:
        constexpr modint& to_modint() {return static_cast<modint&>(*this);}
        constexpr modint const& to_modint() const {return static_cast<modint const&>(*this);}
    };
    template<typename modint>
    concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
    template<modint_type modint>
    decltype(std::cin)& operator >> (decltype(std::cin) &in, modint &x) {
        typename modint::UInt r;
        auto &res = in >> r;
        x.setr(r);
        return res;
    }
    template<modint_type modint>
    decltype(std::cout)& operator << (decltype(std::cout) &out, modint const& x) {
        return out << x.getr();
    }

    template<auto m>
    struct modint: modint_base<modint<m>, decltype(m)> {
        using Base = modint_base<modint<m>, decltype(m)>;
        using Base::Base;
        static constexpr Base::Int mod() {return m;}
        static constexpr Base::UInt remod() {return m;}
        auto getr() const {return Base::r;}
    };

    template<typename Int = int>
    struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
        using Base = modint_base<dynamic_modint<Int>, Int>;
        using Base::Base;

        static Base::UInt m_reduce(Base::UInt2 ab) {
            if(mod() % 2 == 0) [[unlikely]] {
                return typename Base::UInt(ab % mod());
            } else {
                typename Base::UInt2 m = typename Base::UInt(ab) * imod();
                return typename Base::UInt((ab + m * mod()) >> Base::bits);
            }
        }
        static Base::UInt m_transform(Base::UInt a) {
            if(mod() % 2 == 0) [[unlikely]] {
                return a;
            } else {
                return m_reduce(a * pw128());
            }
        }
        dynamic_modint& operator *= (const dynamic_modint &t) {
            Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
            return *this;
        }
        void setr(Base::UInt rr) {
            Base::r = m_transform(rr);
        }
        Base::UInt getr() const {
            typename Base::UInt res = m_reduce(Base::r);
            return std::min(res, res - mod());
        }
        static Int mod() {return m;}
        static Int remod() {return 2 * m;}
        static Base::UInt imod() {return im;}
        static Base::UInt2 pw128() {return r2;}
        static void switch_mod(Int nm) {
            m = nm;
            im = m % 2 ? inv2(-m) : 0;
            r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
        }

        // Wrapper for temp switching
        auto static with_mod(Int tmp, auto callback) {
            struct scoped {
                Int prev = mod();
                ~scoped() {switch_mod(prev);}
            } _;
            switch_mod(tmp);
            return callback();
        }
    private:
        static thread_local Int m;
        static thread_local Base::UInt im, r2;
    };
    template<typename Int>
    Int thread_local dynamic_modint<Int>::m = 1;
    template<typename Int>
    dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
    template<typename Int>
    dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}

#line 1 "cp-algo/util/checkpoint.hpp"


#line 1 "cp-algo/util/big_alloc.hpp"



#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>

// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#  define CP_ALGO_USE_MMAP 1
#  include <sys/mman.h>
#else
#  define CP_ALGO_USE_MMAP 0
#endif

namespace cp_algo {
    template <typename T, size_t Align = 32>
    class big_alloc {
        static_assert( Align >= alignof(void*), "Align must be at least pointer-size");
        static_assert(std::popcount(Align) == 1, "Align must be a power of two");
    public:
        using value_type = T;
        template <class U> struct rebind { using other = big_alloc<U, Align>; };
        constexpr bool operator==(const big_alloc&) const = default;
        constexpr bool operator!=(const big_alloc&) const = default;

        big_alloc() noexcept = default;
        template <typename U, std::size_t A>
        big_alloc(const big_alloc<U, A>&) noexcept {}

        [[nodiscard]] T* allocate(std::size_t n) {
            std::size_t padded = round_up(n * sizeof(T));
            std::size_t align = std::max<std::size_t>(alignof(T),  Align);
#if CP_ALGO_USE_MMAP
            if (padded >= MEGABYTE) {
                void* raw = mmap(nullptr, padded,
                                PROT_READ | PROT_WRITE,
                                MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
                madvise(raw, padded, MADV_HUGEPAGE);
                madvise(raw, padded, MADV_POPULATE_WRITE);
                return static_cast<T*>(raw);
            }
#endif
            return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
        }

        void deallocate(T* p, std::size_t n) noexcept {
            if (!p) return;
            std::size_t padded = round_up(n * sizeof(T));
            std::size_t align  = std::max<std::size_t>(alignof(T),  Align);
    #if CP_ALGO_USE_MMAP
            if (padded >= MEGABYTE) { munmap(p, padded); return; }
    #endif
            ::operator delete(p, padded, std::align_val_t(align));
        }

    private:
        static constexpr std::size_t MEGABYTE = 1 << 20;
        static constexpr std::size_t round_up(std::size_t x) noexcept {
            return (x + Align - 1) / Align * Align;
        }
    };

    template<typename T> using big_vector = std::vector<T, big_alloc<T>>;
    template<typename T> using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
    template<typename T> using big_deque = std::deque<T, big_alloc<T>>;
    template<typename T> using big_stack = std::stack<T, big_deque<T>>;
    template<typename T> using big_queue = std::queue<T, big_deque<T>>;
    template<typename T> using big_priority_queue = std::priority_queue<T, big_vector<T>>;
    template<typename T> using big_forward_list = std::forward_list<T, big_alloc<T>>;
    using big_string = big_basic_string<char>;

    template<typename Key, typename Value, typename Compare = std::less<Key>>
    using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
    template<typename T, typename Compare = std::less<T>>
    using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
    template<typename T, typename Compare = std::less<T>>
    using big_set = std::set<T, Compare, big_alloc<T>>;
    template<typename Ref, typename V = void>

    using big_generator = std::generator<Ref, V, big_alloc<std::byte>>;
}

// Deduction guide to make elements_of with big_generator default to big_alloc
namespace std::ranges {
    template<typename Ref, typename V>
    elements_of(cp_algo::big_generator<Ref, V>&&) -> elements_of<cp_algo::big_generator<Ref, V>&&, cp_algo::big_alloc<std::byte>>;
}


#line 5 "cp-algo/util/checkpoint.hpp"
#include <chrono>
#line 8 "cp-algo/util/checkpoint.hpp"
namespace cp_algo {
#ifdef CP_ALGO_CHECKPOINT
    big_map<big_string, double> checkpoints;
    double last;
#endif
    template<bool final = false>
    void checkpoint([[maybe_unused]] auto const& _msg) {
#ifdef CP_ALGO_CHECKPOINT
        big_string msg = _msg;
        double now = (double)clock() / CLOCKS_PER_SEC;
        double delta = now - last;
        last = now;
        if(msg.size() && !final) {
            checkpoints[msg] += delta;
        }
        if(final) {
            for(auto const& [key, value] : checkpoints) {
                std::cerr << key << ": " << value * 1000 << " ms\n";
            }
            std::cerr << "Total: " << now * 1000 << " ms\n";
        }
#endif
    }
    template<bool final = false>
    void checkpoint() {
        checkpoint<final>("");
    }
}

#line 1 "cp-algo/random/rng.hpp"


#line 4 "cp-algo/random/rng.hpp"
#include <random>
namespace cp_algo::random {
    std::mt19937_64 gen(
        std::chrono::steady_clock::now().time_since_epoch().count()
    );
    uint64_t rng() {
        return gen();
    }
}

#line 1 "cp-algo/math/cvector.hpp"


#line 1 "cp-algo/util/simd.hpp"


#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>

#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif

#define CP_ALGO_SIMD_PRAGMA_PUSH \
    _Pragma("GCC push_options") \
    CP_ALGO_SIMD_AVX2_TARGET

CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
    template<typename T, size_t len>
    using simd [[gnu::vector_size(len * sizeof(T))]] = T;
    using u64x8 = simd<uint64_t, 8>;
    using u32x16 = simd<uint32_t, 16>;
    using i64x4 = simd<int64_t, 4>;
    using u64x4 = simd<uint64_t, 4>;
    using u32x8 = simd<uint32_t, 8>;
    using u16x16 = simd<uint16_t, 16>;
    using i32x4 = simd<int32_t, 4>;
    using u32x4 = simd<uint32_t, 4>;
    using u16x8 = simd<uint16_t, 8>;
    using u16x4 = simd<uint16_t, 4>;
    using i16x4 = simd<int16_t, 4>;
    using u8x32 = simd<uint8_t, 32>;
    using u8x8 = simd<uint8_t, 8>;
    using u8x4 = simd<uint8_t, 4>;
    using dx4 = simd<double, 4>;

    inline dx4 abs(dx4 a) {
        return dx4{
            std::abs(a[0]),
            std::abs(a[1]),
            std::abs(a[2]),
            std::abs(a[3])
        };
    }

    // https://stackoverflow.com/a/77376595
    // works for ints in (-2^51, 2^51)
    static constexpr dx4 magic = dx4() + (3ULL << 51);
    inline i64x4 lround(dx4 x) {
        return i64x4(x + magic) - i64x4(magic);
    }
    inline dx4 to_double(i64x4 x) {
        return dx4(x + i64x4(magic)) - magic;
    }

    inline dx4 round(dx4 a) {
        return dx4{
            std::nearbyint(a[0]),
            std::nearbyint(a[1]),
            std::nearbyint(a[2]),
            std::nearbyint(a[3])
        };
    }

    inline u64x4 low32(u64x4 x) {
        return x & uint32_t(-1);
    }
    inline auto swap_bytes(auto x) {
        return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
    }
    inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
        auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
        x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
#else
        auto x_ninv = u64x4(u32x8(low32(x)) * imod);
        x += x_ninv * uint64_t(mod);
#endif
        return swap_bytes(x);
    }

    inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
        return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
#else
        return montgomery_reduce(x * y, mod, imod);
#endif
    }
    inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
        return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
               u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
    }
    inline dx4 rotate_right(dx4 x) {
        static constexpr u64x4 shuffler = {3, 0, 1, 2};
        return __builtin_shuffle(x, shuffler);
    }

    template<std::size_t Align = 32>
    inline bool is_aligned(const auto* p) noexcept {
        return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
    }

    template<class Target>
    inline Target& vector_cast(auto &&p) {
        return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
    }
}
#pragma GCC pop_options

#line 1 "cp-algo/util/complex.hpp"


#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
    // Custom implementation, since std::complex is UB on non-floating types
    template<typename T>
    struct complex {
        using value_type = T;
        T x, y;
        inline constexpr complex(): x(), y() {}
        inline constexpr complex(T const& x): x(x), y() {}
        inline constexpr complex(T const& x, T const& y): x(x), y(y) {}
        inline complex& operator *= (T const& t) {x *= t; y *= t; return *this;}
        inline complex& operator /= (T const& t) {x /= t; y /= t; return *this;}
        inline complex operator * (T const& t) const {return complex(*this) *= t;}
        inline complex operator / (T const& t) const {return complex(*this) /= t;}
        inline complex& operator += (complex const& t) {x += t.x; y += t.y; return *this;}
        inline complex& operator -= (complex const& t) {x -= t.x; y -= t.y; return *this;}
        inline complex operator * (complex const& t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
        inline complex operator / (complex const& t) const {return *this * t.conj() / t.norm();}
        inline complex operator + (complex const& t) const {return complex(*this) += t;}
        inline complex operator - (complex const& t) const {return complex(*this) -= t;}
        inline complex& operator *= (complex const& t) {return *this = *this * t;}
        inline complex& operator /= (complex const& t) {return *this = *this / t;}
        inline complex operator - () const {return {-x, -y};}
        inline complex conj() const {return {x, -y};}
        inline T norm() const {return x * x + y * y;}
        inline T abs() const {return std::sqrt(norm());}
        inline T const real() const {return x;}
        inline T const imag() const {return y;}
        inline T& real() {return x;}
        inline T& imag() {return y;}
        inline static constexpr complex polar(T r, T theta) {return {T(r * cos(theta)), T(r * sin(theta))};}
        inline auto operator <=> (complex const& t) const = default;
    };
    template<typename T> inline complex<T> conj(complex<T> const& x) {return x.conj();}
    template<typename T> inline T norm(complex<T> const& x) {return x.norm();}
    template<typename T> inline T abs(complex<T> const& x) {return x.abs();}
    template<typename T> inline T& real(complex<T> &x) {return x.real();}
    template<typename T> inline T& imag(complex<T> &x) {return x.imag();}
    template<typename T> inline T const real(complex<T> const& x) {return x.real();}
    template<typename T> inline T const imag(complex<T> const& x) {return x.imag();}
    template<typename T>
    inline constexpr complex<T> polar(T r, T theta) {
        return complex<T>::polar(r, theta);
    }
    template<typename T>
    inline std::ostream& operator << (std::ostream &out, complex<T> const& x) {
        return out << x.real() << ' ' << x.imag();
    }
}
#pragma GCC pop_options

#line 7 "cp-algo/math/cvector.hpp"
#include <ranges>
#line 9 "cp-algo/math/cvector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace stdx = std::experimental;
namespace cp_algo::math::fft {
    static constexpr size_t flen = 4;
    using ftype = double;
    using vftype = dx4;
    using point = complex<ftype>;
    using vpoint = complex<vftype>;
    static constexpr vftype vz = {};
    vpoint vi(vpoint const& r) {
        return {-imag(r), real(r)};
    }

    struct cvector {
        big_vector<vpoint> r;
        cvector(size_t n) {
            n = std::max(flen, std::bit_ceil(n));
            r.resize(n / flen);
            checkpoint("cvector create");
        }

        vpoint& at(size_t k) {return r[k / flen];}
        vpoint at(size_t k) const {return r[k / flen];}
        template<class pt = point>
        inline void set(size_t k, pt const& t) {
            if constexpr(std::is_same_v<pt, point>) {
                real(r[k / flen])[k % flen] = real(t);
                imag(r[k / flen])[k % flen] = imag(t);
            } else {
                at(k) = t;
            }
        }
        template<class pt = point>
        inline pt get(size_t k) const {
            if constexpr(std::is_same_v<pt, point>) {
                return {real(r[k / flen])[k % flen], imag(r[k / flen])[k % flen]};
            } else {
                return at(k);
            }
        }

        size_t size() const {
            return flen * r.size();
        }
        static constexpr size_t eval_arg(size_t n) {
            if(n < pre_evals) {
                return eval_args[n];
            } else {
                return eval_arg(n / 2) | (n & 1) << (std::bit_width(n) - 1);
            }
        }
        static constexpr point eval_point(size_t n) {
            if(n % 2) {
                return -eval_point(n - 1);
            } else if(n % 4) {
                return eval_point(n - 2) * point(0, 1);
            } else if(n / 4 < pre_evals) {
                return evalp[n / 4];
            } else {
                return polar<ftype>(1., std::numbers::pi / (ftype)std::bit_floor(n) * (ftype)eval_arg(n));
            }
        }
        static constexpr std::array<point, 32> roots = []() {
            std::array<point, 32> res;
            for(size_t i = 2; i < 32; i++) {
                res[i] = polar<ftype>(1., std::numbers::pi / (1ull << (i - 2)));
            }
            return res;
        }();
        static constexpr point root(size_t n) {
            return roots[std::bit_width(n)];
        }
        template<int step>
        static void exec_on_eval(size_t n, size_t k, auto &&callback) {
            callback(k, root(4 * step * n) * eval_point(step * k));
        }
        template<int step>
        static void exec_on_evals(size_t n, auto &&callback) {
            point factor = root(4 * step * n);
            for(size_t i = 0; i < n; i++) {
                callback(i, factor * eval_point(step * i));
            }
        }

        static void do_dot_iter(point rt, vpoint& Bv, vpoint const& Av, vpoint& res) {
            res += Av * Bv;
            real(Bv) = rotate_right(real(Bv));
            imag(Bv) = rotate_right(imag(Bv));
            auto x = real(Bv)[0], y = imag(Bv)[0];
            real(Bv)[0] = x * real(rt) - y * imag(rt);
            imag(Bv)[0] = x * imag(rt) + y * real(rt);
        }

        void dot(cvector const& t) {
            size_t n = this->size();
            exec_on_evals<1>(n / flen, [&](size_t k, point rt) __attribute__((always_inline)) {
                k *= flen;
                auto [Ax, Ay] = at(k);
                auto Bv = t.at(k);
                vpoint res = vz;
                for (size_t i = 0; i < flen; i++) {
                    vpoint Av = vpoint(vz + Ax[i], vz + Ay[i]);
                    do_dot_iter(rt, Bv, Av, res);
                }
                set(k, res);
            });
            checkpoint("dot");
        }
        template<bool partial = true>
        void ifft() {
            size_t n = size();
            if constexpr (!partial) {
                point pi(0, 1);
                exec_on_evals<4>(n / 4, [&](size_t k, point rt) __attribute__((always_inline)) {
                    k *= 4;
                    point v1 = conj(rt);
                    point v2 = v1 * v1;
                    point v3 = v1 * v2;
                    auto A = get(k);
                    auto B = get(k + 1);
                    auto C = get(k + 2);
                    auto D = get(k + 3);
                    set(k, (A + B) + (C + D));
                    set(k + 2, ((A + B) - (C + D)) * v2);
                    set(k + 1, ((A - B) - pi * (C - D)) * v1);
                    set(k + 3, ((A - B) + pi * (C - D)) * v3);
                });
            }
            bool parity = std::countr_zero(n) % 2;
            if(parity) {
                exec_on_evals<2>(n / (2 * flen), [&](size_t k, point rt) __attribute__((always_inline)) {
                    k *= 2 * flen;
                    vpoint cvrt = {vz + real(rt), vz - imag(rt)};
                    auto B = at(k) - at(k + flen);
                    at(k) += at(k + flen);
                    at(k + flen) = B * cvrt;
                });
            }

            for(size_t leaf = 3 * flen; leaf < n; leaf += 4 * flen) {
                size_t level = std::countr_one(leaf + 3);
                for(size_t lvl = 4 + parity; lvl <= level; lvl += 2) {
                    size_t i = (1 << lvl) / 4;
                    exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) __attribute__((always_inline)) {
                        k <<= lvl;
                        vpoint v1 = {vz + real(rt), vz - imag(rt)};
                        vpoint v2 = v1 * v1;
                        vpoint v3 = v1 * v2;
                        for(size_t j = k; j < k + i; j += flen) {
                            auto A = at(j);
                            auto B = at(j + i);
                            auto C = at(j + 2 * i);
                            auto D = at(j + 3 * i);
                            at(j) = ((A + B) + (C + D));
                            at(j + 2 * i) = ((A + B) - (C + D)) * v2;
                            at(j +     i) = ((A - B) - vi(C - D)) * v1;
                            at(j + 3 * i) = ((A - B) + vi(C - D)) * v3;
                        }
                    });
                }
            }
            checkpoint("ifft");
            for(size_t k = 0; k < n; k += flen) {
                if constexpr (partial) {
                    set(k, get<vpoint>(k) /= vz + ftype(n / flen));
                } else {
                    set(k, get<vpoint>(k) /= vz + ftype(n));
                }
            }
        }
        template<bool partial = true>
        void fft() {
            size_t n = size();
            bool parity = std::countr_zero(n) % 2;
            for(size_t leaf = 0; leaf < n; leaf += 4 * flen) {
                size_t level = std::countr_zero(n + leaf);
                level -= level % 2 != parity;
                for(size_t lvl = level; lvl >= 4; lvl -= 2) {
                    size_t i = (1 << lvl) / 4;
                    exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) __attribute__((always_inline)) {
                        k <<= lvl;
                        vpoint v1 = {vz + real(rt), vz + imag(rt)};
                        vpoint v2 = v1 * v1;
                        vpoint v3 = v1 * v2;
                        for(size_t j = k; j < k + i; j += flen) {
                            auto A = at(j);
                            auto B = at(j + i) * v1;
                            auto C = at(j + 2 * i) * v2;
                            auto D = at(j + 3 * i) * v3;
                            at(j)         = (A + C) + (B + D);
                            at(j + i)     = (A + C) - (B + D);
                            at(j + 2 * i) = (A - C) + vi(B - D);
                            at(j + 3 * i) = (A - C) - vi(B - D);
                        }
                    });
                }
            }
            if(parity) {
                exec_on_evals<2>(n / (2 * flen), [&](size_t k, point rt) __attribute__((always_inline)) {
                    k *= 2 * flen;
                    vpoint vrt = {vz + real(rt), vz + imag(rt)};
                    auto t = at(k + flen) * vrt;
                    at(k + flen) = at(k) - t;
                    at(k) += t;
                });
            }
            if constexpr (!partial) {
                point pi(0, 1);
                exec_on_evals<4>(n / 4, [&](size_t k, point rt) __attribute__((always_inline)) {
                    k *= 4;
                    point v1 = rt;
                    point v2 = v1 * v1;
                    point v3 = v1 * v2;
                    auto A = get(k);
                    auto B = get(k + 1) * v1;
                    auto C = get(k + 2) * v2;
                    auto D = get(k + 3) * v3;
                    set(k, (A + C) + (B + D));
                    set(k + 1, (A + C) - (B + D));
                    set(k + 2, (A - C) + pi * (B - D));
                    set(k + 3, (A - C) - pi * (B - D));
                });
            }
            checkpoint("fft");
        }
        static constexpr size_t pre_evals = 1 << 16;
        static const std::array<size_t, pre_evals> eval_args;
        static const std::array<point, pre_evals> evalp;
    };

    const std::array<size_t, cvector::pre_evals> cvector::eval_args = []() {
        std::array<size_t, pre_evals> res = {};
        for(size_t i = 1; i < pre_evals; i++) {
            res[i] = res[i >> 1] | (i & 1) << (std::bit_width(i) - 1);
        }
        return res;
    }();
    const std::array<point, cvector::pre_evals> cvector::evalp = []() {
        std::array<point, pre_evals> res = {};
        res[0] = 1;
        for(size_t n = 1; n < pre_evals; n++) {
            res[n] = polar<ftype>(1., std::numbers::pi * ftype(eval_args[n]) / ftype(4 * std::bit_floor(n)));
        }
        return res;
    }();
}
#pragma GCC pop_options

