cp-algo/math/poly/transform.hpp

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• Last update: 2026-09-10 02:45:56+02:00
• Include: #include "cp-algo/math/poly/transform.hpp"

Depends on

• cp-algo/math/combinatorics.hpp
• cp-algo/math/common.hpp
• cp-algo/math/cvector.hpp
• cp-algo/math/dft.hpp
• cp-algo/math/factorials.hpp
• cp-algo/math/fft.hpp
• cp-algo/math/poly/base.hpp
• cp-algo/math/poly/calculus.hpp
• cp-algo/math/poly/series/inv.hpp
• cp-algo/number_theory/modint.hpp
• cp-algo/random/rng.hpp
• cp-algo/util/big_alloc.hpp
• cp-algo/util/bump_alloc.hpp
• cp-algo/util/checkpoint.hpp
• cp-algo/util/complex.hpp
• cp-algo/util/simd.hpp

Required by

• cp-algo/math/poly.hpp
• cp-algo/math/poly/chirpz.hpp
• cp-algo/math/poly/compose.hpp
• tests/fps.cpp
• tests/poly.cpp
• tests/poly_series.cpp
• tests/power.cpp

Verified with

• Bell Number (verify/poly/bell.test.cpp)
• Bernoulli Number (verify/poly/bernoulli.test.cpp)
• Composition of Formal Power Series (verify/poly/compose.test.cpp)
• Polynomial Interpolation (Geometric Sequence) (verify/poly/inter_geo.test.cpp)
• Multipoint Evaluation (Geometric Sequence) (verify/poly/multieval_geo.test.cpp)
• Prefix Sum of Polynomial (verify/poly/prefix_sum.test.cpp)
• Polynomial Taylor Shift (verify/poly/taylor.test.cpp)

Code

#ifndef CP_ALGO_MATH_POLY_TRANSFORM_HPP
#define CP_ALGO_MATH_POLY_TRANSFORM_HPP
#include "calculus.hpp"
#include "series/inv.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
    // Multiply coefficient k by c^k, i.e. substitute cx for x.
    template<typename T>
    poly_t<T> mulx(poly_t<T> p, std::type_identity_t<T> c) {
        T cur = 1;
        for(auto &x: p.a) {x *= cur; cur *= c;}
        p.normalize();
        return p;
    }
    // Multiply coefficient k by c^{k(k+1)/2} (chirp factors).
    template<typename T>
    poly_t<T> mulx_sq(poly_t<T> p, std::type_identity_t<T> c) {
        T cur = 1, total = 1;
        for(auto &x: p.a) {x *= total; cur *= c; total *= cur;}
        p.normalize();
        return p;
    }
    template<typename T>
    poly_t<T> invborel(poly_t<T> p) { // a[k] *= k!
        for(int i = 0; i <= p.deg(); i++) {p.a[i] *= fact<T>(i);}
        return p;
    }
    template<typename T>
    poly_t<T> borel(poly_t<T> p) { // a[k] /= k!
        for(int i = 0; i <= p.deg(); i++) {p.a[i] *= rfact<T>(i);}
        return p;
    }
    template<typename T>
    poly_t<T> expx(size_t n) {return borel(poly_t<T>::ones(n));}

    template<typename T>
    poly_t<T> log1px(size_t n) {
        typename poly_t<T>::Vector a(n);
        for(size_t i = 1; i < n; i++) {a[i] = (i & 1 ? small_inv<T>(i) : -small_inv<T>(i));}
        return a;
    }
    template<typename T>
    poly_t<T> log1mx(size_t n) {
        return n ? -integr(poly_t<T>::ones(n - 1)) : poly_t<T>{};
    }
    // Cross-correlation, with the reversed second argument.
    template<typename T>
    poly_t<T> corr(poly_t<T> a, poly_t<T> b) {
        b.reverse();
        a *= b;
        return a;
    }
    // Coefficient k is sum_i a[i+k] b[i].
    template<typename T>
    poly_t<T> forward_corr(poly_t<T> a, poly_t<T> b) {
        if(b.is_zero()) {return {};}
        int d = b.deg();
        return corr(std::move(a), std::move(b)).div_xk(d);
    }
    // Dot product of b with every full-length substring of a.
    template<typename T>
    poly_t<T> inner_corr(poly_t<T> a, poly_t<T> b) {
        if(b.is_zero() || a.deg() < b.deg()) {return {};}
        int d = b.deg(), n = a.deg() - d + 1;
        size_t N = std::max(fft::flen, std::bit_ceil(a.a.size()));
        std::ranges::reverse(b.a);
        a.a.resize(N);
        b.a.resize(N);
        fft::cyclic_mul(a.a, b.a, N);
        a.substr_inplace(d, n);
        return a;
    }
    template<typename T>
    poly_t<T> apply_diff(poly_t<T> p, poly_t<T> g) { // g(D) p(x)
        return borel(forward_corr(invborel(std::move(p)), std::move(g)));
    }
    template<typename T>
    poly_t<T> shift(poly_t<T> p, std::type_identity_t<T> c) { // p(x+c)
        auto g = mulx(expx<T>(p.deg() + 1), c);
        return apply_diff(std::move(p), std::move(g));
    }
    // q(0) = 0 and q(x+1) - q(x) = p(x).
    template<typename T>
    poly_t<T> prefix_sum(poly_t<T> p) {
        auto g = inv(expx<T>(p.deg() + 2).div_xk(1), p.deg() + 1);
        return integr(apply_diff(std::move(p), std::move(g)));
    }
}
#pragma GCC pop_options
#endif // CP_ALGO_MATH_POLY_TRANSFORM_HPP

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


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


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


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


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


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


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


#include <functional>
#include <cstdint>
#include <cassert>
#include <bit>
#include <vector>
#include <algorithm>
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

    // Nonnegative 64-bit exponents, with an associative operation and its identity.
    // Windows >1 precompute odd powers only when that saves operations.
    template<int window = 1>
    auto bpow(auto const& x, auto n, auto const& one, auto op) {
        static_assert(window >= 1 && window <= 6);
        if constexpr(window > 1) {
            if(n == 0) {return one;}
            int bits = std::bit_width(uint64_t(n));
            auto low_bit = [&](int high) {
                int low = std::max(0, high - window + 1);
                while(!((n >> low) & 1)) {low++;}
                return low;
            };
            int first = low_bit(bits - 1);
            int cost = (1 << (window - 1)) + first;
            for(int j = first - 1; j >= 0;) {
                if(!((n >> j) & 1)) {j--;}
                else {cost++; j = low_bit(j) - 1;}
            }
            // Do not pay for the table when binary powering uses fewer operations.
            if(cost >= bits + std::popcount(uint64_t(n)) - 2) {return bpow<1>(x, n, one, op);}
            using T = std::decay_t<decltype(x)>;
            std::vector<T> odd;
            odd.reserve(1 << (window - 1));
            odd.push_back(x);
            auto square = op(x, x);
            while(odd.size() < size_t(1 << (window - 1))) {odd.push_back(op(odd.back(), square));}
            auto ans = odd[(n >> first) / 2];
            for(int j = first - 1; j >= 0;) {
                if(!((n >> j) & 1)) {ans = op(ans, ans); j--;}
                else {
                    int low = low_bit(j), length = j - low + 1;
                    auto digit = (n >> low) & ((1u << length) - 1);
                    for(int i = 0; i < length; i++) {ans = op(ans, ans);}
                    ans = op(ans, odd[digit / 2]);
                    j = low - 1;
                }
            }
            return ans;
        } else {
            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;
        }
    }
    template<int window = 1>
    auto bpow(auto x, auto n, auto ans) {
        return bpow<window>(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()));
        }
        constexpr 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>
#line 10 "cp-algo/util/big_alloc.hpp"
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#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);
                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>>;
}