#line 9 "cp-algo/math/fft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft {
    template<modint_type base>
    struct dft {
        cvector A, B;
        static base factor, ifactor;
        using Int2 = base::Int2;
        static bool _init;
        static int split() {
            static const int splt = int(std::sqrt(base::mod())) + 1;
            return splt;
        }
        static uint32_t mod, imod;

        static void init() {
            if(!_init) {
                factor = 1 + random::rng() % (base::mod() - 1);
                ifactor = base(1) / factor;
                mod = base::mod();
                imod = -inv2<uint32_t>(base::mod());
                _init = true;
            }
        }

        static std::pair<vftype, vftype> 
        do_split(auto const& a, size_t idx, u64x4 mul) {
            if(idx >= std::size(a)) {
                return std::pair{vftype(), vftype()};
            }
            u64x4 au = {
                idx < std::size(a) ? a[idx].getr() : 0,
                idx + 1 < std::size(a) ? a[idx + 1].getr() : 0,
                idx + 2 < std::size(a) ? a[idx + 2].getr() : 0,
                idx + 3 < std::size(a) ? a[idx + 3].getr() : 0
            };
            au = montgomery_mul(au, mul, mod, imod);
            au = au >= base::mod() ? au - base::mod() : au;
            auto ai = to_double(i64x4(au >= base::mod() / 2 ? au - base::mod() : au));
            auto quo = round(ai / split());
            return std::pair{ai - quo * split(), quo};
        }

        dft(size_t n): A(n), B(n) {init();}
        dft(auto const& a, size_t n, bool partial = true): A(n), B(n) {
            init();
            base b2x32 = bpow(base(2), 32);
            u64x4 cur = {
                (bpow(factor, 1) * b2x32).getr(),
                (bpow(factor, 2) * b2x32).getr(),
                (bpow(factor, 3) * b2x32).getr(),
                (bpow(factor, 4) * b2x32).getr()
            };
            u64x4 step4 = u64x4{} + (bpow(factor, 4) * b2x32).getr();
            u64x4 stepn = u64x4{} + (bpow(factor, n) * b2x32).getr();
            for(size_t i = 0; i < std::min(n, std::size(a)); i += flen) {
                auto [rai, qai] = do_split(a, i, cur);
                auto [rani, qani] = do_split(a, n + i, montgomery_mul(cur, stepn, mod, imod));
                A.at(i) = vpoint(rai, rani);
                B.at(i) = vpoint(qai, qani);
                cur = montgomery_mul(cur, step4, mod, imod);
            }
            checkpoint("dft init");
            if(n) {
                if(partial) {
                    A.fft();
                    B.fft();
                } else {
                    A.template fft<false>();
                    B.template fft<false>();
                }
            }
        }
        static void do_dot_iter(point rt, vpoint& Cv, vpoint& Dv, vpoint const& Av, vpoint const& Bv, vpoint& AC, vpoint& AD, vpoint& BC, vpoint& BD) {
            AC += Av * Cv; AD += Av * Dv;
            BC += Bv * Cv; BD += Bv * Dv;
            real(Cv) = rotate_right(real(Cv));
            imag(Cv) = rotate_right(imag(Cv));
            real(Dv) = rotate_right(real(Dv));
            imag(Dv) = rotate_right(imag(Dv));
            auto cx = real(Cv)[0], cy = imag(Cv)[0];
            auto dx = real(Dv)[0], dy = imag(Dv)[0];
            real(Cv)[0] = cx * real(rt) - cy * imag(rt);
            imag(Cv)[0] = cx * imag(rt) + cy * real(rt);
            real(Dv)[0] = dx * real(rt) - dy * imag(rt);
            imag(Dv)[0] = dx * imag(rt) + dy * real(rt);
        }

        template<bool overwrite = true, bool partial = true>
        void dot(auto const& C, auto const& D, auto &Aout, auto &Bout, auto &Cout) const {
            cvector::exec_on_evals<1>(A.size() / flen, [&](size_t k, point rt) __attribute__((always_inline)) {
                k *= flen;
                vpoint AC, AD, BC, BD;
                AC = AD = BC = BD = vz;
                auto Cv = C.at(k), Dv = D.at(k);
                if constexpr(partial) {
                    auto [Ax, Ay] = A.at(k);
                    auto [Bx, By] = B.at(k);
                    for (size_t i = 0; i < flen; i++) {
                        vpoint Av = {vz + Ax[i], vz + Ay[i]}, Bv = {vz + Bx[i], vz + By[i]};
                        do_dot_iter(rt, Cv, Dv, Av, Bv, AC, AD, BC, BD);
                    }
                } else {
                    AC = A.at(k) * Cv;
                    AD = A.at(k) * Dv;
                    BC = B.at(k) * Cv;
                    BD = B.at(k) * Dv;
                }
                if constexpr (overwrite) {
                    Aout.at(k) = AC;
                    Cout.at(k) = AD + BC;
                    Bout.at(k) = BD;
                } else {
                    Aout.at(k) += AC;
                    Cout.at(k) += AD + BC;
                    Bout.at(k) += BD;
                }
            });
            checkpoint("dot");
        }

        void dot(auto &&C, auto const& D) {
            dot(C, D, A, B, C);
        }

        static void do_recover_iter(size_t idx, auto A, auto B, auto C, auto mul, uint64_t splitsplit, auto &res) {
            auto A0 = lround(A), A1 = lround(C), A2 = lround(B);
            auto Ai = A0 + A1 * split() + A2 * splitsplit + uint64_t(base::modmod());
            auto Au = montgomery_reduce(u64x4(Ai), mod, imod);
            Au = montgomery_mul(Au, mul, mod, imod);
            Au = Au >= base::mod() ? Au - base::mod() : Au;
            for(size_t j = 0; j < flen; j++) {
                res[idx + j].setr(typename base::UInt(Au[j]));
            }
        }

        void recover_mod(auto &&C, auto &res, size_t k) {
            size_t check = (k + flen - 1) / flen * flen;
            assert(res.size() >= check);
            size_t n = A.size();
            auto const splitsplit = base(split() * split()).getr();
            base b2x32 = bpow(base(2), 32);
            base b2x64 = bpow(base(2), 64);
            u64x4 cur = {
                (bpow(ifactor, 2) * b2x64).getr(),
                (bpow(ifactor, 3) * b2x64).getr(),
                (bpow(ifactor, 4) * b2x64).getr(),
                (bpow(ifactor, 5) * b2x64).getr()
            };
            u64x4 step4 = u64x4{} + (bpow(ifactor, 4) * b2x32).getr();
            u64x4 stepn = u64x4{} + (bpow(ifactor, n) * b2x32).getr();
            for(size_t i = 0; i < std::min(n, k); i += flen) {
                auto [Ax, Ay] = A.at(i);
                auto [Bx, By] = B.at(i);
                auto [Cx, Cy] = C.at(i);
                do_recover_iter(i, Ax, Bx, Cx, cur, splitsplit, res);
                if(i + n < k) {
                    do_recover_iter(i + n, Ay, By, Cy, montgomery_mul(cur, stepn, mod, imod), splitsplit, res);
                }
                cur = montgomery_mul(cur, step4, mod, imod);
            }
            checkpoint("recover mod");
        }

        void mul(auto &&C, auto const& D, auto &res, size_t k) {
            assert(A.size() == C.size());
            size_t n = A.size();
            if(!n) {
                res = {};
                return;
            }
            dot(C, D);
            A.ifft();
            B.ifft();
            C.ifft();
            recover_mod(C, res, k);
        }
        void mul_inplace(auto &&B, auto& res, size_t k) {
            mul(B.A, B.B, res, k);
        }
        void mul(auto const& B, auto& res, size_t k) {
            mul(cvector(B.A), B.B, res, k);
        }
        big_vector<base> operator *= (dft &B) {
            big_vector<base> res(2 * A.size());
            mul_inplace(B, res, 2 * A.size());
            return res;
        }
        big_vector<base> operator *= (dft const& B) {
            big_vector<base> res(2 * A.size());
            mul(B, res, 2 * A.size());
            return res;
        }
        auto operator * (dft const& B) const {
            return dft(*this) *= B;
        }

        point operator [](int i) const {return A.get(i);}
    };
    template<modint_type base> base dft<base>::factor = 1;
    template<modint_type base> base dft<base>::ifactor = 1;
    template<modint_type base> bool dft<base>::_init = false;
    template<modint_type base> uint32_t dft<base>::mod = {};
    template<modint_type base> uint32_t dft<base>::imod = {};
    
    void mul_slow(auto &a, auto const& b, size_t k) {
        if(std::empty(a) || std::empty(b)) {
            a.clear();
        } else {
            size_t n = std::min(k, std::size(a));
            size_t m = std::min(k, std::size(b));
            a.resize(k);
            for(int j = int(k - 1); j >= 0; j--) {
                a[j] *= b[0];
                for(int i = std::max(j - (int)n, 0) + 1; i < std::min(j + 1, (int)m); i++) {
                    a[j] += a[j - i] * b[i];
                }
            }
        }
    }
    size_t com_size(size_t as, size_t bs) {
        if(!as || !bs) {
            return 0;
        }
        return std::max(flen, std::bit_ceil(as + bs - 1) / 2);
    }
    void mul_truncate(auto &a, auto const& b, size_t k) {
        using base = std::decay_t<decltype(a[0])>;
        if(std::min({k, std::size(a), std::size(b)}) < magic) {
            mul_slow(a, b, k);
            return;
        }
        auto n = std::max(flen, std::bit_ceil(
            std::min(k, std::size(a)) + std::min(k, std::size(b)) - 1
        ) / 2);
        auto A = dft<base>(a | std::views::take(k), n);
        auto B = dft<base>(b | std::views::take(k), n);
        a.resize((k + flen - 1) / flen * flen);
        A.mul_inplace(B, a, k);
        a.resize(k);
    }

    // store mod x^n-k in first half, x^n+k in second half
    void mod_split(auto &&x, size_t n, auto k) {
        using base = std::decay_t<decltype(k)>;
        dft<base>::init();
        assert(std::size(x) == 2 * n);
        u64x4 cur = u64x4{} + (k * bpow(base(2), 32)).getr();
        for(size_t i = 0; i < n; i += flen) {
            u64x4 xl = {
                x[i].getr(),
                x[i + 1].getr(),
                x[i + 2].getr(),
                x[i + 3].getr()
            };
            u64x4 xr = {
                x[n + i].getr(),
                x[n + i + 1].getr(),
                x[n + i + 2].getr(),
                x[n + i + 3].getr()
            };
            xr = montgomery_mul(xr, cur, dft<base>::mod, dft<base>::imod);
            xr = xr >= base::mod() ? xr - base::mod() : xr;
            auto t = xr;
            xr = xl - t;
            xl += t;
            xl = xl >= base::mod() ? xl - base::mod() : xl;
            xr = xr >= base::mod() ? xr + base::mod() : xr;
            for(size_t k = 0; k < flen; k++) {
                x[i + k].setr(typename base::UInt(xl[k]));
                x[n + i + k].setr(typename base::UInt(xr[k]));
            }
        }
        cp_algo::checkpoint("mod split");
    }
    void cyclic_mul(auto &a, auto &&b, size_t k) {
        assert(std::popcount(k) == 1);
        assert(std::size(a) == std::size(b) && std::size(a) == k);
        using base = std::decay_t<decltype(a[0])>;
        dft<base>::init();
        if(k <= (1 << 16)) {
            big_vector<base> ap(begin(a), end(a));
            mul_truncate(ap, b, 2 * k);
            mod_split(ap, k, bpow(dft<base>::factor, k));
            std::ranges::copy(ap | std::views::take(k), begin(a));
            return;
        }
        k /= 2;
        auto factor = bpow(dft<base>::factor, k);
        mod_split(a, k, factor);
        mod_split(b, k, factor);
        auto la = std::span(a).first(k);
        auto lb = std::span(b).first(k);
        auto ra = std::span(a).last(k);
        auto rb = std::span(b).last(k);
        cyclic_mul(la, lb, k);
        auto A = dft<base>(ra, k / 2);
        auto B = dft<base>(rb, k / 2);
        A.mul_inplace(B, ra, k);
        base i2 = base(2).inv();
        factor = factor.inv() * i2;
        for(size_t i = 0; i < k; i++) {
            auto t = (a[i] + a[i + k]) * i2;
            a[i + k] = (a[i] - a[i + k]) * factor;
            a[i] = t;
        }
        cp_algo::checkpoint("mod join");
    }
    auto make_copy(auto &&x) {
        return x;
    }
    void cyclic_mul(auto &a, auto const& b, size_t k) {
        return cyclic_mul(a, make_copy(b), k);
    }
    void mul(auto &a, auto &&b) {
        size_t N = size(a) + size(b);
        if(N > (1 << 20)) {
            N--;
            size_t NN = std::bit_ceil(N);
            a.resize(NN);
            b.resize(NN);
            cyclic_mul(a, b, NN);
            a.resize(N);
        } else {
            mul_truncate(a, b, N - 1);
        }
    }
    void mul(auto &a, auto const& b) {
        size_t N = size(a) + size(b);
        if(N > (1 << 20)) {
            mul(a, make_copy(b));
        } else {
            mul_truncate(a, b, N - 1);
        }
    }
}
#pragma GCC pop_options

#line 6 "cp-algo/math/poly/impl/euclid.hpp"
#include <algorithm>
#include <numeric>
#line 11 "cp-algo/math/poly/impl/euclid.hpp"
#include <list>
CP_ALGO_SIMD_PRAGMA_PUSH
// operations related to gcd and Euclidean algo
namespace cp_algo::math::poly::impl {
    template<typename poly>
    using gcd_result = std::pair<
        std::list<std::decay_t<poly>>,
        linfrac<std::decay_t<poly>>>;

    template<typename poly>
    gcd_result<poly> half_gcd(poly &&A, poly &&B) {
        assert(A.deg() >= B.deg());
        size_t m = size(A.a) / 2;
        if(B.deg() < (int)m) {
            return {};
        }
        auto [ai, R] = A.divmod(B);
        std::tie(A, B) = {B, R};
        std::list a = {ai};
        auto T = -linfrac(ai).adj();

        auto advance = [&](size_t k) {
            auto [ak, Tk] = half_gcd(A.div_xk(k), B.div_xk(k));
            a.splice(end(a), ak);
            T.prepend(Tk);
            return Tk;
        };
        advance(m).apply(A, B);
        if constexpr (std::is_reference_v<poly>) {
            advance(2 * m - A.deg()).apply(A, B);
        } else {
            advance(2 * m - A.deg());
        }
        return {std::move(a), std::move(T)};
    }
    template<typename poly>
    gcd_result<poly> full_gcd(poly &&A, poly &&B) {
        using poly_t = std::decay_t<poly>;
        std::list<poly_t> ak;
        big_vector<linfrac<poly_t>> trs;
        while(!B.is_zero()) {
            auto [a0, R] = A.divmod(B);
            ak.push_back(a0);
            trs.push_back(-linfrac(a0).adj());
            std::tie(A, B) = {B, R};

            auto [a, Tr] = half_gcd(A, B);
            ak.splice(end(ak), a);
            trs.push_back(Tr);
        }
        return {ak, std::accumulate(rbegin(trs), rend(trs), linfrac<poly_t>{}, std::multiplies{})};
    }

    // computes product of linfrac on [L, R)
    auto convergent(auto L, auto R) {
        using poly = decltype(L)::value_type;
        if(R == next(L)) {
            return linfrac(*L);
        } else {
            int s = std::transform_reduce(L, R, 0, std::plus{}, std::mem_fn(&poly::deg));
            auto M = L;
            for(int c = M->deg(); 2 * c <= s; M++) {
                c += next(M)->deg();
            }
            return convergent(L, M) * convergent(M, R);
        }
    }
    template<typename poly>
    poly min_rec(poly const& p, size_t d) {
        auto R2 = p.mod_xk(d).reversed(d), R1 = poly::xk(d);
        if(R2.is_zero()) {
            return poly(1);
        }
        auto [a, Tr] = full_gcd(R1, R2);
        a.emplace_back();
        auto pref = begin(a);
        for(int delta = (int)d - a.front().deg(); delta >= 0; pref++) {
            delta -= pref->deg() + next(pref)->deg();
        }
        return convergent(begin(a), pref).a;
    }

    template<typename poly>
    std::optional<poly> inv_mod(poly p, poly q) {
        assert(!q.is_zero());
        auto [a, Tr] = full_gcd(q, p);
        if(q.deg() != 0) {
            return std::nullopt;
        }
        return Tr.b / q[0];
    }
}
#pragma GCC pop_options

#line 1 "cp-algo/math/poly/impl/div.hpp"


#line 6 "cp-algo/math/poly/impl/div.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
// operations related to polynomial division
namespace cp_algo::math::poly::impl {
    auto divmod_slow(auto const& p, auto const& q) {
        auto R = p;
        auto D = decltype(p){};
        auto q_lead_inv = q.lead().inv();
        while(R.deg() >= q.deg()) {
            D.a.push_back(R.lead() * q_lead_inv);
            if(D.lead() != 0) {
                for(size_t i = 1; i <= q.a.size(); i++) {
                    R.a[R.a.size() - i] -= D.lead() * q.a[q.a.size() - i];
                }
            }
            R.a.pop_back();
        }
        std::ranges::reverse(D.a);
        R.normalize();
        return std::array{D, R};
    }
    template<typename poly>
    auto divmod_hint(poly const& p, poly const& q, poly const& qri) {
        assert(!q.is_zero());
        int d = p.deg() - q.deg();
        if(std::min(d, q.deg()) < magic) {
            return divmod_slow(p, q);
        }
        poly D;
        if(d >= 0) {
            D = (p.reversed().mod_xk(d + 1) * qri.mod_xk(d + 1)).mod_xk(d + 1).reversed(d + 1);
        }
        return std::array{D, p - D * q};
    }
    auto divmod(auto const& p, auto const& q) {
        assert(!q.is_zero());
        int d = p.deg() - q.deg();
        if(std::min(d, q.deg()) < magic) {
            return divmod_slow(p, q);
        }
        return divmod_hint(p, q, q.reversed().inv(d + 1));
    }

    template<typename poly>
    poly powmod_hint(poly const& p, int64_t k, poly const& md, poly const& mdri) {
        return bpow(p % md, k, poly(1), [&](auto const& p, auto const& q){
            return divmod_hint(p * q, md, mdri)[1];
        });
    }
    template<typename poly>
    auto powmod(poly const& p, int64_t k, poly const& md) {
        int d = md.deg();
        if(p == poly::xk(1) && false) { // does it actually speed anything up?..
            if(k < md.deg()) {
                return poly::xk(k);
            } else {
                auto mdr = md.reversed();
                return (mdr.inv(k - md.deg() + 1, md.deg()) * mdr).reversed(md.deg());
            }
        }
        if(md == poly::xk(d)) {
            return p.pow(k, d);
        }
        if(md == poly::xk(d) - poly(1)) {
            return p.powmod_circular(k, d);
        }
        return powmod_hint(p, k, md, md.reversed().inv(md.deg() + 1));
    }
    template<typename poly>
    poly& inv_inplace(poly& q, int64_t k, size_t n) {
        using poly_t = std::decay_t<poly>;
        using base = poly_t::base;
        if(k <= std::max<int64_t>(n, size(q.a))) {
            return q.inv_inplace(k + n).div_xk_inplace(k);
        }
        if(k % 2) {
            return inv_inplace(q, k - 1, n + 1).div_xk_inplace(1);
        }
        auto [q0, q1] = q.bisect();
        auto qq = q0 * q0 - (q1 * q1).mul_xk_inplace(1);
        inv_inplace(qq, k / 2 - q.deg() / 2, (n + 1) / 2 + q.deg() / 2);
        size_t N = fft::com_size(size(q0.a), size(qq.a));
        auto q0f = fft::dft<base>(q0.a, N);
        auto q1f = fft::dft<base>(q1.a, N);
        auto qqf = fft::dft<base>(qq.a, N);
        size_t M = q0.deg() + (n + 1) / 2;
        typename poly::Vector A, B;
        A.resize((M + fft::flen - 1) / fft::flen * fft::flen);
        B.resize((M + fft::flen - 1) / fft::flen * fft::flen);
        q0f.mul(qqf, A, M);
        q1f.mul_inplace(qqf, B, M);
        q.a.resize(n + 1);
        for(size_t i = 0; i < n; i += 2) {
            q.a[i] = A[q0.deg() + i / 2];
            q.a[i + 1] = -B[q0.deg() + i / 2];
        }
        q.a.pop_back();
        q.normalize();
        return q;
    }
    template<typename poly>
    poly& inv_inplace(poly& p, size_t n) {
        using poly_t = std::decay_t<poly>;
        using base = poly_t::base;
        if(n == 1) {
            return p = base(1) / p[0];
        }
        // Q(-x) = P0(x^2) + xP1(x^2)
        auto [q0, q1] = p.bisect(n);
        
        size_t N = fft::com_size((n + 1) / 2, (n + 1) / 2);
        
        auto q0f = fft::dft<base>(q0.a, N);
        auto q1f = fft::dft<base>(q1.a, N);

        // Q(x)*Q(-x) = Q0(x^2)^2 - x^2 Q1(x^2)^2
        auto qq = poly_t(q0f * q0f) - poly_t(q1f * q1f).mul_xk_inplace(1);

        inv_inplace(qq, (n + 1) / 2);
        auto qqf = fft::dft<base>(qq.a, N);
        
        typename poly::Vector A, B;
        A.resize(((n + 1) / 2 + fft::flen - 1) / fft::flen * fft::flen);
        B.resize(((n + 1) / 2 + fft::flen - 1) / fft::flen * fft::flen);
        q0f.mul(qqf, A, (n + 1) / 2);
        q1f.mul_inplace(qqf, B, (n + 1) / 2);
        p.a.resize(n + 1);
        for(size_t i = 0; i < n; i += 2) {
            p.a[i] = A[i / 2];
            p.a[i + 1] = -B[i / 2];
        }
        p.a.pop_back();
        p.normalize();
        return p;
    }
}
#pragma GCC pop_options

#line 1 "cp-algo/math/combinatorics.hpp"


#line 7 "cp-algo/math/combinatorics.hpp"
namespace cp_algo::math {
    // fact/rfact/small_inv are caching
    // Beware of usage with dynamic mod
    template<typename T>
    T fact(auto n) {
        static big_vector<T> F(maxn);
        static bool init = false;
        if(!init) {
            F[0] = T(1);
            for(int i = 1; i < maxn; i++) {
                F[i] = F[i - 1] * T(i);
            }
            init = true;
        }
        return F[n];
    }
    // Only works for modint types
    template<typename T>
    T rfact(auto n) {
        static big_vector<T> F(maxn);
        static bool init = false;
        if(!init) {
            int t = (int)std::min<int64_t>(T::mod(), maxn) - 1;
            F[t] = T(1) / fact<T>(t);
            for(int i = t - 1; i >= 0; i--) {
                F[i] = F[i + 1] * T(i + 1);
            }
            init = true;
        }
        return F[n];
    }
    template<typename T, int base>
    T pow_fixed(int n) {
        static big_vector<T> prec_low(1 << 16);
        static big_vector<T> prec_high(1 << 16);
        static bool init = false;
        if(!init) {
            init = true;
            prec_low[0] = prec_high[0] = T(1);
            T step_low = T(base);
            T step_high = bpow(T(base), 1 << 16);
            for(int i = 1; i < (1 << 16); i++) {
                prec_low[i] = prec_low[i - 1] * step_low;
                prec_high[i] = prec_high[i - 1] * step_high;
            }
        }
        return prec_low[n & 0xFFFF] * prec_high[n >> 16];
    }
    template<typename T>
    big_vector<T> bulk_invs(auto const& args) {
        big_vector<T> res(std::size(args), args[0]);
        for(size_t i = 1; i < std::size(args); i++) {
            res[i] = res[i - 1] * args[i];
        }
        auto all_invs = T(1) / res.back();
        for(size_t i = std::size(args) - 1; i > 0; i--) {
            res[i] = all_invs * res[i - 1];
            all_invs *= args[i];
        }
        res[0] = all_invs;
        return res;
    }
    template<typename T>
    T small_inv(auto n) {
        static auto F = bulk_invs<T>(std::views::iota(1, maxn));
        return F[n - 1];
    }
    template<typename T>
    T binom_large(T n, auto r) {
        assert(r < maxn);
        T ans = 1;
        for(decltype(r) i = 0; i < r; i++) {
            ans = ans * T(n - i) * small_inv<T>(i + 1);
        }
        return ans;
    }
    template<typename T>
    T binom(auto n, auto r) {
        if(r < 0 || r > n) {
            return T(0);
        } else if(n >= maxn) {
            return binom_large(T(n), r);
        } else {
            return fact<T>(n) * rfact<T>(r) * rfact<T>(n - r);
        }
    }
}

#line 1 "cp-algo/number_theory/discrete_sqrt.hpp"


#line 6 "cp-algo/number_theory/discrete_sqrt.hpp"
namespace cp_algo::math {
    // https://en.wikipedia.org/wiki/Berlekamp-Rabin_algorithm
    template<modint_type base>
    std::optional<base> sqrt(base b) {
        if(b == base(0)) {
            return base(0);
        } else if(bpow(b, (b.mod() - 1) / 2) != base(1)) {
            return std::nullopt;
        } else {
            while(true) {
                base z = random::rng();
                if(z * z == b) {
                    return z;
                }
                lin<base> x(1, z, b); // x + z (mod x^2 - b)
                x = bpow(x, (b.mod() - 1) / 2, lin<base>(0, 1, b));
                if(x.a != base(0)) {
                    return x.a.inv();
                }
            }
        }
    }
}

#line 15 "cp-algo/math/poly.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
    template<typename T>
    struct poly_t {
        using Vector = big_vector<T>;
        using base = T;
        Vector a;
        
        poly_t& normalize() {
            while(deg() >= 0 && lead() == base(0)) {
                a.pop_back();
            }
            return *this;
        }
        
        poly_t(){}
        poly_t(T a0): a{a0} {normalize();}
        poly_t(Vector const& t): a(t) {normalize();}
        poly_t(Vector &&t): a(std::move(t)) {normalize();}
        
        poly_t& negate_inplace() {
            std::ranges::transform(a, begin(a), std::negate{});
            return *this;
        }
        poly_t operator -() const {
            return poly_t(*this).negate_inplace();
        }
        poly_t& operator += (poly_t const& t) {
            a.resize(std::max(size(a), size(t.a)));
            std::ranges::transform(a, t.a, begin(a), std::plus{});
            return normalize();
        }
        poly_t& operator -= (poly_t const& t) {
            a.resize(std::max(size(a), size(t.a)));
            std::ranges::transform(a, t.a, begin(a), std::minus{});
            return normalize();
        }
        poly_t operator + (poly_t const& t) const {return poly_t(*this) += t;}
        poly_t operator - (poly_t const& t) const {return poly_t(*this) -= t;}
        
        poly_t& mod_xk_inplace(size_t k) {
            a.resize(std::min(size(a), k));
            return normalize();
        }
        poly_t& mul_xk_inplace(size_t k) {
            a.insert(begin(a), k, T(0));
            return normalize();
        }
        poly_t& div_xk_inplace(int64_t k) {
            if(k < 0) {
                return mul_xk_inplace(-k);
            }
            a.erase(begin(a), begin(a) + std::min<size_t>(k, size(a)));
            return normalize();
        }
        poly_t &substr_inplace(size_t l, size_t k) {
            return mod_xk_inplace(l + k).div_xk_inplace(l);
        }
        poly_t mod_xk(size_t k) const {return poly_t(*this).mod_xk_inplace(k);}
        poly_t mul_xk(size_t k) const {return poly_t(*this).mul_xk_inplace(k);}
        poly_t div_xk(int64_t k) const {return poly_t(*this).div_xk_inplace(k);}
        poly_t substr(size_t l, size_t k) const {return poly_t(*this).substr_inplace(l, k);}
        
        poly_t& operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
        poly_t operator * (const poly_t &t) const {return poly_t(*this) *= t;}

        poly_t& operator /= (const poly_t &t) {return *this = divmod(t)[0];}
        poly_t& operator %= (const poly_t &t) {return *this = divmod(t)[1];}
        poly_t operator / (poly_t const& t) const {return poly_t(*this) /= t;}
        poly_t operator % (poly_t const& t) const {return poly_t(*this) %= t;}

        poly_t& operator *= (T const& x) {
            for(auto &it: a) {
                it *= x;
            }
            return normalize();
        }
        poly_t& operator /= (T const& x) {return *this *= x.inv();}
        poly_t operator * (T const& x) const {return poly_t(*this) *= x;}
        poly_t operator / (T const& x) const {return poly_t(*this) /= x;}
        
        poly_t& reverse(size_t n) {
            a.resize(n);
            std::ranges::reverse(a);
            return normalize();
        }
        poly_t& reverse() {return reverse(size(a));}
        poly_t reversed(size_t n) const {return poly_t(*this).reverse(n);}
        poly_t reversed() const {return poly_t(*this).reverse();}
        
        std::array<poly_t, 2> divmod(poly_t const& b) const {
            return poly::impl::divmod(*this, b);
        }
        
        // reduces A/B to A'/B' such that
        // deg B' < deg A / 2
        static std::pair<std::list<poly_t>, linfrac<poly_t>> half_gcd(auto &&A, auto &&B) {
            return poly::impl::half_gcd(A, B);
        }
        // reduces A / B to gcd(A, B) / 0
        static std::pair<std::list<poly_t>, linfrac<poly_t>> full_gcd(auto &&A, auto &&B) {
            return poly::impl::full_gcd(A, B);
        }
        static poly_t gcd(poly_t &&A, poly_t &&B) {
            full_gcd(A, B);
            return A;
        }
        
        // Returns a (non-monic) characteristic polynomial
        // of the minimum linear recurrence for the sequence
        poly_t min_rec(size_t d) const {
            return poly::impl::min_rec(*this, d);
        }
        
        // calculate inv to *this modulo t
        std::optional<poly_t> inv_mod(poly_t const& t) const {
            return poly::impl::inv_mod(*this, t);
        };
        
        poly_t negx() const { // A(x) -> A(-x)
            auto res = *this;
            for(int i = 1; i <= deg(); i += 2) {
                res.a[i] = -res[i];
            }
            return res;
        }
        
        void print(int n) const {
            for(int i = 0; i < n; i++) {
                std::cout << (*this)[i] << ' ';
            }
            std::cout << "\n";
        }
        
        void print() const {
            print(deg() + 1);
        }
        
        T eval(T x) const { // evaluates in single point x
            T res(0);
            for(int i = deg(); i >= 0; i--) {
                res *= x;
                res += a[i];
            }
            return res;
        }
        
        T lead() const { // leading coefficient
            assert(!is_zero());
            return a.back();
        }
        
        int deg() const { // degree, -1 for P(x) = 0
            return (int)a.size() - 1;
        }
        
        bool is_zero() const {
            return a.empty();
        }
        
        T operator [](int idx) const {
            return idx < 0 || idx > deg() ? T(0) : a[idx];
        }
        
        T& coef(size_t idx) { // mutable reference at coefficient
            return a[idx];
        }
        
        bool operator == (const poly_t &t) const {return a == t.a;}
        bool operator != (const poly_t &t) const {return a != t.a;}
        
        poly_t& deriv_inplace(int k = 1) {
            if(deg() + 1 < k) {
                return *this = poly_t{};
            }
            for(int i = k; i <= deg(); i++) {
                a[i - k] = fact<T>(i) * rfact<T>(i - k) * a[i];
            }
            a.resize(deg() + 1 - k);
            return *this;
        }
        poly_t deriv(int k = 1) const { // calculate derivative
            return poly_t(*this).deriv_inplace(k);
        }

        poly_t& integr_inplace() {
            a.push_back(0);
            for(int i = deg() - 1; i >= 0; i--) {
                a[i + 1] = a[i] * small_inv<T>(i + 1);
            }
            a[0] = 0;
            return *this;
        }
        poly_t integr() const { // calculate integral with C = 0
            Vector res(deg() + 2);
            for(int i = 0; i <= deg(); i++) {
                res[i + 1] = a[i] * small_inv<T>(i + 1);
            }
            return res;
        }
        
        size_t trailing_xk() const { // Let p(x) = x^k * t(x), return k
            if(is_zero()) {
                return -1;
            }
            int res = 0;
            while(a[res] == T(0)) {
                res++;
            }
            return res;
        }
        
        // calculate log p(x) mod x^n
        poly_t& log_inplace(size_t n) {
            assert(a[0] == T(1));
            mod_xk_inplace(n);
            return (inv_inplace(n) *= mod_xk_inplace(n).deriv()).mod_xk_inplace(n - 1).integr_inplace();
        }
        poly_t log(size_t n) const {
            return poly_t(*this).log_inplace(n);
        }
        
        poly_t& mul_truncate(poly_t const& t, size_t k) {
            fft::mul_truncate(a, t.a, k);
            return normalize();
        }

        poly_t& exp_inplace(size_t n) {
            if(is_zero()) {
                return *this = T(1);
            }
            assert(a[0] == T(0));
            a[0] = 1;
            size_t a = 1;
            while(a < n) {
                poly_t C = log(2 * a).div_xk_inplace(a) - substr(a, 2 * a);
                *this -= C.mul_truncate(*this, a).mul_xk_inplace(a);
                a *= 2;
            }
            return mod_xk_inplace(n);
        }

        poly_t exp(size_t n) const { // calculate exp p(x) mod x^n
            return poly_t(*this).exp_inplace(n);
        }
        
        poly_t pow_bin(int64_t k, size_t n) const { // O(n log n log k)
            if(k == 0) {
                return poly_t(1).mod_xk(n);
            } else {
                auto t = pow(k / 2, n);
                t = (t * t).mod_xk(n);
                return (k % 2 ? *this * t : t).mod_xk(n);
            }
        }

        poly_t circular_closure(size_t m) const {
            if(deg() == -1) {
                return *this;
            }
            auto t = *this;
            for(size_t i = t.deg(); i >= m; i--) {
                t.a[i - m] += t.a[i];
            }
            t.a.resize(std::min(t.a.size(), m));
            return t;
        }

        static poly_t mul_circular(poly_t const& a, poly_t const& b, size_t m) {
            return (a.circular_closure(m) * b.circular_closure(m)).circular_closure(m);
        }

        poly_t powmod_circular(int64_t k, size_t m) const {
            if(k == 0) {
                return poly_t(1);
            } else {
                auto t = powmod_circular(k / 2, m);
                t = mul_circular(t, t, m);
                if(k % 2) {
                    t = mul_circular(t, *this, m);
                }
                return t;
            }
        }
        
        poly_t powmod(int64_t k, poly_t const& md) const {
            return poly::impl::powmod(*this, k, md);
        }
        
        // O(d * n) with the derivative trick from
        // https://codeforces.com/blog/entry/73947?#comment-581173
        poly_t pow_dn(int64_t k, size_t n) const {
            if(n == 0) {
                return poly_t(T(0));
            }
            assert((*this)[0] != T(0));
            Vector Q(n);
            Q[0] = bpow(a[0], k);
            auto a0inv = a[0].inv();
            for(int i = 1; i < (int)n; i++) {
                for(int j = 1; j <= std::min(deg(), i); j++) {
                    Q[i] += a[j] * Q[i - j] * (T(k) * T(j) - T(i - j));
                }
                Q[i] *= small_inv<T>(i) * a0inv;
            }
            return Q;
        }
        
        // calculate p^k(n) mod x^n in O(n log n)
        // might be quite slow due to high constant
        poly_t pow(int64_t k, size_t n) const {
            if(is_zero()) {
                return k ? *this : poly_t(1);
            }
            size_t i = trailing_xk();
            if(i > 0) {
                return k >= int64_t(n + i - 1) / (int64_t)i ? poly_t(T(0)) : div_xk(i).pow(k, n - i * k).mul_xk(i * k);
            }
            if(std::min(deg(), (int)n) <= magic) {
                return pow_dn(k, n);
            }
            if(k <= magic) {
                return pow_bin(k, n);
            }
            T j = a[i];
            poly_t t = *this / j;
            return bpow(j, k) * (t.log(n) * T(k)).exp(n).mod_xk(n);
        }
        