#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 u8x16 = simd<uint8_t, 16>;
    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);
            prepare_roots(n / 16);
            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 if(n / 4 - pre_evals < extra.size()) {
                return extra[n / 4 - pre_evals];
            } 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");
        }
        // normalize=false leaves the inverse-transform scale for the caller.
        template<bool partial = true, bool normalize = true>
        void ifft() {
            size_t n = size();
            if constexpr (!partial) {
                prepare_roots(n / 4);
                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;
                });
            }

            transform<true>(n, parity);
            checkpoint("ifft");
            if constexpr(normalize) {
                auto scale = vz + ftype(partial ? flen : 1) / ftype(n);
                for(size_t k = 0; k < n; k += flen) {
                    set(k, get<vpoint>(k) * scale);
                }
            }
        }
        template<bool partial = true>
        void fft() {
            size_t n = size();
            prepare_roots(n / (partial ? 16 : 4));
            bool parity = std::countr_zero(n) % 2;
            transform<false>(n, parity);
            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) {
                prepare_roots(n / 4);
                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;
    private:
        // Tile two radix-four stages together before descending into each child.
        template<bool inverse>
        void transform(size_t n, bool parity) {
            auto butterfly = [&](size_t offset, size_t length, size_t begin, size_t end) __attribute__((always_inline)) {
                size_t i = length / 4;
                point rt = root(16 * n / length) * eval_point(4 * offset / length);
                vpoint v1 = {vz + real(rt), inverse ? vz - imag(rt) : vz + imag(rt)};
                vpoint v2 = v1 * v1, v3 = v1 * v2;
                for(size_t j = offset + begin; j < offset + end; j += flen) {
                    auto A = at(j), B = at(j+i), C = at(j+2*i), D = at(j+3*i);
                    if constexpr(inverse) {
                        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;
                    } else {
                        B = B*v1; C = C*v2; D = D*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);
                    }
                }
            };
            auto recurse = [&](auto &&self, size_t offset, size_t length) -> void {
                if(length < 4 * flen) {return;}
                if(length >= (1 << 15)) {
                    size_t step = length / 16;
                    if constexpr(inverse) {
                        for(size_t t = 0; t < 16; t++) {self(self, offset + t*step, step);}
                    }
                    for(size_t j = 0; j < step; j += 256) {
                        size_t end = std::min(step, j+256);
                        if constexpr(inverse) {
                            for(size_t t=0;t<4;t++) {butterfly(offset+t*length/4, length/4, j,end);}
                            for(size_t t=0;t<4;t++) {butterfly(offset,length,j+t*step,end+t*step);}
                        } else {
                            for(size_t t=0;t<4;t++) {butterfly(offset,length,j+t*step,end+t*step);}
                            for(size_t t=0;t<4;t++) {butterfly(offset+t*length/4,length/4,j,end);}
                        }
                    }
                    if constexpr(!inverse) {
                        for(size_t t = 0; t < 16; t++) {self(self, offset + t*step, step);}
                    }
                } else {
                    if constexpr(inverse) {
                        for(size_t leaf = offset + 3 * flen; leaf < offset + length; leaf += 4 * flen) {
                            size_t level = std::min<size_t>(std::countr_one(leaf + 3), std::countr_zero(length));
                            for(size_t lvl = 4 + parity; lvl <= level; lvl += 2) {
                                size_t len = size_t(1) << lvl;
                                butterfly(leaf / len * len, len, 0, len / 4);
                            }
                        }
                    } else {
                        for(size_t leaf = offset; leaf < offset + length; leaf += 4 * flen) {
                            size_t level = std::min<size_t>(std::countr_zero(n + leaf), std::countr_zero(length));
                            level -= level % 2 != parity;
                            for(size_t lvl = level; lvl >= 4; lvl -= 2) {
                                size_t len = size_t(1) << lvl;
                                butterfly(leaf / len * len, len, 0, len / 4);
                            }
                        }
                    }
                }
            };
            // Radix two is performed separately at the leaves.
            recurse(recurse, 0, n);
        }
        static big_vector<point> extra;
        // Keep the usual table small; cache additional roots for large transforms.
        static void prepare_roots(size_t n) {
            if(n <= pre_evals + extra.size()) {return;}
            size_t old = extra.size();
            extra.resize(std::bit_ceil(n) - pre_evals);
            for(size_t i = old; i < extra.size(); i++) {
                size_t j = 4 * (i + pre_evals);
                extra[i] = polar<ftype>(1., std::numbers::pi / (ftype)std::bit_floor(j) * (ftype)eval_arg(j));
            }
        }
    };