        // returns std::nullopt if undefined
        std::optional<poly_t> sqrt(size_t n) const {
            if(is_zero()) {
                return *this;
            }
            size_t i = trailing_xk();
            if(i % 2) {
                return std::nullopt;
            } else if(i > 0) {
                auto ans = div_xk(i).sqrt(n - i / 2);
                return ans ? ans->mul_xk(i / 2) : ans;
            }
            auto st = math::sqrt((*this)[0]);
            if(st) {
                poly_t ans = *st;
                size_t a = 1;
                while(a < n) {
                    a *= 2;
                    ans -= (ans - mod_xk(a) * ans.inv(a)).mod_xk(a) / 2;
                }
                return ans.mod_xk(n);
            }
            return std::nullopt;
        }
        
        poly_t mulx(T a) const { // component-wise multiplication with a^k
            T cur = 1;
            poly_t res(*this);
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= cur;
                cur *= a;
            }
            return res;
        }

        poly_t mulx_sq(T a) const { // component-wise multiplication with a^{k choose 2}
            T cur = 1, total = 1;
            poly_t res(*this);
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= total;
                cur *= a;
                total *= cur;
            }
            return res;
        }

        // be mindful of maxn, as the function
        // requires multiplying polynomials of size deg() and n+deg()!
        poly_t chirpz(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
            if(is_zero()) {
                return Vector(n, 0);
            }
            if(z == T(0)) {
                Vector ans(n, (*this)[0]);
                if(n > 0) {
                    ans[0] = accumulate(begin(a), end(a), T(0));
                }
                return ans;
            }
            auto A = mulx_sq(z.inv());
            auto B = ones(n+deg()).mulx_sq(z);
            return semicorr(B, A).mod_xk(n).mulx_sq(z.inv());
        }

        // res[i] = prod_{1 <= j <= i} 1/(1 - z^j)
        static auto _1mzk_prod_inv(T z, int n) {
            Vector res(n, 1), zk(n);
            zk[0] = 1;
            for(int i = 1; i < n; i++) {
                zk[i] = zk[i - 1] * z;
                res[i] = res[i - 1] * (T(1) - zk[i]);
            }
            res.back() = res.back().inv();
            for(int i = n - 2; i >= 0; i--) {
                res[i] = (T(1) - zk[i+1]) * res[i+1];
            }
            return res;
        }
        
        // prod_{0 <= j < n} (1 - z^j x)
        static auto _1mzkx_prod(T z, int n) {
            if(n == 1) {
                return poly_t(Vector{1, -1});
            } else {
                auto t = _1mzkx_prod(z, n / 2);
                t *= t.mulx(bpow(z, n / 2));
                if(n % 2) {
                    t *= poly_t(Vector{1, -bpow(z, n - 1)});
                }
                return t;
            }
        }

        poly_t chirpz_inverse(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
            if(is_zero()) {
                return {};
            }
            if(z == T(0)) {
                if(n == 1) {
                    return *this;
                } else {
                    return Vector{(*this)[1], (*this)[0] - (*this)[1]};
                }
            }
            Vector y(n);
            for(int i = 0; i < n; i++) {
                y[i] = (*this)[i];
            }
            auto prods_pos = _1mzk_prod_inv(z, n);
            auto prods_neg = _1mzk_prod_inv(z.inv(), n);

            T zn = bpow(z, n-1).inv();
            T znk = 1;
            for(int i = 0; i < n; i++) {
                y[i] *= znk * prods_neg[i] * prods_pos[(n - 1) - i];
                znk *= zn;
            }

            poly_t p_over_q = poly_t(y).chirpz(z, n);
            poly_t q = _1mzkx_prod(z, n);

            return (p_over_q * q).mod_xk_inplace(n).reverse(n);
        }

        static poly_t build(big_vector<poly_t> &res, int v, auto L, auto R) { // builds evaluation tree for (x-a1)(x-a2)...(x-an)
            if(R - L == 1) {
                return res[v] = Vector{-*L, 1};
            } else {
                auto M = L + (R - L) / 2;
                return res[v] = build(res, 2 * v, L, M) * build(res, 2 * v + 1, M, R);
            }
        }

        poly_t to_newton(big_vector<poly_t> &tree, int v, auto l, auto r) {
            if(r - l == 1) {
                return *this;
            } else {
                auto m = l + (r - l) / 2;
                auto A = (*this % tree[2 * v]).to_newton(tree, 2 * v, l, m);
                auto B = (*this / tree[2 * v]).to_newton(tree, 2 * v + 1, m, r);
                return A + B.mul_xk(m - l);
            }
        }

        poly_t to_newton(Vector p) {
            if(is_zero()) {
                return *this;
            }
            size_t n = p.size();
            big_vector<poly_t> tree(4 * n);
            build(tree, 1, begin(p), end(p));
            return to_newton(tree, 1, begin(p), end(p));
        }

        Vector eval(big_vector<poly_t> &tree, int v, auto l, auto r) { // auxiliary evaluation function
            if(r - l == 1) {
                return {eval(*l)};
            } else {
                auto m = l + (r - l) / 2;
                auto A = (*this % tree[2 * v]).eval(tree, 2 * v, l, m);
                auto B = (*this % tree[2 * v + 1]).eval(tree, 2 * v + 1, m, r);
                A.insert(end(A), begin(B), end(B));
                return A;
            }
        }
        
        Vector eval(Vector x) { // evaluate polynomial in (x1, ..., xn)
            size_t n = x.size();
            if(is_zero()) {
                return Vector(n, T(0));
            }
            big_vector<poly_t> tree(4 * n);
            build(tree, 1, begin(x), end(x));
            return eval(tree, 1, begin(x), end(x));
        }
        
        poly_t inter(big_vector<poly_t> &tree, int v, auto ly, auto ry) { // auxiliary interpolation function
            if(ry - ly == 1) {
                return {*ly / a[0]};
            } else {
                auto my = ly + (ry - ly) / 2;
                auto A = (*this % tree[2 * v]).inter(tree, 2 * v, ly, my);
                auto B = (*this % tree[2 * v + 1]).inter(tree, 2 * v + 1, my, ry);
                return A * tree[2 * v + 1] + B * tree[2 * v];
            }
        }
        
        static auto inter(Vector x, Vector y) { // interpolates minimum polynomial from (xi, yi) pairs
            size_t n = x.size();
            big_vector<poly_t> tree(4 * n);
            return build(tree, 1, begin(x), end(x)).deriv().inter(tree, 1, begin(y), end(y));
        }

        static auto resultant(poly_t a, poly_t b) { // computes resultant of a and b
            if(b.is_zero()) {
                return 0;
            } else if(b.deg() == 0) {
                return bpow(b.lead(), a.deg());
            } else {
                int pw = a.deg();
                a %= b;
                pw -= a.deg();
                auto mul = bpow(b.lead(), pw) * T((b.deg() & a.deg() & 1) ? -1 : 1);
                auto ans = resultant(b, a);
                return ans * mul;
            }
        }
                
        static poly_t xk(size_t n) { // P(x) = x^n
            return poly_t(T(1)).mul_xk(n);
        }
        
        static poly_t ones(size_t n) { // P(x) = 1 + x + ... + x^{n-1} 
            return Vector(n, 1);
        }
        
        static poly_t expx(size_t n) { // P(x) = e^x (mod x^n)
            return ones(n).borel();
        }

        static poly_t log1px(size_t n) { // P(x) = log(1+x) (mod x^n)
            Vector coeffs(n, 0);
            for(size_t i = 1; i < n; i++) {
                coeffs[i] = (i & 1 ? T(i).inv() : -T(i).inv());
            }
            return coeffs;
        }

        static poly_t log1mx(size_t n) { // P(x) = log(1-x) (mod x^n)
            return -ones(n).integr();
        }
        
        // [x^k] (a corr b) = sum_{i} a{(k-m)+i}*bi
        static poly_t corr(poly_t const& a, poly_t const& b) { // cross-correlation
            return a * b.reversed();
        }

        // [x^k] (a semicorr b) = sum_i a{i+k} * b{i}
        static poly_t semicorr(poly_t const& a, poly_t const& b) {
            return corr(a, b).div_xk(b.deg());
        }
        
        poly_t invborel() const { // ak *= k!
            auto res = *this;
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= fact<T>(i);
            }
            return res;
        }
        
        poly_t borel() const { // ak /= k!
            auto res = *this;
            for(int i = 0; i <= deg(); i++) {
                res.coef(i) *= rfact<T>(i);
            }
            return res;
        }
        
        poly_t shift(T a) const { // P(x + a)
            return semicorr(invborel(), expx(deg() + 1).mulx(a)).borel();
        }
        
        poly_t x2() { // P(x) -> P(x^2)
            Vector res(2 * a.size());
            for(size_t i = 0; i < a.size(); i++) {
                res[2 * i] = a[i];
            }
            return res;
        }
        
        // Return {P0, P1}, where P(x) = P0(x) + xP1(x)
        std::array<poly_t, 2> bisect(size_t n) const {
            n = std::min(n, size(a));
            Vector res[2];
            for(size_t i = 0; i < n; i++) {
                res[i % 2].push_back(a[i]);
            }
            return {res[0], res[1]};
        }
        std::array<poly_t, 2> bisect() const {
            return bisect(size(a));
        }
        
        // Find [x^k] P / Q
        static T kth_rec_inplace(poly_t &P, poly_t &Q, int64_t k) {
            while(k > Q.deg()) {
                size_t n = Q.a.size();
                auto [Q0, Q1] = Q.bisect();
                auto [P0, P1] = P.bisect();
                
                size_t N = fft::com_size((n + 1) / 2, (n + 1) / 2);
                
                auto Q0f = fft::dft<T>(Q0.a, N);
                auto Q1f = fft::dft<T>(Q1.a, N);
                auto P0f = fft::dft<T>(P0.a, N);
                auto P1f = fft::dft<T>(P1.a, N);
                
                Q = poly_t(Q0f * Q0f) -= poly_t(Q1f * Q1f).mul_xk_inplace(1);
                if(k % 2) {
                    P = poly_t(Q0f *= P1f) -= poly_t(Q1f *= P0f);
                } else {
                    P = poly_t(Q0f *= P0f) -= poly_t(Q1f *= P1f).mul_xk_inplace(1);
                }
                k /= 2;
            }
            return (P *= Q.inv_inplace(Q.deg() + 1))[(int)k];
        }
        static T kth_rec(poly_t const& P, poly_t const& Q, int64_t k) {
            return kth_rec_inplace(poly_t(P), poly_t(Q), k);
        }

        // inverse series mod x^n
        poly_t& inv_inplace(size_t n) {
            return poly::impl::inv_inplace(*this, n);
        }
        poly_t inv(size_t n) const {
            return poly_t(*this).inv_inplace(n);
        }
        // [x^k]..[x^{k+n-1}] of inv()
        // supports negative k if k+n >= 0
        poly_t& inv_inplace(int64_t k, size_t n) {
            return poly::impl::inv_inplace(*this, k, n);
        }
        poly_t inv(int64_t k, size_t n) const {
            return poly_t(*this).inv_inplace(k, n);
        }
        
        // compute A(B(x)) mod x^n in O(n^2)
        static poly_t compose(poly_t A, poly_t B, int n) {
            int q = std::sqrt(n);
            big_vector<poly_t> Bk(q);
            auto Bq = B.pow(q, n);
            Bk[0] = poly_t(T(1));
            for(int i = 1; i < q; i++) {
                Bk[i] = (Bk[i - 1] * B).mod_xk(n);
            }
            poly_t Bqk(1);
            poly_t ans;
            for(int i = 0; i <= n / q; i++) {
                poly_t cur;
                for(int j = 0; j < q; j++) {
                    cur += Bk[j] * A[i * q + j];
                }
                ans += (Bqk * cur).mod_xk(n);
                Bqk = (Bqk * Bq).mod_xk(n);
            }
            return ans;
        }
        
        // compute A(B(x)) mod x^n in O(sqrt(pqn log^3 n))
        // preferrable when p = deg A and q = deg B
        // are much less than n
        static poly_t compose_large(poly_t A, poly_t B, int n) {
            if(B[0] != T(0)) {
                return compose_large(A.shift(B[0]), B - B[0], n);
            }
            
            int q = std::sqrt(n);
            auto [B0, B1] = std::make_pair(B.mod_xk(q), B.div_xk(q));
            
            B0 = B0.div_xk(1);
            big_vector<poly_t> pw(A.deg() + 1);
            auto getpow = [&](int k) {
                return pw[k].is_zero() ? pw[k] = B0.pow(k, n - k) : pw[k];
            };
            
            std::function<poly_t(poly_t const&, int, int)> compose_dac = [&getpow, &compose_dac](poly_t const& f, int m, int N) {
                if(f.deg() <= 0) {
                    return f;
                }
                int k = m / 2;
                auto [f0, f1] = std::make_pair(f.mod_xk(k), f.div_xk(k));
                auto [A, B] = std::make_pair(compose_dac(f0, k, N), compose_dac(f1, m - k, N - k));
                return (A + (B.mod_xk(N - k) * getpow(k).mod_xk(N - k)).mul_xk(k)).mod_xk(N);
            };
            
            int r = n / q;
            auto Ar = A.deriv(r);
            auto AB0 = compose_dac(Ar, Ar.deg() + 1, n);
            
            auto Bd = B0.mul_xk(1).deriv();
            
            poly_t ans = T(0);
            
            big_vector<poly_t> B1p(r + 1);
            B1p[0] = poly_t(T(1));
            for(int i = 1; i <= r; i++) {
                B1p[i] = (B1p[i - 1] * B1.mod_xk(n - i * q)).mod_xk(n - i * q);
            }
            while(r >= 0) {
                ans += (AB0.mod_xk(n - r * q) * rfact<T>(r) * B1p[r]).mul_xk(r * q).mod_xk(n);
                r--;
                if(r >= 0) {
                    AB0 = ((AB0 * Bd).integr() + A[r] * fact<T>(r)).mod_xk(n);
                }
            }
            
            return ans;
        }
    };
    template<typename base>
    static auto operator * (const auto& a, const poly_t<base>& b) {
        return b * a;
    }
};
#pragma GCC pop_options

#ifndef CP_ALGO_MATH_POLY_HPP
#define CP_ALGO_MATH_POLY_HPP
#include "poly/impl/euclid.hpp"
#include "poly/impl/div.hpp"
#include "combinatorics.hpp"
#include "../number_theory/discrete_sqrt.hpp"
#include "fft.hpp"
#include <functional>
#include <algorithm>
#include <iostream>
#include <optional>
#include <utility>
#include <vector>
#include <list>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<typename T>struct poly_t{using Vector=big_vector<T>;using base=T;Vector a;poly_t&normalize(){while(deg()>=0&&lead()==base(0)){a.pop_back();}return*this;}poly_t(){}poly_t(T a0):a{a0}{normalize();}poly_t(Vector const&t):a(t){normalize();}poly_t(Vector&&t):a(std::move(t)){normalize();}poly_t&negate_inplace(){std::ranges::transform(a,begin(a),std::negate{});return*this;}poly_t operator-()const{return poly_t(*this).negate_inplace();}poly_t&operator+=(poly_t const&t){a.resize(std::max(size(a),size(t.a)));std::ranges::transform(a,t.a,begin(a),std::plus{});return normalize();}poly_t&operator-=(poly_t const&t){a.resize(std::max(size(a),size(t.a)));std::ranges::transform(a,t.a,begin(a),std::minus{});return normalize();}poly_t operator+(poly_t const&t)const{return poly_t(*this)+=t;}poly_t operator-(poly_t const&t)const{return poly_t(*this)-=t;}poly_t&mod_xk_inplace(size_t k){a.resize(std::min(size(a),k));return normalize();}poly_t&mul_xk_inplace(size_t k){a.insert(begin(a),k,T(0));return normalize();}poly_t&div_xk_inplace(int64_t k){if(k<0){return mul_xk_inplace(-k);}a.erase(begin(a),begin(a)+std::min<size_t>(k,size(a)));return normalize();}poly_t&substr_inplace(size_t l,size_t k){return mod_xk_inplace(l+k).div_xk_inplace(l);}poly_t mod_xk(size_t k)const{return poly_t(*this).mod_xk_inplace(k);}poly_t mul_xk(size_t k)const{return poly_t(*this).mul_xk_inplace(k);}poly_t div_xk(int64_t k)const{return poly_t(*this).div_xk_inplace(k);}poly_t substr(size_t l,size_t k)const{return poly_t(*this).substr_inplace(l,k);}poly_t&operator*=(const poly_t&t){fft::mul(a,t.a);normalize();return*this;}poly_t operator*(const poly_t&t)const{return poly_t(*this)*=t;}poly_t&operator/=(const poly_t&t){return*this=divmod(t)[0];}poly_t&operator%=(const poly_t&t){return*this=divmod(t)[1];}poly_t operator/(poly_t const&t)const{return poly_t(*this)/=t;}poly_t operator%(poly_t const&t)const{return poly_t(*this)%=t;}poly_t&operator*=(T const&x){for(auto&it:a){it*=x;}return normalize();}poly_t&operator/=(T const&x){return*this*=x.inv();}poly_t operator*(T const&x)const{return poly_t(*this)*=x;}poly_t operator/(T const&x)const{return poly_t(*this)/=x;}poly_t&reverse(size_t n){a.resize(n);std::ranges::reverse(a);return normalize();}poly_t&reverse(){return reverse(size(a));}poly_t reversed(size_t n)const{return poly_t(*this).reverse(n);}poly_t reversed()const{return poly_t(*this).reverse();}std::array<poly_t,2>divmod(poly_t const&b)const{return poly::impl::divmod(*this,b);}static std::pair<std::list<poly_t>,linfrac<poly_t>>half_gcd(auto&&A,auto&&B){return poly::impl::half_gcd(A,B);}static std::pair<std::list<poly_t>,linfrac<poly_t>>full_gcd(auto&&A,auto&&B){return poly::impl::full_gcd(A,B);}static poly_t gcd(poly_t&&A,poly_t&&B){full_gcd(A,B);return A;}poly_t min_rec(size_t d)const{return poly::impl::min_rec(*this,d);}std::optional<poly_t>inv_mod(poly_t const&t)const{return poly::impl::inv_mod(*this,t);};poly_t negx()const{auto res=*this;for(int i=1;i<=deg();i+=2){res.a[i]=-res[i];}return res;}void print(int n)const{for(int i=0;i<n;i++){std::cout<<(*this)[i]<<' ';}std::cout<<"\n";}void print()const{print(deg()+1);}T eval(T x)const{T res(0);for(int i=deg();i>=0;i--){res*=x;res+=a[i];}return res;}T lead()const{assert(!is_zero());return a.back();}int deg()const{return(int)a.size()-1;}bool is_zero()const{return a.empty();}T operator[](int idx)const{return idx<0||idx>deg()?T(0):a[idx];}T&coef(size_t idx){return a[idx];}bool operator==(const poly_t&t)const{return a==t.a;}bool operator!=(const poly_t&t)const{return a!=t.a;}poly_t&deriv_inplace(int k=1){if(deg()+1<k){return*this=poly_t{};}for(int i=k;i<=deg();i++){a[i-k]=fact<T>(i)*rfact<T>(i-k)*a[i];}a.resize(deg()+1-k);return*this;}poly_t deriv(int k=1)const{return poly_t(*this).deriv_inplace(k);}poly_t&integr_inplace(){a.push_back(0);for(int i=deg()-1;i>=0;i--){a[i+1]=a[i]*small_inv<T>(i+1);}a[0]=0;return*this;}poly_t integr()const{Vector res(deg()+2);for(int i=0;i<=deg();i++){res[i+1]=a[i]*small_inv<T>(i+1);}return res;}size_t trailing_xk()const{if(is_zero()){return-1;}int res=0;while(a[res]==T(0)){res++;}return res;}poly_t&log_inplace(size_t n){assert(a[0]==T(1));mod_xk_inplace(n);return(inv_inplace(n)*=mod_xk_inplace(n).deriv()).mod_xk_inplace(n-1).integr_inplace();}poly_t log(size_t n)const{return poly_t(*this).log_inplace(n);}poly_t&mul_truncate(poly_t const&t,size_t k){fft::mul_truncate(a,t.a,k);return normalize();}poly_t&exp_inplace(size_t n){if(is_zero()){return*this=T(1);}assert(a[0]==T(0));a[0]=1;size_t a=1;while(a<n){poly_t C=log(2*a).div_xk_inplace(a)-substr(a,2*a);*this-=C.mul_truncate(*this,a).mul_xk_inplace(a);a*=2;}return mod_xk_inplace(n);}poly_t exp(size_t n)const{return poly_t(*this).exp_inplace(n);}poly_t pow_bin(int64_t k,size_t n)const{if(k==0){return poly_t(1).mod_xk(n);}else{auto t=pow(k/2,n);t=(t*t).mod_xk(n);return(k%2?*this*t:t).mod_xk(n);}}poly_t circular_closure(size_t m)const{if(deg()==-1){return*this;}auto t=*this;for(size_t i=t.deg();i>=m;i--){t.a[i-m]+=t.a[i];}t.a.resize(std::min(t.a.size(),m));return t;}static poly_t mul_circular(poly_t const&a,poly_t const&b,size_t m){return(a.circular_closure(m)*b.circular_closure(m)).circular_closure(m);}poly_t powmod_circular(int64_t k,size_t m)const{if(k==0){return poly_t(1);}else{auto t=powmod_circular(k/2,m);t=mul_circular(t,t,m);if(k%2){t=mul_circular(t,*this,m);}return t;}}poly_t powmod(int64_t k,poly_t const&md)const{return poly::impl::powmod(*this,k,md);}poly_t pow_dn(int64_t k,size_t n)const{if(n==0){return poly_t(T(0));}assert((*this)[0]!=T(0));Vector Q(n);Q[0]=bpow(a[0],k);auto a0inv=a[0].inv();for(int i=1;i<(int)n;i++){for(int j=1;j<=std::min(deg(),i);j++){Q[i]+=a[j]*Q[i-j]*(T(k)*T(j)-T(i-j));}Q[i]*=small_inv<T>(i)*a0inv;}return Q;}poly_t pow(int64_t k,size_t n)const{if(is_zero()){return k?*this:poly_t(1);}size_t i=trailing_xk();if(i>0){return k>=int64_t(n+i-1)/(int64_t)i?poly_t(T(0)):div_xk(i).pow(k,n-i*k).mul_xk(i*k);}if(std::min(deg(),(int)n)<=magic){return pow_dn(k,n);}if(k<=magic){return pow_bin(k,n);}T j=a[i];poly_t t=*this/j;return bpow(j,k)*(t.log(n)*T(k)).exp(n).mod_xk(n);}std::optional<poly_t>sqrt(size_t n)const{if(is_zero()){return*this;}size_t i=trailing_xk();if(i%2){return std::nullopt;}else if(i>0){auto ans=div_xk(i).sqrt(n-i/2);return ans?ans->mul_xk(i/2):ans;}auto st=math::sqrt((*this)[0]);if(st){poly_t ans=*st;size_t a=1;while(a<n){a*=2;ans-=(ans-mod_xk(a)*ans.inv(a)).mod_xk(a)/2;}return ans.mod_xk(n);}return std::nullopt;}poly_t mulx(T a)const{T cur=1;poly_t res(*this);for(int i=0;i<=deg();i++){res.coef(i)*=cur;cur*=a;}return res;}poly_t mulx_sq(T a)const{T cur=1,total=1;poly_t res(*this);for(int i=0;i<=deg();i++){res.coef(i)*=total;cur*=a;total*=cur;}return res;}poly_t chirpz(T z,int n)const{if(is_zero()){return Vector(n,0);}if(z==T(0)){Vector ans(n,(*this)[0]);if(n>0){ans[0]=accumulate(begin(a),end(a),T(0));}return ans;}auto A=mulx_sq(z.inv());auto B=ones(n+deg()).mulx_sq(z);return semicorr(B,A).mod_xk(n).mulx_sq(z.inv());}static auto _1mzk_prod_inv(T z,int n){Vector res(n,1),zk(n);zk[0]=1;for(int i=1;i<n;i++){zk[i]=zk[i-1]*z;res[i]=res[i-1]*(T(1)-zk[i]);}res.back()=res.back().inv();for(int i=n-2;i>=0;i--){res[i]=(T(1)-zk[i+1])*res[i+1];}return res;}static auto _1mzkx_prod(T z,int n){if(n==1){return poly_t(Vector{1,-1});}else{auto t=_1mzkx_prod(z,n/2);t*=t.mulx(bpow(z,n/2));if(n%2){t*=poly_t(Vector{1,-bpow(z,n-1)});}return t;}}poly_t chirpz_inverse(T z,int n)const{if(is_zero()){return{};}if(z==T(0)){if(n==1){return*this;}else{return Vector{(*this)[1],(*this)[0]-(*this)[1]};}}Vector y(n);for(int i=0;i<n;i++){y[i]=(*this)[i];}auto prods_pos=_1mzk_prod_inv(z,n);auto prods_neg=_1mzk_prod_inv(z.inv(),n);T zn=bpow(z,n-1).inv();T znk=1;for(int i=0;i<n;i++){y[i]*=znk*prods_neg[i]*prods_pos[(n-1)-i];znk*=zn;}poly_t p_over_q=poly_t(y).chirpz(z,n);poly_t q=_1mzkx_prod(z,n);return(p_over_q*q).mod_xk_inplace(n).reverse(n);}static poly_t build(big_vector<poly_t>&res,int v,auto L,auto R){if(R-L==1){return res[v]=Vector{-*L,1};}else{auto M=L+(R-L)/2;return res[v]=build(res,2*v,L,M)*build(res,2*v+1,M,R);}}poly_t to_newton(big_vector<poly_t>&tree,int v,auto l,auto r){if(r-l==1){return*this;}else{auto m=l+(r-l)/2;auto A=(*this%tree[2*v]).to_newton(tree,2*v,l,m);auto B=(*this/tree[2*v]).to_newton(tree,2*v+1,m,r);return A+B.mul_xk(m-l);}}poly_t to_newton(Vector p){if(is_zero()){return*this;}size_t n=p.size();big_vector<poly_t>tree(4*n);build(tree,1,begin(p),end(p));return to_newton(tree,1,begin(p),end(p));}Vector eval(big_vector<poly_t>&tree,int v,auto l,auto r){if(r-l==1){return{eval(*l)};}else{auto m=l+(r-l)/2;auto A=(*this%tree[2*v]).eval(tree,2*v,l,m);auto B=(*this%tree[2*v+1]).eval(tree,2*v+1,m,r);A.insert(end(A),begin(B),end(B));return A;}}Vector eval(Vector x){size_t n=x.size();if(is_zero()){return Vector(n,T(0));}big_vector<poly_t>tree(4*n);build(tree,1,begin(x),end(x));return eval(tree,1,begin(x),end(x));}poly_t inter(big_vector<poly_t>&tree,int v,auto ly,auto ry){if(ry-ly==1){return{*ly/a[0]};}else{auto my=ly+(ry-ly)/2;auto A=(*this%tree[2*v]).inter(tree,2*v,ly,my);auto B=(*this%tree[2*v+1]).inter(tree,2*v+1,my,ry);return A*tree[2*v+1]+B*tree[2*v];}}static auto inter(Vector x,Vector y){size_t n=x.size();big_vector<poly_t>tree(4*n);return build(tree,1,begin(x),end(x)).deriv().inter(tree,1,begin(y),end(y));}static auto resultant(poly_t a,poly_t b){if(b.is_zero()){return 0;}else if(b.deg()==0){return bpow(b.lead(),a.deg());}else{int pw=a.deg();a%=b;pw-=a.deg();auto mul=bpow(b.lead(),pw)*T((b.deg()&a.deg()&1)?-1:1);auto ans=resultant(b,a);return ans*mul;}}static poly_t xk(size_t n){return poly_t(T(1)).mul_xk(n);}static poly_t ones(size_t n){return Vector(n,1);}static poly_t expx(size_t n){return ones(n).borel();}static poly_t log1px(size_t n){Vector coeffs(n,0);for(size_t i=1;i<n;i++){coeffs[i]=(i&1?T(i).inv():-T(i).inv());}return coeffs;}static poly_t log1mx(size_t n){return-ones(n).integr();}static poly_t corr(poly_t const&a,poly_t const&b){return a*b.reversed();}static poly_t semicorr(poly_t const&a,poly_t const&b){return corr(a,b).div_xk(b.deg());}poly_t invborel()const{auto res=*this;for(int i=0;i<=deg();i++){res.coef(i)*=fact<T>(i);}return res;}poly_t borel()const{auto res=*this;for(int i=0;i<=deg();i++){res.coef(i)*=rfact<T>(i);}return res;}poly_t shift(T a)const{return semicorr(invborel(),expx(deg()+1).mulx(a)).borel();}poly_t x2(){Vector res(2*a.size());for(size_t i=0;i<a.size();i++){res[2*i]=a[i];}return res;}std::array<poly_t,2>bisect(size_t n)const{n=std::min(n,size(a));Vector res[2];for(size_t i=0;i<n;i++){res[i%2].push_back(a[i]);}return{res[0],res[1]};}std::array<poly_t,2>bisect()const{return bisect(size(a));}static T kth_rec_inplace(poly_t&P,poly_t&Q,int64_t k){while(k>Q.deg()){size_t n=Q.a.size();auto[Q0,Q1]=Q.bisect();auto[P0,P1]=P.bisect();size_t N=fft::com_size((n+1)/2,(n+1)/2);auto Q0f=fft::dft<T>(Q0.a,N);auto Q1f=fft::dft<T>(Q1.a,N);auto P0f=fft::dft<T>(P0.a,N);auto P1f=fft::dft<T>(P1.a,N);Q=poly_t(Q0f*Q0f)-=poly_t(Q1f*Q1f).mul_xk_inplace(1);if(k%2){P=poly_t(Q0f*=P1f)-=poly_t(Q1f*=P0f);}else{P=poly_t(Q0f*=P0f)-=poly_t(Q1f*=P1f).mul_xk_inplace(1);}k/=2;}return(P*=Q.inv_inplace(Q.deg()+1))[(int)k];}static T kth_rec(poly_t const&P,poly_t const&Q,int64_t k){return kth_rec_inplace(poly_t(P),poly_t(Q),k);}poly_t&inv_inplace(size_t n){return poly::impl::inv_inplace(*this,n);}poly_t inv(size_t n)const{return poly_t(*this).inv_inplace(n);}poly_t&inv_inplace(int64_t k,size_t n){return poly::impl::inv_inplace(*this,k,n);}poly_t inv(int64_t k,size_t n)const{return poly_t(*this).inv_inplace(k,n);}static poly_t compose(poly_t A,poly_t B,int n){int q=std::sqrt(n);big_vector<poly_t>Bk(q);auto Bq=B.pow(q,n);Bk[0]=poly_t(T(1));for(int i=1;i<q;i++){Bk[i]=(Bk[i-1]*B).mod_xk(n);}poly_t Bqk(1);poly_t ans;for(int i=0;i<=n/q;i++){poly_t cur;for(int j=0;j<q;j++){cur+=Bk[j]*A[i*q+j];}ans+=(Bqk*cur).mod_xk(n);Bqk=(Bqk*Bq).mod_xk(n);}return ans;}static poly_t compose_large(poly_t A,poly_t B,int n){if(B[0]!=T(0)){return compose_large(A.shift(B[0]),B-B[0],n);}int q=std::sqrt(n);auto[B0,B1]=std::make_pair(B.mod_xk(q),B.div_xk(q));B0=B0.div_xk(1);big_vector<poly_t>pw(A.deg()+1);auto getpow=[&](int k){return pw[k].is_zero()?pw[k]=B0.pow(k,n-k):pw[k];};std::function<poly_t(poly_t const&,int,int)>compose_dac=[&getpow,&compose_dac](poly_t const&f,int m,int N){if(f.deg()<=0){return f;}int k=m/2;auto[f0,f1]=std::make_pair(f.mod_xk(k),f.div_xk(k));auto[A,B]=std::make_pair(compose_dac(f0,k,N),compose_dac(f1,m-k,N-k));return(A+(B.mod_xk(N-k)*getpow(k).mod_xk(N-k)).mul_xk(k)).mod_xk(N);};int r=n/q;auto Ar=A.deriv(r);auto AB0=compose_dac(Ar,Ar.deg()+1,n);auto Bd=B0.mul_xk(1).deriv();poly_t ans=T(0);big_vector<poly_t>B1p(r+1);B1p[0]=poly_t(T(1));for(int i=1;i<=r;i++){B1p[i]=(B1p[i-1]*B1.mod_xk(n-i*q)).mod_xk(n-i*q);}while(r>=0){ans+=(AB0.mod_xk(n-r*q)*rfact<T>(r)*B1p[r]).mul_xk(r*q).mod_xk(n);r--;if(r>=0){AB0=((AB0*Bd).integr()+A[r]*fact<T>(r)).mod_xk(n);}}return ans;}};template<typename base>static auto operator*(const auto&a,const poly_t<base>&b){return b*a;}};
#pragma GCC pop_options
#endif