    big_vector<point> cvector::extra;

    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/dft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft {
    // Twist coefficients by a random factor, split near sqrt(mod), and pack pairs of halves.
    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 * (1.0 / 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(0), B(0) {
            // Construct split coefficients once instead of zeroing both buffers first.
            A.r.clear(); B.r.clear();
            size_t blocks = std::max(flen, std::bit_ceil(n)) / flen;
            A.r.reserve(blocks); B.r.reserve(blocks);
            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.r.emplace_back(rai, rani);
                B.r.emplace_back(qai, qani);
                cur = montgomery_mul(cur, step4, mod, imod);
            }
            A.r.resize(blocks); B.r.resize(blocks);
            checkpoint("dft init");
            if(n) {
                if(partial) {
                    A.fft();
                    B.fft();
                } else {
                    A.template fft<false>();
                    B.template fft<false>();
                }
            }
        }
        // Multiply split evaluations; Cout collects the mixed low/high terms.
        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);
                    // Precompute wrapped coefficients, then select each rotation with SIMD shuffles.
                    vpoint vrt = {vz + real(rt), vz + imag(rt)};
                    auto Cr = Cv * vrt, Dr = Dv * vrt;
                    auto iter = [&]<int i>() __attribute__((always_inline)) {
                        auto wrap = [&](vftype original, vftype rotated) {
                            if constexpr(i == 0) {return original;}
                            else {return __builtin_shufflevector(rotated, original, 4 - i, 5 - i, 6 - i, 7 - i);}
                        };
                        vpoint Cw = {wrap(real(Cv), real(Cr)), wrap(imag(Cv), imag(Cr))};
                        vpoint Dw = {wrap(real(Dv), real(Dr)), wrap(imag(Dv), imag(Dr))};
                        vpoint Av = {vz + Ax[i], vz + Ay[i]}, Bv = {vz + Bx[i], vz + By[i]};
                        AC += Av * Cw; AD += Av * Dw;
                        BC += Bv * Cw; BD += Bv * Dw;
                    };
                    iter.template operator()<0>();
                    iter.template operator()<1>();
                    iter.template operator()<2>();
                    iter.template operator()<3>();
                } 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);
            // Center signed lifts in the unsigned Montgomery input range [0, mod*2^32).
            auto Ai = A0 + A1 * split() + A2 * splitsplit + (uint64_t(base::mod()) << 31);
            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]));
            }
        }

        // Round the convolutions and undo twisting, optionally including the inverse-FFT scale.
        template<bool normalized = true>
        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 scale = vz + ftype(flen) / ftype(n);
            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 get = [&](auto const& x) {
                    if constexpr(normalized) {return x.at(i);}
                    else {return x.at(i) * scale;}
                };
                auto [Ax, Ay] = get(A);
                auto [Bx, By] = get(B);
                auto [Cx, Cy] = get(C);
                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);
            // Normalize during recovery to avoid another pass over the buffers.
            A.template ifft<true, false>();
            B.template ifft<true, false>();
            C.template ifft<true, false>();
            recover_mod<false>(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 = {};