#line 1 "cp-algo/math/poly.hpp"
#line 1 "cp-algo/math/poly/impl/euclid.hpp"
#line 1 "cp-algo/math/affine.hpp"
#include <optional>
#include <utility>
#include <cassert>
#include <tuple>
namespace cp_algo::math{template<typename base>struct lin{base a=1,b=0;std::optional<base>c;lin(){}lin(base b):a(0),b(b){}lin(base a,base b):a(a),b(b){}lin(base a,base b,base _c):a(a),b(b),c(_c){}lin operator*(const lin&t){assert(c&&t.c&&*c==*t.c);return{a*t.b+b*t.a,b*t.b+a*t.a*(*c),*c};}lin apply(lin const&t)const{return{a*t.a,a*t.b+b};}void prepend(lin const&t){*this=t.apply(*this);}base eval(base x)const{return a*x+b;}};template<typename base>struct linfrac{base a,b,c,d;linfrac():a(1),b(0),c(0),d(1){}linfrac(base a):a(a),b(1),c(1),d(0){}linfrac(base a,base b,base c,base d):a(a),b(b),c(c),d(d){}linfrac operator*(linfrac t)const{return t.prepend(linfrac(*this));}linfrac operator-()const{return{-a,-b,-c,-d};}linfrac adj()const{return{d,-b,-c,a};}linfrac&prepend(linfrac const&t){t.apply(a,c);t.apply(b,d);return*this;}void apply(base&A,base&B)const{std::tie(A,B)=std::pair{a*A+b*B,c*A+d*B};}};}
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#line 6 "cp-algo/math/common.hpp"
#include <bit>
namespace cp_algo::math{
#ifdef CP_ALGO_MAXN
const int maxn=CP_ALGO_MAXN;
#else
const int maxn=1<<19;
#endif
const int magic=64;auto bpow(auto const&x,auto n,auto const&one,auto op){if(n==0){return one;}auto ans=x;for(int j=std::bit_width<uint64_t>(n)-2;~j;j--){ans=op(ans,ans);if((n>>j)&1){ans=op(ans,x);}}return ans;}auto bpow(auto x,auto n,auto ans){return bpow(x,n,ans,std::multiplies{});}template<typename T>T bpow(T const&x,auto n){return bpow(x,n,T(1));}inline constexpr auto inv2(auto x){assert(x%2);std::make_unsigned_t<decltype(x)>y=1;while(y*x!=1){y*=2-x*y;}return y;}}
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math{template<typename modint,typename _Int>struct modint_base{using Int=_Int;using UInt=std::make_unsigned_t<Int>;static constexpr size_t bits=sizeof(Int)*8;using Int2=std::conditional_t<bits<=32,int64_t,__int128_t>;using UInt2=std::conditional_t<bits<=32,uint64_t,__uint128_t>;constexpr static Int mod(){return modint::mod();}constexpr static Int remod(){return modint::remod();}constexpr static UInt2 modmod(){return UInt2(mod())*mod();}constexpr modint_base()=default;constexpr modint_base(Int2 rr){to_modint().setr(UInt((rr+modmod())%mod()));}modint inv()const{return bpow(to_modint(),mod()-2);}modint operator-()const{modint neg;neg.r=std::min(-r,remod()-r);return neg;}modint&operator/=(const modint&t){return to_modint()*=t.inv();}modint&operator*=(const modint&t){r=UInt(UInt2(r)*t.r%mod());return to_modint();}modint&operator+=(const modint&t){r+=t.r;r=std::min(r,r-remod());return to_modint();}modint&operator-=(const modint&t){r-=t.r;r=std::min(r,r+remod());return to_modint();}modint operator+(const modint&t)const{return modint(to_modint())+=t;}modint operator-(const modint&t)const{return modint(to_modint())-=t;}modint operator*(const modint&t)const{return modint(to_modint())*=t;}modint operator/(const modint&t)const{return modint(to_modint())/=t;}auto operator==(const modint&t)const{return to_modint().getr()==t.getr();}auto operator!=(const modint&t)const{return to_modint().getr()!=t.getr();}auto operator<=(const modint&t)const{return to_modint().getr()<=t.getr();}auto operator>=(const modint&t)const{return to_modint().getr()>=t.getr();}auto operator<(const modint&t)const{return to_modint().getr()<t.getr();}auto operator>(const modint&t)const{return to_modint().getr()>t.getr();}Int rem()const{UInt R=to_modint().getr();return R-(R>(UInt)mod()/2)*mod();}constexpr void setr(UInt rr){r=rr;}constexpr UInt getr()const{return r;}static uint64_t modmod8(){return uint64_t(8*modmod());}void add_unsafe(UInt t){r+=t;}void pseudonormalize(){r=std::min(r,r-modmod8());}modint const&normalize(){if(r>=(UInt)mod()){r%=mod();}return to_modint();}void setr_direct(UInt rr){r=rr;}UInt getr_direct()const{return r;}protected:UInt r;private:constexpr modint&to_modint(){return static_cast<modint&>(*this);}constexpr modint const&to_modint()const{return static_cast<modint const&>(*this);}};template<typename modint>concept modint_type=std::is_base_of_v<modint_base<modint,typename modint::Int>,modint>;template<modint_type modint>decltype(std::cin)&operator>>(decltype(std::cin)&in,modint&x){typename modint::UInt r;auto&res=in>>r;x.setr(r);return res;}template<modint_type modint>decltype(std::cout)&operator<<(decltype(std::cout)&out,modint const&x){return out<<x.getr();}template<auto m>struct modint:modint_base<modint<m>,decltype(m)>{using Base=modint_base<modint<m>,decltype(m)>;using Base::Base;static constexpr Base::Int mod(){return m;}static constexpr Base::UInt remod(){return m;}auto getr()const{return Base::r;}};template<typename Int=int>struct dynamic_modint:modint_base<dynamic_modint<Int>,Int>{using Base=modint_base<dynamic_modint<Int>,Int>;using Base::Base;static Base::UInt m_reduce(Base::UInt2 ab){if(mod()%2==0)[[unlikely]]{return typename Base::UInt(ab%mod());}else{typename Base::UInt2 m=typename Base::UInt(ab)*imod();return typename Base::UInt((ab+m*mod())>>Base::bits);}}static Base::UInt m_transform(Base::UInt a){if(mod()%2==0)[[unlikely]]{return a;}else{return m_reduce(a*pw128());}}dynamic_modint&operator*=(const dynamic_modint&t){Base::r=m_reduce(typename Base::UInt2(Base::r)*t.r);return*this;}void setr(Base::UInt rr){Base::r=m_transform(rr);}Base::UInt getr()const{typename Base::UInt res=m_reduce(Base::r);return std::min(res,res-mod());}static Int mod(){return m;}static Int remod(){return 2*m;}static Base::UInt imod(){return im;}static Base::UInt2 pw128(){return r2;}static void switch_mod(Int nm){m=nm;im=m%2?inv2(-m):0;r2=static_cast<Base::UInt>(static_cast<Base::UInt2>(-1)%m+1);}auto static with_mod(Int tmp,auto callback){struct scoped{Int prev=mod();~scoped(){switch_mod(prev);}}_;switch_mod(tmp);return callback();}private:static thread_local Int m;static thread_local Base::UInt im,r2;};template<typename Int>Int thread_local dynamic_modint<Int>::m=1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im=-1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2=0;}
#line 1 "cp-algo/util/checkpoint.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#  define CP_ALGO_USE_MMAP 1
#  include <sys/mman.h>
#else
#  define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo{template<typename T,size_t Align=32>class big_alloc{static_assert(Align>=alignof(void*),"Align must be at least pointer-size");static_assert(std::popcount(Align)==1,"Align must be a power of two");public:using value_type=T;template<class U>struct rebind{using other=big_alloc<U,Align>;};constexpr bool operator==(const big_alloc&)const=default;constexpr bool operator!=(const big_alloc&)const=default;big_alloc()noexcept=default;template<typename U,std::size_t A>big_alloc(const big_alloc<U,A>&)noexcept{}[[nodiscard]]T*allocate(std::size_t n){std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0);madvise(raw,padded,MADV_HUGEPAGE);madvise(raw,padded,MADV_POPULATE_WRITE);return static_cast<T*>(raw);}
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align)));}void deallocate(T*p,std::size_t n)noexcept{if(!p)return;std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){munmap(p,padded);return;}
#endif
::operator delete(p,padded,std::align_val_t(align));}private:static constexpr std::size_t MEGABYTE=1<<20;static constexpr std::size_t round_up(std::size_t x)noexcept{return(x+Align-1)/Align*Align;}};template<typename T>using big_vector=std::vector<T,big_alloc<T>>;template<typename T>using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>;template<typename T>using big_deque=std::deque<T,big_alloc<T>>;template<typename T>using big_stack=std::stack<T,big_deque<T>>;template<typename T>using big_queue=std::queue<T,big_deque<T>>;template<typename T>using big_priority_queue=std::priority_queue<T,big_vector<T>>;template<typename T>using big_forward_list=std::forward_list<T,big_alloc<T>>;using big_string=big_basic_string<char>;template<typename Key,typename Value,typename Compare=std::less<Key>>using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>;template<typename T,typename Compare=std::less<T>>using big_multiset=std::multiset<T,Compare,big_alloc<T>>;template<typename T,typename Compare=std::less<T>>using big_set=std::set<T,Compare,big_alloc<T>>;template<typename Ref,typename V=void>using big_generator=std::generator<Ref,V,big_alloc<std::byte>>;}namespace std::ranges{template<typename Ref,typename V>elements_of(cp_algo::big_generator<Ref,V>&&)->elements_of<cp_algo::big_generator<Ref,V>&&,cp_algo::big_alloc<std::byte>>;}
#line 5 "cp-algo/util/checkpoint.hpp"
#include <chrono>
#line 8 "cp-algo/util/checkpoint.hpp"
namespace cp_algo{
#ifdef CP_ALGO_CHECKPOINT
big_map<big_string,double>checkpoints;double last;
#endif
template<bool final=false>void checkpoint([[maybe_unused]]auto const&_msg){
#ifdef CP_ALGO_CHECKPOINT
big_string msg=_msg;double now=(double)clock()/CLOCKS_PER_SEC;double delta=now-last;last=now;if(msg.size()&&!final){checkpoints[msg]+=delta;}if(final){for(auto const&[key,value]:checkpoints){std::cerr<<key<<": "<<value*1000<<" ms\n";}std::cerr<<"Total: "<<now*1000<<" ms\n";}
#endif
}template<bool final=false>void checkpoint(){checkpoint<final>("");}}
#line 1 "cp-algo/random/rng.hpp"
#line 4 "cp-algo/random/rng.hpp"
#include <random>
namespace cp_algo::random{std::mt19937_64 gen(std::chrono::steady_clock::now().time_since_epoch().count());uint64_t rng(){return gen();}}
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH  _Pragma("GCC push_options")  CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo{template<typename T,size_t len>using simd[[gnu::vector_size(len*sizeof(T))]]=T;using u64x8=simd<uint64_t,8>;using u32x16=simd<uint32_t,16>;using i64x4=simd<int64_t,4>;using u64x4=simd<uint64_t,4>;using u32x8=simd<uint32_t,8>;using u16x16=simd<uint16_t,16>;using i32x4=simd<int32_t,4>;using u32x4=simd<uint32_t,4>;using u16x8=simd<uint16_t,8>;using u16x4=simd<uint16_t,4>;using i16x4=simd<int16_t,4>;using u8x32=simd<uint8_t,32>;using u8x8=simd<uint8_t,8>;using u8x4=simd<uint8_t,4>;using dx4=simd<double,4>;inline dx4 abs(dx4 a){return dx4{std::abs(a[0]),std::abs(a[1]),std::abs(a[2]),std::abs(a[3])};}static constexpr dx4 magic=dx4()+(3ULL<<51);inline i64x4 lround(dx4 x){return i64x4(x+magic)-i64x4(magic);}inline dx4 to_double(i64x4 x){return dx4(x+i64x4(magic))-magic;}inline dx4 round(dx4 a){return dx4{std::nearbyint(a[0]),std::nearbyint(a[1]),std::nearbyint(a[2]),std::nearbyint(a[3])};}inline u64x4 low32(u64x4 x){return x&uint32_t(-1);}inline auto swap_bytes(auto x){return decltype(x)(__builtin_shufflevector(u32x8(x),u32x8(x),1,0,3,2,5,4,7,6));}inline u64x4 montgomery_reduce(u64x4 x,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
auto x_ninv=u64x4(_mm256_mul_epu32(__m256i(x),__m256i()+imod));x+=u64x4(_mm256_mul_epu32(__m256i(x_ninv),__m256i()+mod));
#else
auto x_ninv=u64x4(u32x8(low32(x))*imod);x+=x_ninv*uint64_t(mod);
#endif
return swap_bytes(x);}inline u64x4 montgomery_mul(u64x4 x,u64x4 y,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x),__m256i(y))),mod,imod);
#else
return montgomery_reduce(x*y,mod,imod);
#endif
}inline u32x8 montgomery_mul(u32x8 x,u32x8 y,uint32_t mod,uint32_t imod){return u32x8(montgomery_mul(u64x4(x),u64x4(y),mod,imod))|u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)),u64x4(swap_bytes(y)),mod,imod)));}inline dx4 rotate_right(dx4 x){static constexpr u64x4 shuffler={3,0,1,2};return __builtin_shuffle(x,shuffler);}template<std::size_t Align=32>inline bool is_aligned(const auto*p)noexcept{return(reinterpret_cast<std::uintptr_t>(p)%Align)==0;}template<class Target>inline Target&vector_cast(auto&&p){return*reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));}}
#pragma GCC pop_options
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo{template<typename T>struct complex{using value_type=T;T x,y;inline constexpr complex():x(),y(){}inline constexpr complex(T const&x):x(x),y(){}inline constexpr complex(T const&x,T const&y):x(x),y(y){}inline complex&operator*=(T const&t){x*=t;y*=t;return*this;}inline complex&operator/=(T const&t){x/=t;y/=t;return*this;}inline complex operator*(T const&t)const{return complex(*this)*=t;}inline complex operator/(T const&t)const{return complex(*this)/=t;}inline complex&operator+=(complex const&t){x+=t.x;y+=t.y;return*this;}inline complex&operator-=(complex const&t){x-=t.x;y-=t.y;return*this;}inline complex operator*(complex const&t)const{return{x*t.x-y*t.y,x*t.y+y*t.x};}inline complex operator/(complex const&t)const{return*this*t.conj()/t.norm();}inline complex operator+(complex const&t)const{return complex(*this)+=t;}inline complex operator-(complex const&t)const{return complex(*this)-=t;}inline complex&operator*=(complex const&t){return*this=*this*t;}inline complex&operator/=(complex const&t){return*this=*this/t;}inline complex operator-()const{return{-x,-y};}inline complex conj()const{return{x,-y};}inline T norm()const{return x*x+y*y;}inline T abs()const{return std::sqrt(norm());}inline T const real()const{return x;}inline T const imag()const{return y;}inline T&real(){return x;}inline T&imag(){return y;}inline static constexpr complex polar(T r,T theta){return{T(r*cos(theta)),T(r*sin(theta))};}inline auto operator<=>(complex const&t)const=default;};template<typename T>inline complex<T>conj(complex<T>const&x){return x.conj();}template<typename T>inline T norm(complex<T>const&x){return x.norm();}template<typename T>inline T abs(complex<T>const&x){return x.abs();}template<typename T>inline T&real(complex<T>&x){return x.real();}template<typename T>inline T&imag(complex<T>&x){return x.imag();}template<typename T>inline T const real(complex<T>const&x){return x.real();}template<typename T>inline T const imag(complex<T>const&x){return x.imag();}template<typename T>inline constexpr complex<T>polar(T r,T theta){return complex<T>::polar(r,theta);}template<typename T>inline std::ostream&operator<<(std::ostream&out,complex<T>const&x){return out<<x.real()<<' '<<x.imag();}}
#pragma GCC pop_options
#line 7 "cp-algo/math/cvector.hpp"
#include <ranges>