}
#pragma GCC pop_options

#line 4 "cp-algo/math/fft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft {
    void mul_slow(auto &a, auto const& b, size_t k) {
        if(!std::empty(a) && std::data(a) == std::data(b)) {
            using base = std::decay_t<decltype(a[0])>;
            size_t n = std::min(k, std::size(a)), m = std::min(k, std::size(b));
            if(!m) {a.clear(); return;}
            a.resize(k);
            // Descending output only reads original coefficients at indices <=j.
            for(size_t j = k; j-- > 0;) {
                base sum = 0;
                size_t lo = j >= n ? j + 1 - n : 0, hi = std::min(j + 1, m);
                for(size_t i = lo; i < hi; i++) {
                    if(n == m && i > j - i) {break;}
                    auto term = a[i] * a[j - i];
                    sum += n == m && i != j - i ? term + term : term;
                }
                a[j] = sum;
            }
            return;
        }
        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);
        size_t as = std::min(k, std::size(a)), bs = std::min(k, std::size(b));
        size_t tail = as + bs - 1 - n;
        // Correct a short wrapped tail instead of doubling the FFT size.
        if(tail <= 32 && as <= n && bs <= n) {
            std::array<base, 32> high{};
            for(size_t i = 0; i < tail; i++) {
                for(size_t j = n + i - bs + 1; j < as; j++) {
                    high[i] += a[j] * b[n + i - j];
                }
            }
            auto A = dft<base>(a | std::views::take(k), n / 2);
            if(as == bs && std::data(a) == std::data(b)) {
                a.resize((k + flen - 1) / flen * flen);
                A.mul(A, a, std::min(k, n));
            } else {
                auto B = dft<base>(b | std::views::take(k), n / 2);
                a.resize((k + flen - 1) / flen * flen);
                A.mul_inplace(B, a, std::min(k, n));
            }
            auto wrap = bpow(dft<base>::factor, n);
            for(size_t i = 0; i < tail; i++) {
                a[i] += wrap * high[i];
                if(n + i < k) {a[n + i] = high[i];}
            }
            a.resize(k);
            return;
        }
        auto A = dft<base>(a | std::views::take(k), n);
        if(as == bs && std::data(a) == std::data(b)) {
            a.resize((k + flen - 1) / flen * flen);
            A.mul(A, a, k);
        } else {
            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
    // inverse reconstructs the halves with k = 1/(2 * forward_k).
    template<bool inverse = false>
    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()
            };
            if constexpr(!inverse) {
                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;
            if constexpr(inverse) {
                xl = (xl + (xl & 1) * base::mod()) >> 1;
                xr = montgomery_mul(xr, cur, dft<base>::mod, dft<base>::imod);
                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(inverse ? "mod join" : "mod split");
    }
    // zero_upper skips arithmetic on the known zero padding in the first split.
    void cyclic_mul(auto &a, auto &&b, size_t k, bool zero_upper = false) {
        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();
        bool square = std::data(a) == std::data(b);
        if(k <= (1 << 16)) {
            big_vector<base> ap(begin(a), end(a));
            if(square) {mul_truncate(ap, ap, 2 * k);}
            else {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);
        if(zero_upper) {
            std::ranges::copy(std::span(a).first(k), begin(a) + k);
            if(!square) {std::ranges::copy(std::span(b).first(k), begin(b) + k);}
        } else {
            mod_split(a, k, factor);
            if(!square) {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);
        if(square) {A.mul(A, ra, k);}
        else {
            auto B = dft<base>(rb, k / 2);
            A.mul_inplace(B, ra, k);
        }
        base i2 = base(2).inv();
        factor = factor.inv() * i2;
        mod_split<true>(a, k, factor);
    }
    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);
    }
    namespace impl {
        // Overlap-add for a short fixed operand; every block reuses its transform.
        void mul_unbalanced(auto &a, auto const& b) {
            using base = std::decay_t<decltype(a[0])>;
            auto x = std::span<base const>(a), y = std::span<base const>(b);