#line 9 "cp-algo/math/cvector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace stdx=std::experimental;namespace cp_algo::math::fft{static constexpr size_t flen=4;using ftype=double;using vftype=dx4;using point=complex<ftype>;using vpoint=complex<vftype>;static constexpr vftype vz={};vpoint vi(vpoint const&r){return{-imag(r),real(r)};}struct cvector{big_vector<vpoint>r;cvector(size_t n){n=std::max(flen,std::bit_ceil(n));r.resize(n/flen);checkpoint("cvector create");}vpoint&at(size_t k){return r[k/flen];}vpoint at(size_t k)const{return r[k/flen];}template<class pt=point>inline void set(size_t k,pt const&t){if constexpr(std::is_same_v<pt,point>){real(r[k/flen])[k%flen]=real(t);imag(r[k/flen])[k%flen]=imag(t);}else{at(k)=t;}}template<class pt=point>inline pt get(size_t k)const{if constexpr(std::is_same_v<pt,point>){return{real(r[k/flen])[k%flen],imag(r[k/flen])[k%flen]};}else{return at(k);}}size_t size()const{return flen*r.size();}static constexpr size_t eval_arg(size_t n){if(n<pre_evals){return eval_args[n];}else{return eval_arg(n/2)|(n&1)<<(std::bit_width(n)-1);}}static constexpr point eval_point(size_t n){if(n%2){return-eval_point(n-1);}else if(n%4){return eval_point(n-2)*point(0,1);}else if(n/4<pre_evals){return evalp[n/4];}else{return polar<ftype>(1.,std::numbers::pi/(ftype)std::bit_floor(n)*(ftype)eval_arg(n));}}static constexpr std::array<point,32>roots=[](){std::array<point,32>res;for(size_t i=2;i<32;i++){res[i]=polar<ftype>(1.,std::numbers::pi/(1ull<<(i-2)));}return res;}();static constexpr point root(size_t n){return roots[std::bit_width(n)];}template<int step>static void exec_on_eval(size_t n,size_t k,auto&&callback){callback(k,root(4*step*n)*eval_point(step*k));}template<int step>static void exec_on_evals(size_t n,auto&&callback){point factor=root(4*step*n);for(size_t i=0;i<n;i++){callback(i,factor*eval_point(step*i));}}static void do_dot_iter(point rt,vpoint&Bv,vpoint const&Av,vpoint&res){res+=Av*Bv;real(Bv)=rotate_right(real(Bv));imag(Bv)=rotate_right(imag(Bv));auto x=real(Bv)[0],y=imag(Bv)[0];real(Bv)[0]=x*real(rt)-y*imag(rt);imag(Bv)[0]=x*imag(rt)+y*real(rt);}void dot(cvector const&t){size_t n=this->size();exec_on_evals<1>(n/flen,[&](size_t k,point rt)__attribute__((always_inline)){k*=flen;auto[Ax,Ay]=at(k);auto Bv=t.at(k);vpoint res=vz;for(size_t i=0;i<flen;i++){vpoint Av=vpoint(vz+Ax[i],vz+Ay[i]);do_dot_iter(rt,Bv,Av,res);}set(k,res);});checkpoint("dot");}template<bool partial=true>void ifft(){size_t n=size();if constexpr(!partial){point pi(0,1);exec_on_evals<4>(n/4,[&](size_t k,point rt)__attribute__((always_inline)){k*=4;point v1=conj(rt);point v2=v1*v1;point v3=v1*v2;auto A=get(k);auto B=get(k+1);auto C=get(k+2);auto D=get(k+3);set(k,(A+B)+(C+D));set(k+2,((A+B)-(C+D))*v2);set(k+1,((A-B)-pi*(C-D))*v1);set(k+3,((A-B)+pi*(C-D))*v3);});}bool parity=std::countr_zero(n)%2;if(parity){exec_on_evals<2>(n/(2*flen),[&](size_t k,point rt)__attribute__((always_inline)){k*=2*flen;vpoint cvrt={vz+real(rt),vz-imag(rt)};auto B=at(k)-at(k+flen);at(k)+=at(k+flen);at(k+flen)=B*cvrt;});}for(size_t leaf=3*flen;leaf<n;leaf+=4*flen){size_t level=std::countr_one(leaf+3);for(size_t lvl=4+parity;lvl<=level;lvl+=2){size_t i=(1<<lvl)/4;exec_on_eval<4>(n>>lvl,leaf>>lvl,[&](size_t k,point rt)__attribute__((always_inline)){k<<=lvl;vpoint v1={vz+real(rt),vz-imag(rt)};vpoint v2=v1*v1;vpoint v3=v1*v2;for(size_t j=k;j<k+i;j+=flen){auto A=at(j);auto B=at(j+i);auto C=at(j+2*i);auto D=at(j+3*i);at(j)=((A+B)+(C+D));at(j+2*i)=((A+B)-(C+D))*v2;at(j+i)=((A-B)-vi(C-D))*v1;at(j+3*i)=((A-B)+vi(C-D))*v3;}});}}checkpoint("ifft");for(size_t k=0;k<n;k+=flen){if constexpr(partial){set(k,get<vpoint>(k)/=vz+ftype(n/flen));}else{set(k,get<vpoint>(k)/=vz+ftype(n));}}}template<bool partial=true>void fft(){size_t n=size();bool parity=std::countr_zero(n)%2;for(size_t leaf=0;leaf<n;leaf+=4*flen){size_t level=std::countr_zero(n+leaf);level-=level%2!=parity;for(size_t lvl=level;lvl>=4;lvl-=2){size_t i=(1<<lvl)/4;exec_on_eval<4>(n>>lvl,leaf>>lvl,[&](size_t k,point rt)__attribute__((always_inline)){k<<=lvl;vpoint v1={vz+real(rt),vz+imag(rt)};vpoint v2=v1*v1;vpoint v3=v1*v2;for(size_t j=k;j<k+i;j+=flen){auto A=at(j);auto B=at(j+i)*v1;auto C=at(j+2*i)*v2;auto D=at(j+3*i)*v3;at(j)=(A+C)+(B+D);at(j+i)=(A+C)-(B+D);at(j+2*i)=(A-C)+vi(B-D);at(j+3*i)=(A-C)-vi(B-D);}});}}if(parity){exec_on_evals<2>(n/(2*flen),[&](size_t k,point rt)__attribute__((always_inline)){k*=2*flen;vpoint vrt={vz+real(rt),vz+imag(rt)};auto t=at(k+flen)*vrt;at(k+flen)=at(k)-t;at(k)+=t;});}if constexpr(!partial){point pi(0,1);exec_on_evals<4>(n/4,[&](size_t k,point rt)__attribute__((always_inline)){k*=4;point v1=rt;point v2=v1*v1;point v3=v1*v2;auto A=get(k);auto B=get(k+1)*v1;auto C=get(k+2)*v2;auto D=get(k+3)*v3;set(k,(A+C)+(B+D));set(k+1,(A+C)-(B+D));set(k+2,(A-C)+pi*(B-D));set(k+3,(A-C)-pi*(B-D));});}checkpoint("fft");}static constexpr size_t pre_evals=1<<16;static const std::array<size_t,pre_evals>eval_args;static const std::array<point,pre_evals>evalp;};const std::array<size_t,cvector::pre_evals>cvector::eval_args=[](){std::array<size_t,pre_evals>res={};for(size_t i=1;i<pre_evals;i++){res[i]=res[i>>1]|(i&1)<<(std::bit_width(i)-1);}return res;}();const std::array<point,cvector::pre_evals>cvector::evalp=[](){std::array<point,pre_evals>res={};res[0]=1;for(size_t n=1;n<pre_evals;n++){res[n]=polar<ftype>(1.,std::numbers::pi*ftype(eval_args[n])/ftype(4*std::bit_floor(n)));}return res;}();}
#pragma GCC pop_options
#line 9 "cp-algo/math/fft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft{template<modint_type base>struct dft{cvector A,B;static base factor,ifactor;using Int2=base::Int2;static bool _init;static int split(){static const int splt=int(std::sqrt(base::mod()))+1;return splt;}static uint32_t mod,imod;static void init(){if(!_init){factor=1+random::rng()%(base::mod()-1);ifactor=base(1)/factor;mod=base::mod();imod=-inv2<uint32_t>(base::mod());_init=true;}}static std::pair<vftype,vftype>do_split(auto const&a,size_t idx,u64x4 mul){if(idx>=std::size(a)){return std::pair{vftype(),vftype()};}u64x4 au={idx<std::size(a)?a[idx].getr():0,idx+1<std::size(a)?a[idx+1].getr():0,idx+2<std::size(a)?a[idx+2].getr():0,idx+3<std::size(a)?a[idx+3].getr():0};au=montgomery_mul(au,mul,mod,imod);au=au>=base::mod()?au-base::mod():au;auto ai=to_double(i64x4(au>=base::mod()/2?au-base::mod():au));auto quo=round(ai/split());return std::pair{ai-quo*split(),quo};}dft(size_t n):A(n),B(n){init();}dft(auto const&a,size_t n,bool partial=true):A(n),B(n){init();base b2x32=bpow(base(2),32);u64x4 cur={(bpow(factor,1)*b2x32).getr(),(bpow(factor,2)*b2x32).getr(),(bpow(factor,3)*b2x32).getr(),(bpow(factor,4)*b2x32).getr()};u64x4 step4=u64x4{}+(bpow(factor,4)*b2x32).getr();u64x4 stepn=u64x4{}+(bpow(factor,n)*b2x32).getr();for(size_t i=0;i<std::min(n,std::size(a));i+=flen){auto[rai,qai]=do_split(a,i,cur);auto[rani,qani]=do_split(a,n+i,montgomery_mul(cur,stepn,mod,imod));A.at(i)=vpoint(rai,rani);B.at(i)=vpoint(qai,qani);cur=montgomery_mul(cur,step4,mod,imod);}checkpoint("dft init");if(n){if(partial){A.fft();B.fft();}else{A.template fft<false>();B.template fft<false>();}}}static void do_dot_iter(point rt,vpoint&Cv,vpoint&Dv,vpoint const&Av,vpoint const&Bv,vpoint&AC,vpoint&AD,vpoint&BC,vpoint&BD){AC+=Av*Cv;AD+=Av*Dv;BC+=Bv*Cv;BD+=Bv*Dv;real(Cv)=rotate_right(real(Cv));imag(Cv)=rotate_right(imag(Cv));real(Dv)=rotate_right(real(Dv));imag(Dv)=rotate_right(imag(Dv));auto cx=real(Cv)[0],cy=imag(Cv)[0];auto dx=real(Dv)[0],dy=imag(Dv)[0];real(Cv)[0]=cx*real(rt)-cy*imag(rt);imag(Cv)[0]=cx*imag(rt)+cy*real(rt);real(Dv)[0]=dx*real(rt)-dy*imag(rt);imag(Dv)[0]=dx*imag(rt)+dy*real(rt);}template<bool overwrite=true,bool partial=true>void dot(auto const&C,auto const&D,auto&Aout,auto&Bout,auto&Cout)const{cvector::exec_on_evals<1>(A.size()/flen,[&](size_t k,point rt)__attribute__((always_inline)){k*=flen;vpoint AC,AD,BC,BD;AC=AD=BC=BD=vz;auto Cv=C.at(k),Dv=D.at(k);if constexpr(partial){auto[Ax,Ay]=A.at(k);auto[Bx,By]=B.at(k);for(size_t i=0;i<flen;i++){vpoint Av={vz+Ax[i],vz+Ay[i]},Bv={vz+Bx[i],vz+By[i]};do_dot_iter(rt,Cv,Dv,Av,Bv,AC,AD,BC,BD);}}else{AC=A.at(k)*Cv;AD=A.at(k)*Dv;BC=B.at(k)*Cv;BD=B.at(k)*Dv;}if constexpr(overwrite){Aout.at(k)=AC;Cout.at(k)=AD+BC;Bout.at(k)=BD;}else{Aout.at(k)+=AC;Cout.at(k)+=AD+BC;Bout.at(k)+=BD;}});checkpoint("dot");}void dot(auto&&C,auto const&D){dot(C,D,A,B,C);}static void do_recover_iter(size_t idx,auto A,auto B,auto C,auto mul,uint64_t splitsplit,auto&res){auto A0=lround(A),A1=lround(C),A2=lround(B);auto Ai=A0+A1*split()+A2*splitsplit+uint64_t(base::modmod());auto Au=montgomery_reduce(u64x4(Ai),mod,imod);Au=montgomery_mul(Au,mul,mod,imod);Au=Au>=base::mod()?Au-base::mod():Au;for(size_t j=0;j<flen;j++){res[idx+j].setr(typename base::UInt(Au[j]));}}void recover_mod(auto&&C,auto&res,size_t k){size_t check=(k+flen-1)/flen*flen;assert(res.size()>=check);size_t n=A.size();auto const splitsplit=base(split()*split()).getr();base b2x32=bpow(base(2),32);base b2x64=bpow(base(2),64);u64x4 cur={(bpow(ifactor,2)*b2x64).getr(),(bpow(ifactor,3)*b2x64).getr(),(bpow(ifactor,4)*b2x64).getr(),(bpow(ifactor,5)*b2x64).getr()};u64x4 step4=u64x4{}+(bpow(ifactor,4)*b2x32).getr();u64x4 stepn=u64x4{}+(bpow(ifactor,n)*b2x32).getr();for(size_t i=0;i<std::min(n,k);i+=flen){auto[Ax,Ay]=A.at(i);auto[Bx,By]=B.at(i);auto[Cx,Cy]=C.at(i);do_recover_iter(i,Ax,Bx,Cx,cur,splitsplit,res);if(i+n<k){do_recover_iter(i+n,Ay,By,Cy,montgomery_mul(cur,stepn,mod,imod),splitsplit,res);}cur=montgomery_mul(cur,step4,mod,imod);}checkpoint("recover mod");}void mul(auto&&C,auto const&D,auto&res,size_t k){assert(A.size()==C.size());size_t n=A.size();if(!n){res={};return;}dot(C,D);A.ifft();B.ifft();C.ifft();recover_mod(C,res,k);}void mul_inplace(auto&&B,auto&res,size_t k){mul(B.A,B.B,res,k);}void mul(auto const&B,auto&res,size_t k){mul(cvector(B.A),B.B,res,k);}big_vector<base>operator*=(dft&B){big_vector<base>res(2*A.size());mul_inplace(B,res,2*A.size());return res;}big_vector<base>operator*=(dft const&B){big_vector<base>res(2*A.size());mul(B,res,2*A.size());return res;}auto operator*(dft const&B)const{return dft(*this)*=B;}point operator[](int i)const{return A.get(i);}};template<modint_type base>base dft<base>::factor=1;template<modint_type base>base dft<base>::ifactor=1;template<modint_type base>bool dft<base>::_init=false;template<modint_type base>uint32_t dft<base>::mod={};template<modint_type base>uint32_t dft<base>::imod={};void mul_slow(auto&a,auto const&b,size_t k){if(std::empty(a)||std::empty(b)){a.clear();}else{size_t n=std::min(k,std::size(a));size_t m=std::min(k,std::size(b));a.resize(k);for(int j=int(k-1);j>=0;j--){a[j]*=b[0];for(int i=std::max(j-(int)n,0)+1;i<std::min(j+1,(int)m);i++){a[j]+=a[j-i]*b[i];}}}}size_t com_size(size_t as,size_t bs){if(!as||!bs){return 0;}return std::max(flen,std::bit_ceil(as+bs-1)/2);}void mul_truncate(auto&a,auto const&b,size_t k){using base=std::decay_t<decltype(a[0])>;if(std::min({k,std::size(a),std::size(b)})<magic){mul_slow(a,b,k);return;}auto n=std::max(flen,std::bit_ceil(std::min(k,std::size(a))+std::min(k,std::size(b))-1)/2);auto A=dft<base>(a|std::views::take(k),n);auto B=dft<base>(b|std::views::take(k),n);a.resize((k+flen-1)/flen*flen);A.mul_inplace(B,a,k);a.resize(k);}void mod_split(auto&&x,size_t n,auto k){using base=std::decay_t<decltype(k)>;dft<base>::init();assert(std::size(x)==2*n);u64x4 cur=u64x4{}+(k*bpow(base(2),32)).getr();for(size_t i=0;i<n;i+=flen){u64x4 xl={x[i].getr(),x[i+1].getr(),x[i+2].getr(),x[i+3].getr()};u64x4 xr={x[n+i].getr(),x[n+i+1].getr(),x[n+i+2].getr(),x[n+i+3].getr()};xr=montgomery_mul(xr,cur,dft<base>::mod,dft<base>::imod);xr=xr>=base::mod()?xr-base::mod():xr;auto t=xr;xr=xl-t;xl+=t;xl=xl>=base::mod()?xl-base::mod():xl;xr=xr>=base::mod()?xr+base::mod():xr;for(size_t k=0;k<flen;k++){x[i+k].setr(typename base::UInt(xl[k]));x[n+i+k].setr(typename base::UInt(xr[k]));}}cp_algo::checkpoint("mod split");}void cyclic_mul(auto&a,auto&&b,size_t k){assert(std::popcount(k)==1);assert(std::size(a)==std::size(b)&&std::size(a)==k);using base=std::decay_t<decltype(a[0])>;dft<base>::init();if(k<=(1<<16)){big_vector<base>ap(begin(a),end(a));mul_truncate(ap,b,2*k);mod_split(ap,k,bpow(dft<base>::factor,k));std::ranges::copy(ap|std::views::take(k),begin(a));return;}k/=2;auto factor=bpow(dft<base>::factor,k);mod_split(a,k,factor);mod_split(b,k,factor);auto la=std::span(a).first(k);auto lb=std::span(b).first(k);auto ra=std::span(a).last(k);auto rb=std::span(b).last(k);cyclic_mul(la,lb,k);auto A=dft<base>(ra,k/2);auto B=dft<base>(rb,k/2);A.mul_inplace(B,ra,k);base i2=base(2).inv();factor=factor.inv()*i2;for(size_t i=0;i<k;i++){auto t=(a[i]+a[i+k])*i2;a[i+k]=(a[i]-a[i+k])*factor;a[i]=t;}cp_algo::checkpoint("mod join");}auto make_copy(auto&&x){return x;}void cyclic_mul(auto&a,auto const&b,size_t k){return cyclic_mul(a,make_copy(b),k);}void mul(auto&a,auto&&b){size_t N=size(a)+size(b);if(N>(1<<20)){N--;size_t NN=std::bit_ceil(N);a.resize(NN);b.resize(NN);cyclic_mul(a,b,NN);a.resize(N);}else{mul_truncate(a,b,N-1);}}void mul(auto&a,auto const&b){size_t N=size(a)+size(b);if(N>(1<<20)){mul(a,make_copy(b));}else{mul_truncate(a,b,N-1);}}}
#pragma GCC pop_options
#line 6 "cp-algo/math/poly/impl/euclid.hpp"
#include <algorithm>
#include <numeric>
#line 11 "cp-algo/math/poly/impl/euclid.hpp"
#include <list>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::poly::impl{template<typename poly>using gcd_result=std::pair<std::list<std::decay_t<poly>>,linfrac<std::decay_t<poly>>>;template<typename poly>gcd_result<poly>half_gcd(poly&&A,poly&&B){assert(A.deg()>=B.deg());size_t m=size(A.a)/2;if(B.deg()<(int)m){return{};}auto[ai,R]=A.divmod(B);std::tie(A,B)={B,R};std::list a={ai};auto T=-linfrac(ai).adj();auto advance=[&](size_t k){auto[ak,Tk]=half_gcd(A.div_xk(k),B.div_xk(k));a.splice(end(a),ak);T.prepend(Tk);return Tk;};advance(m).apply(A,B);if constexpr(std::is_reference_v<poly>){advance(2*m-A.deg()).apply(A,B);}else{advance(2*m-A.deg());}return{std::move(a),std::move(T)};}template<typename poly>gcd_result<poly>full_gcd(poly&&A,poly&&B){using poly_t=std::decay_t<poly>;std::list<poly_t>ak;big_vector<linfrac<poly_t>>trs;while(!B.is_zero()){auto[a0,R]=A.divmod(B);ak.push_back(a0);trs.push_back(-linfrac(a0).adj());std::tie(A,B)={B,R};auto[a,Tr]=half_gcd(A,B);ak.splice(end(ak),a);trs.push_back(Tr);}return{ak,std::accumulate(rbegin(trs),rend(trs),linfrac<poly_t>{},std::multiplies{})};}auto convergent(auto L,auto R){using poly=decltype(L)::value_type;if(R==next(L)){return linfrac(*L);}else{int s=std::transform_reduce(L,R,0,std::plus{},std::mem_fn(&poly::deg));auto M=L;for(int c=M->deg();2*c<=s;M++){c+=next(M)->deg();}return convergent(L,M)*convergent(M,R);}}template<typename poly>poly min_rec(poly const&p,size_t d){auto R2=p.mod_xk(d).reversed(d),R1=poly::xk(d);if(R2.is_zero()){return poly(1);}auto[a,Tr]=full_gcd(R1,R2);a.emplace_back();auto pref=begin(a);for(int delta=(int)d-a.front().deg();delta>=0;pref++){delta-=pref->deg()+next(pref)->deg();}return convergent(begin(a),pref).a;}template<typename poly>std::optional<poly>inv_mod(poly p,poly q){assert(!q.is_zero());auto[a,Tr]=full_gcd(q,p);if(q.deg()!=0){return std::nullopt;}return Tr.b/q[0];}}
#pragma GCC pop_options
#line 1 "cp-algo/math/poly/impl/div.hpp"
#line 6 "cp-algo/math/poly/impl/div.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::poly::impl{auto divmod_slow(auto const&p,auto const&q){auto R=p;auto D=decltype(p){};auto q_lead_inv=q.lead().inv();while(R.deg()>=q.deg()){D.a.push_back(R.lead()*q_lead_inv);if(D.lead()!=0){for(size_t i=1;i<=q.a.size();i++){R.a[R.a.size()-i]-=D.lead()*q.a[q.a.size()-i];}}R.a.pop_back();}std::ranges::reverse(D.a);R.normalize();return std::array{D,R};}template<typename poly>auto divmod_hint(poly const&p,poly const&q,poly const&qri){assert(!q.is_zero());int d=p.deg()-q.deg();if(std::min(d,q.deg())<magic){return divmod_slow(p,q);}poly D;if(d>=0){D=(p.reversed().mod_xk(d+1)*qri.mod_xk(d+1)).mod_xk(d+1).reversed(d+1);}return std::array{D,p-D*q};}auto divmod(auto const&p,auto const&q){assert(!q.is_zero());int d=p.deg()-q.deg();if(std::min(d,q.deg())<magic){return divmod_slow(p,q);}return divmod_hint(p,q,q.reversed().inv(d+1));}template<typename poly>poly powmod_hint(poly const&p,int64_t k,poly const&md,poly const&mdri){return bpow(p%md,k,poly(1),[&](auto const&p,auto const&q){return divmod_hint(p*q,md,mdri)[1];});}template<typename poly>auto powmod(poly const&p,int64_t k,poly const&md){int d=md.deg();if(p==poly::xk(1)&&false){if(k<md.deg()){return poly::xk(k);}else{auto mdr=md.reversed();return(mdr.inv(k-md.deg()+1,md.deg())*mdr).reversed(md.deg());}}if(md==poly::xk(d)){return p.pow(k,d);}if(md==poly::xk(d)-poly(1)){return p.powmod_circular(k,d);}return powmod_hint(p,k,md,md.reversed().inv(md.deg()+1));}template<typename poly>poly&inv_inplace(poly&q,int64_t k,size_t n){using poly_t=std::decay_t<poly>;using base=poly_t::base;if(k<=std::max<int64_t>(n,size(q.a))){return q.inv_inplace(k+n).div_xk_inplace(k);}if(k%2){return inv_inplace(q,k-1,n+1).div_xk_inplace(1);}auto[q0,q1]=q.bisect();auto qq=q0*q0-(q1*q1).mul_xk_inplace(1);inv_inplace(qq,k/2-q.deg()/2,(n+1)/2+q.deg()/2);size_t N=fft::com_size(size(q0.a),size(qq.a));auto q0f=fft::dft<base>(q0.a,N);auto q1f=fft::dft<base>(q1.a,N);auto qqf=fft::dft<base>(qq.a,N);size_t M=q0.deg()+(n+1)/2;typename poly::Vector A,B;A.resize((M+fft::flen-1)/fft::flen*fft::flen);B.resize((M+fft::flen-1)/fft::flen*fft::flen);q0f.mul(qqf,A,M);q1f.mul_inplace(qqf,B,M);q.a.resize(n+1);for(size_t i=0;i<n;i+=2){q.a[i]=A[q0.deg()+i/2];q.a[i+1]=-B[q0.deg()+i/2];}q.a.pop_back();q.normalize();return q;}template<typename poly>poly&inv_inplace(poly&p,size_t n){using poly_t=std::decay_t<poly>;using base=poly_t::base;if(n==1){return p=base(1)/p[0];}auto[q0,q1]=p.bisect(n);size_t N=fft::com_size((n+1)/2,(n+1)/2);auto q0f=fft::dft<base>(q0.a,N);auto q1f=fft::dft<base>(q1.a,N);auto qq=poly_t(q0f*q0f)-poly_t(q1f*q1f).mul_xk_inplace(1);inv_inplace(qq,(n+1)/2);auto qqf=fft::dft<base>(qq.a,N);typename poly::Vector A,B;A.resize(((n+1)/2+fft::flen-1)/fft::flen*fft::flen);B.resize(((n+1)/2+fft::flen-1)/fft::flen*fft::flen);q0f.mul(qqf,A,(n+1)/2);q1f.mul_inplace(qqf,B,(n+1)/2);p.a.resize(n+1);for(size_t i=0;i<n;i+=2){p.a[i]=A[i/2];p.a[i+1]=-B[i/2];}p.a.pop_back();p.normalize();return p;}}
#pragma GCC pop_options
#line 1 "cp-algo/math/combinatorics.hpp"
#line 7 "cp-algo/math/combinatorics.hpp"
namespace cp_algo::math{template<typename T>T fact(auto n){static big_vector<T>F(maxn);static bool init=false;if(!init){F[0]=T(1);for(int i=1;i<maxn;i++){F[i]=F[i-1]*T(i);}init=true;}return F[n];}template<typename T>T rfact(auto n){static big_vector<T>F(maxn);static bool init=false;if(!init){int t=(int)std::min<int64_t>(T::mod(),maxn)-1;F[t]=T(1)/fact<T>(t);for(int i=t-1;i>=0;i--){F[i]=F[i+1]*T(i+1);}init=true;}return F[n];}template<typename T,int base>T pow_fixed(int n){static big_vector<T>prec_low(1<<16);static big_vector<T>prec_high(1<<16);static bool init=false;if(!init){init=true;prec_low[0]=prec_high[0]=T(1);T step_low=T(base);T step_high=bpow(T(base),1<<16);for(int i=1;i<(1<<16);i++){prec_low[i]=prec_low[i-1]*step_low;prec_high[i]=prec_high[i-1]*step_high;}}return prec_low[n&0xFFFF]*prec_high[n>>16];}template<typename T>big_vector<T>bulk_invs(auto const&args){big_vector<T>res(std::size(args),args[0]);for(size_t i=1;i<std::size(args);i++){res[i]=res[i-1]*args[i];}auto all_invs=T(1)/res.back();for(size_t i=std::size(args)-1;i>0;i--){res[i]=all_invs*res[i-1];all_invs*=args[i];}res[0]=all_invs;return res;}template<typename T>T small_inv(auto n){static auto F=bulk_invs<T>(std::views::iota(1,maxn));return F[n-1];}template<typename T>T binom_large(T n,auto r){assert(r<maxn);T ans=1;for(decltype(r)i=0;i<r;i++){ans=ans*T(n-i)*small_inv<T>(i+1);}return ans;}template<typename T>T binom(auto n,auto r){if(r<0||r>n){return T(0);}else if(n>=maxn){return binom_large(T(n),r);}else{return fact<T>(n)*rfact<T>(r)*rfact<T>(n-r);}}}
#line 1 "cp-algo/number_theory/discrete_sqrt.hpp"
#line 6 "cp-algo/number_theory/discrete_sqrt.hpp"
namespace cp_algo::math{template<modint_type base>std::optional<base>sqrt(base b){if(b==base(0)){return base(0);}else if(bpow(b,(b.mod()-1)/2)!=base(1)){return std::nullopt;}else{while(true){base z=random::rng();if(z*z==b){return z;}lin<base>x(1,z,b);x=bpow(x,(b.mod()-1)/2,lin<base>(0,1,b));if(x.a!=base(0)){return x.a.inv();}}}}}
#line 15 "cp-algo/math/poly.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<typename T>struct poly_t{using Vector=big_vector<T>;using base=T;Vector a;poly_t&normalize(){while(deg()>=0&&lead()==base(0)){a.pop_back();}return*this;}poly_t(){}poly_t(T a0):a{a0}{normalize();}poly_t(Vector const&t):a(t){normalize();}poly_t(Vector&&t):a(std::move(t)){normalize();}poly_t&negate_inplace(){std::ranges::transform(a,begin(a),std::negate{});return*this;}poly_t operator-()const{return poly_t(*this).negate_inplace();}poly_t&operator+=(poly_t const&t){a.resize(std::max(size(a),size(t.a)));std::ranges::transform(a,t.a,begin(a),std::plus{});return normalize();}poly_t&operator-=(poly_t const&t){a.resize(std::max(size(a),size(t.a)));std::ranges::transform(a,t.a,begin(a),std::minus{});return normalize();}poly_t operator+(poly_t const&t)const{return poly_t(*this)+=t;}poly_t operator-(poly_t const&t)const{return poly_t(*this)-=t;}poly_t&mod_xk_inplace(size_t k){a.resize(std::min(size(a),k));return normalize();}poly_t&mul_xk_inplace(size_t k){a.insert(begin(a),k,T(0));return normalize();}poly_t&div_xk_inplace(int64_t k){if(k<0){return mul_xk_inplace(-k);}a.erase(begin(a),begin(a)+std::min<size_t>(k,size(a)));return normalize();}poly_t&substr_inplace(size_t l,size_t k){return mod_xk_inplace(l+k).div_xk_inplace(l);}poly_t mod_xk(size_t k)const{return poly_t(*this).mod_xk_inplace(k);}poly_t mul_xk(size_t k)const{return poly_t(*this).mul_xk_inplace(k);}poly_t div_xk(int64_t k)const{return poly_t(*this).div_xk_inplace(k);}poly_t substr(size_t l,size_t k)const{return poly_t(*this).substr_inplace(l,k);}poly_t&operator*=(const poly_t&t){fft::mul(a,t.a);normalize();return*this;}poly_t operator*(const poly_t&t)const{return poly_t(*this)*=t;}poly_t&operator/=(const poly_t&t){return*this=divmod(t)[0];}poly_t&operator%=(const poly_t&t){return*this=divmod(t)[1];}poly_t operator/(poly_t const&t)const{return poly_t(*this)/=t;}poly_t operator%(poly_t const&t)const{return poly_t(*this)%=t;}poly_t&operator*=(T const&x){for(auto&it:a){it*=x;}return normalize();}poly_t&operator/=(T const&x){return*this*=x.inv();}poly_t operator*(T const&x)const{return poly_t(*this)*=x;}poly_t operator/(T const&x)const{return poly_t(*this)/=x;}poly_t&reverse(size_t n){a.resize(n);std::ranges::reverse(a);return normalize();}poly_t&reverse(){return reverse(size(a));}poly_t reversed(size_t n)const{return poly_t(*this).reverse(n);}poly_t reversed()const{return poly_t(*this).reverse();}std::array<poly_t,2>divmod(poly_t const&b)const{return poly::impl::divmod(*this,b);}static std::pair<std::list<poly_t>,linfrac<poly_t>>half_gcd(auto&&A,auto&&B){return poly::impl::half_gcd(A,B);}static std::pair<std::list<poly_t>,linfrac<poly_t>>full_gcd(auto&&A,auto&&B){return poly::impl::full_gcd(A,B);}static poly_t gcd(poly_t&&A,poly_t&&B){full_gcd(A,B);return A;}poly_t min_rec(size_t d)const{return poly::impl::min_rec(*this,d);}std::optional<poly_t>inv_mod(poly_t const&t)const{return poly::impl::inv_mod(*this,t);};poly_t negx()const{auto res=*this;for(int i=1;i<=deg();i+=2){res.a[i]=-res[i];}return res;}void print(int n)const{for(int i=0;i<n;i++){std::cout<<(*this)[i]<<' ';}std::cout<<"\n";}void print()const{print(deg()+1);}T eval(T x)const{T res(0);for(int i=deg();i>=0;i--){res*=x;res+=a[i];}return res;}T lead()const{assert(!is_zero());return a.back();}int deg()const{return(int)a.size()-1;}bool is_zero()const{return a.empty();}T operator[](int idx)const{return idx<0||idx>deg()?T(0):a[idx];}T&coef(size_t idx){return a[idx];}bool operator==(const poly_t&t)const{return a==t.a;}bool operator!=(const poly_t&t)const{return a!=t.a;}poly_t&deriv_inplace(int k=1){if(deg()+1<k){return*this=poly_t{};}for(int i=k;i<=deg();i++){a[i-k]=fact<T>(i)*rfact<T>(i-k)*a[i];}a.resize(deg()+1-k);return*this;}poly_t deriv(int k=1)const{return poly_t(*this).deriv_inplace(k);}poly_t&integr_inplace(){a.push_back(0);for(int i=deg()-1;i>=0;i--){a[i+1]=a[i]*small_inv<T>(i+1);}a[0]=0;return*this;}poly_t integr()const{Vector res(deg()+2);for(int i=0;i<=deg();i++){res[i+1]=a[i]*small_inv<T>(i+1);}return res;}size_t trailing_xk()const{if(is_zero()){return-1;}int res=0;while(a[res]==T(0)){res++;}return res;}poly_t&log_inplace(size_t n){assert(a[0]==T(1));mod_xk_inplace(n);return(inv_inplace(n)*=mod_xk_inplace(n).deriv()).mod_xk_inplace(n-1).integr_inplace();}poly_t log(size_t n)const{return poly_t(*this).log_inplace(n);}poly_t&mul_truncate(poly_t const&t,size_t k){fft::mul_truncate(a,t.a,k);return normalize();}poly_t&exp_inplace(size_t n){if(is_zero()){return*this=T(1);}assert(a[0]==T(0));a[0]=1;size_t a=1;while(a<n){poly_t C=log(2*a).div_xk_inplace(a)-substr(a,2*a);*this-=C.mul_truncate(*this,a).mul_xk_inplace(a);a*=2;}return mod_xk_inplace(n);}poly_t exp(size_t n)const{return poly_t(*this).exp_inplace(n);}poly_t pow_bin(int64_t k,size_t n)const{if(k==0){return poly_t(1).mod_xk(n);}else{auto t=pow(k/2,n);t=(t*t).mod_xk(n);return(k%2?*this*t:t).mod_xk(n);}}poly_t circular_closure(size_t m)const{if(deg()==-1){return*this;}auto t=*this;for(size_t i=t.deg();i>=m;i--){t.a[i-m]+=t.a[i];}t.a.resize(std::min(t.a.size(),m));return t;}static poly_t mul_circular(poly_t const&a,poly_t const&b,size_t m){return(a.circular_closure(m)*b.circular_closure(m)).circular_closure(m);}poly_t powmod_circular(int64_t k,size_t m)const{if(k==0){return poly_t(1);}else{auto t=powmod_circular(k/2,m);t=mul_circular(t,t,m);if(k%2){t=mul_circular(t,*this,m);}return t;}}poly_t powmod(int64_t k,poly_t const&md)const{return poly::impl::powmod(*this,k,md);}poly_t pow_dn(int64_t k,size_t n)const{if(n==0){return poly_t(T(0));}assert((*this)[0]!=T(0));Vector Q(n);Q[0]=bpow(a[0],k);auto a0inv=a[0].inv();for(int i=1;i<(int)n;i++){for(int j=1;j<=std::min(deg(),i);j++){Q[i]+=a[j]*Q[i-j]*(T(k)*T(j)-T(i-j));}Q[i]*=small_inv<T>(i)*a0inv;}return Q;}poly_t pow(int64_t k,size_t n)const{if(is_zero()){return k?*this:poly_t(1);}size_t i=trailing_xk();if(i>0){return k>=int64_t(n+i-1)/(int64_t)i?poly_t(T(0)):div_xk(i).pow(k,n-i*k).mul_xk(i*k);}if(std::min(deg(),(int)n)<=magic){return pow_dn(k,n);}if(k<=magic){return pow_bin(k,n);}T j=a[i];poly_t t=*this/j;return bpow(j,k)*(t.log(n)*T(k)).exp(n).mod_xk(n);}std::optional<poly_t>sqrt(size_t n)const{if(is_zero()){return*this;}size_t i=trailing_xk();if(i%2){return std::nullopt;}else if(i>0){auto ans=div_xk(i).sqrt(n-i/2);return ans?ans->mul_xk(i/2):ans;}auto st=math::sqrt((*this)[0]);if(st){poly_t ans=*st;size_t a=1;while(a<n){a*=2;ans-=(ans-mod_xk(a)*ans.inv(a)).mod_xk(a)/2;}return ans.mod_xk(n);}return std::nullopt;}poly_t mulx(T a)const{T cur=1;poly_t res(*this);for(int i=0;i<=deg();i++){res.coef(i)*=cur;cur*=a;}return res;}poly_t mulx_sq(T a)const{T cur=1,total=1;poly_t res(*this);for(int i=0;i<=deg();i++){res.coef(i)*=total;cur*=a;total*=cur;}return res;}poly_t chirpz(T z,int n)const{if(is_zero()){return Vector(n,0);}if(z==T(0)){Vector ans(n,(*this)[0]);if(n>0){ans[0]=accumulate(begin(a),end(a),T(0));}return ans;}auto A=mulx_sq(z.inv());auto B=ones(n+deg()).mulx_sq(z);return semicorr(B,A).mod_xk(n).mulx_sq(z.inv());}static auto _1mzk_prod_inv(T z,int n){Vector res(n,1),zk(n);zk[0]=1;for(int i=1;i<n;i++){zk[i]=zk[i-1]*z;res[i]=res[i-1]*(T(1)-zk[i]);}res.back()=res.back().inv();for(int i=n-2;i>=0;i--){res[i]=(T(1)-zk[i+1])*res[i+1];}return res;}static auto _1mzkx_prod(T z,int n){if(n==1){return poly_t(Vector{1,-1});}else{auto t=_1mzkx_prod(z,n/2);t*=t.mulx(bpow(z,n/2));if(n%2){t*=poly_t(Vector{1,-bpow(z,n-1)});}return t;}}poly_t chirpz_inverse(T z,int n)const{if(is_zero()){return{};}if(z==T(0)){if(n==1){return*this;}else{return Vector{(*this)[1],(*this)[0]-(*this)[1]};}}Vector y(n);for(int i=0;i<n;i++){y[i]=(*this)[i];}auto prods_pos=_1mzk_prod_inv(z,n);auto prods_neg=_1mzk_prod_inv(z.inv(),n);T zn=bpow(z,n-1).inv();T znk=1;for(int i=0;i<n;i++){y[i]*=znk*prods_neg[i]*prods_pos[(n-1)-i];znk*=zn;}poly_t p_over_q=poly_t(y).chirpz(z,n);poly_t q=_1mzkx_prod(z,n);return(p_over_q*q).mod_xk_inplace(n).reverse(n);}static poly_t build(big_vector<poly_t>&res,int v,auto L,auto R){if(R-L==1){return res[v]=Vector{-*L,1};}else{auto M=L+(R-L)/2;return res[v]=build(res,2*v,L,M)*build(res,2*v+1,M,R);}}poly_t to_newton(big_vector<poly_t>&tree,int v,auto l,auto r){if(r-l==1){return*this;}else{auto m=l+(r-l)/2;auto A=(*this%tree[2*v]).to_newton(tree,2*v,l,m);auto B=(*this/tree[2*v]).to_newton(tree,2*v+1,m,r);return A+B.mul_xk(m-l);}}poly_t to_newton(Vector p){if(is_zero()){return*this;}size_t n=p.size();big_vector<poly_t>tree(4*n);build(tree,1,begin(p),end(p));return to_newton(tree,1,begin(p),end(p));}Vector eval(big_vector<poly_t>&tree,int v,auto l,auto r){if(r-l==1){return{eval(*l)};}else{auto m=l+(r-l)/2;auto A=(*this%tree[2*v]).eval(tree,2*v,l,m);auto B=(*this%tree[2*v+1]).eval(tree,2*v+1,m,r);A.insert(end(A),begin(B),end(B));return A;}}Vector eval(Vector x){size_t n=x.size();if(is_zero()){return Vector(n,T(0));}big_vector<poly_t>tree(4*n);build(tree,1,begin(x),end(x));return eval(tree,1,begin(x),end(x));}poly_t inter(big_vector<poly_t>&tree,int v,auto ly,auto ry){if(ry-ly==1){return{*ly/a[0]};}else{auto my=ly+(ry-ly)/2;auto A=(*this%tree[2*v]).inter(tree,2*v,ly,my);auto B=(*this%tree[2*v+1]).inter(tree,2*v+1,my,ry);return A*tree[2*v+1]+B*tree[2*v];}}static auto inter(Vector x,Vector y){size_t n=x.size();big_vector<poly_t>tree(4*n);return build(tree,1,begin(x),end(x)).deriv().inter(tree,1,begin(y),end(y));}static auto resultant(poly_t a,poly_t b){if(b.is_zero()){return 0;}else if(b.deg()==0){return bpow(b.lead(),a.deg());}else{int pw=a.deg();a%=b;pw-=a.deg();auto mul=bpow(b.lead(),pw)*T((b.deg()&a.deg()&1)?-1:1);auto ans=resultant(b,a);return ans*mul;}}static poly_t xk(size_t n){return poly_t(T(1)).mul_xk(n);}static poly_t ones(size_t n){return Vector(n,1);}static poly_t expx(size_t n){return ones(n).borel();}static poly_t log1px(size_t n){Vector coeffs(n,0);for(size_t i=1;i<n;i++){coeffs[i]=(i&1?T(i).inv():-T(i).inv());}return coeffs;}static poly_t log1mx(size_t n){return-ones(n).integr();}static poly_t corr(poly_t const&a,poly_t const&b){return a*b.reversed();}static poly_t semicorr(poly_t const&a,poly_t const&b){return corr(a,b).div_xk(b.deg());}poly_t invborel()const{auto res=*this;for(int i=0;i<=deg();i++){res.coef(i)*=fact<T>(i);}return res;}poly_t borel()const{auto res=*this;for(int i=0;i<=deg();i++){res.coef(i)*=rfact<T>(i);}return res;}poly_t shift(T a)const{return semicorr(invborel(),expx(deg()+1).mulx(a)).borel();}poly_t x2(){Vector res(2*a.size());for(size_t i=0;i<a.size();i++){res[2*i]=a[i];}return res;}std::array<poly_t,2>bisect(size_t n)const{n=std::min(n,size(a));Vector res[2];for(size_t i=0;i<n;i++){res[i%2].push_back(a[i]);}return{res[0],res[1]};}std::array<poly_t,2>bisect()const{return bisect(size(a));}static T kth_rec_inplace(poly_t&P,poly_t&Q,int64_t k){while(k>Q.deg()){size_t n=Q.a.size();auto[Q0,Q1]=Q.bisect();auto[P0,P1]=P.bisect();size_t N=fft::com_size((n+1)/2,(n+1)/2);auto Q0f=fft::dft<T>(Q0.a,N);auto Q1f=fft::dft<T>(Q1.a,N);auto P0f=fft::dft<T>(P0.a,N);auto P1f=fft::dft<T>(P1.a,N);Q=poly_t(Q0f*Q0f)-=poly_t(Q1f*Q1f).mul_xk_inplace(1);if(k%2){P=poly_t(Q0f*=P1f)-=poly_t(Q1f*=P0f);}else{P=poly_t(Q0f*=P0f)-=poly_t(Q1f*=P1f).mul_xk_inplace(1);}k/=2;}return(P*=Q.inv_inplace(Q.deg()+1))[(int)k];}static T kth_rec(poly_t const&P,poly_t const&Q,int64_t k){return kth_rec_inplace(poly_t(P),poly_t(Q),k);}poly_t&inv_inplace(size_t n){return poly::impl::inv_inplace(*this,n);}poly_t inv(size_t n)const{return poly_t(*this).inv_inplace(n);}poly_t&inv_inplace(int64_t k,size_t n){return poly::impl::inv_inplace(*this,k,n);}poly_t inv(int64_t k,size_t n)const{return poly_t(*this).inv_inplace(k,n);}static poly_t compose(poly_t A,poly_t B,int n){int q=std::sqrt(n);big_vector<poly_t>Bk(q);auto Bq=B.pow(q,n);Bk[0]=poly_t(T(1));for(int i=1;i<q;i++){Bk[i]=(Bk[i-1]*B).mod_xk(n);}poly_t Bqk(1);poly_t ans;for(int i=0;i<=n/q;i++){poly_t cur;for(int j=0;j<q;j++){cur+=Bk[j]*A[i*q+j];}ans+=(Bqk*cur).mod_xk(n);Bqk=(Bqk*Bq).mod_xk(n);}return ans;}static poly_t compose_large(poly_t A,poly_t B,int n){if(B[0]!=T(0)){return compose_large(A.shift(B[0]),B-B[0],n);}int q=std::sqrt(n);auto[B0,B1]=std::make_pair(B.mod_xk(q),B.div_xk(q));B0=B0.div_xk(1);big_vector<poly_t>pw(A.deg()+1);auto getpow=[&](int k){return pw[k].is_zero()?pw[k]=B0.pow(k,n-k):pw[k];};std::function<poly_t(poly_t const&,int,int)>compose_dac=[&getpow,&compose_dac](poly_t const&f,int m,int N){if(f.deg()<=0){return f;}int k=m/2;auto[f0,f1]=std::make_pair(f.mod_xk(k),f.div_xk(k));auto[A,B]=std::make_pair(compose_dac(f0,k,N),compose_dac(f1,m-k,N-k));return(A+(B.mod_xk(N-k)*getpow(k).mod_xk(N-k)).mul_xk(k)).mod_xk(N);};int r=n/q;auto Ar=A.deriv(r);auto AB0=compose_dac(Ar,Ar.deg()+1,n);auto Bd=B0.mul_xk(1).deriv();poly_t ans=T(0);big_vector<poly_t>B1p(r+1);B1p[0]=poly_t(T(1));for(int i=1;i<=r;i++){B1p[i]=(B1p[i-1]*B1.mod_xk(n-i*q)).mod_xk(n-i*q);}while(r>=0){ans+=(AB0.mod_xk(n-r*q)*rfact<T>(r)*B1p[r]).mul_xk(r*q).mod_xk(n);r--;if(r>=0){AB0=((AB0*Bd).integr()+A[r]*fact<T>(r)).mod_xk(n);}}return ans;}};template<typename base>static auto operator*(const auto&a,const poly_t<base>&b){return b*a;}};
#pragma GCC pop_options

Back to top page