            if(x.size() < y.size()) {std::swap(x, y);}
            constexpr size_t length = 1 << 15;
            size_t step = length - y.size() + 1;
            auto fixed = dft<base>(y, length / 2);
            std::decay_t<decltype(a)> result(x.size() + y.size() - 1);
            big_vector<base> work(length);
            for(size_t start = 0; start < x.size(); start += step) {
                size_t count = std::min(step, x.size() - start);
                auto block = dft<base>(x.subspan(start, count), length / 2);
                size_t need = count + y.size() - 1;
                block.mul(fixed, work, need);
                for(size_t i = 0; i < need; i++) {result[start + i] += work[i];}
            }
            a = std::move(result);
        }
    }
    void mul(auto &a, auto &&b) {
        if(std::empty(a) || std::empty(b)) {a.clear(); return;}
        bool square = std::data(a) == std::data(b) && std::size(a) == std::size(b);
        if(!square && std::data(a) == std::data(b)) {
            auto copy = make_copy(b);
            return mul(a, copy);
        }
        size_t small = std::min(size(a), size(b)), large = std::max(size(a), size(b));
        if(small >= magic && small <= 4096 && large >= (1 << 20) && large / small >= 64) {
            return impl::mul_unbalanced(a, b);
        }
        using base = std::decay_t<decltype(a[0])>;
        size_t N = size(a) + size(b);
        if(N > (1 << 20)) {
            N--;
            size_t NN = std::bit_ceil(N);
            bool zero_upper = std::max(size(a), size(b)) <= NN / 2;
            a.resize(NN);
            // Compute the negative branch before the positive branch consumes the inputs.
            // Only the result needs the upper half; b never needs duplicated padding.
            if(zero_upper && !square) {
                size_t half = NN / 2;
                b.resize(half);
                auto lo = std::span(a).first(half), hi = std::span(a).last(half);
                {
                    auto A = dft<base>(lo, half / 2);
                    auto B = dft<base>(b, half / 2);
                    A.mul_inplace(B, hi, half);
                }
                cyclic_mul(lo, b, half);
                mod_split<true>(a, half, (base(2) * bpow(dft<base>::factor, half)).inv());
            } else {
                if(!square) {b.resize(NN);}
                cyclic_mul(a, b, NN, zero_upper);
            }
            a.resize(N);
        } else {
            mul_truncate(a, b, N - 1);
        }
    }
    void mul(auto &a, auto const& b) {
        if(std::empty(a) || std::empty(b)) {a.clear(); return;}
        size_t small = std::min(size(a), size(b)), large = std::max(size(a), size(b));
        if(small >= magic && small <= 4096 && large >= (1 << 20) && large / small >= 64) {
            return impl::mul_unbalanced(a, b);
        }
        size_t N = size(a) + size(b);
        if(N > (1 << 20)) {
            if(std::data(a) == std::data(b) && std::size(a) == std::size(b)) {mul(a, a);}
            else {mul(a, make_copy(b));}
        } else {
            mul_truncate(a, b, N - 1);
        }
    }
}
#pragma GCC pop_options

#line 4 "cp-algo/math/poly/base.hpp"
#include <array>
#line 6 "cp-algo/math/poly/base.hpp"
#include <utility>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
    template<typename T> struct poly_t;
    template<typename T>
    std::array<poly_t<T>, 2> divmod(poly_t<T> p, poly_t<T> const& q);
    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() = default;
        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;
        }
        friend poly_t operator -(poly_t p) {p.negate_inplace(); return p;}
        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();
        }
        friend poly_t operator + (poly_t p, poly_t const& t) {p += t; return p;}
        friend poly_t operator - (poly_t p, poly_t const& t) {p -= t; return p;}

        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) {
            if(is_zero()) {return *this;}
            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 substr(0, k);}
        poly_t mod_xk(size_t k) && {mod_xk_inplace(k); return std::move(*this);}
        poly_t mul_xk(size_t k) const & {auto p = *this; p.mul_xk_inplace(k); return p;}
        poly_t mul_xk(size_t k) && {mul_xk_inplace(k); return std::move(*this);}
        poly_t div_xk(int64_t k) const & {return k < 0 ? mul_xk(-k) : substr(k, a.size());}
        poly_t div_xk(int64_t k) && {div_xk_inplace(k); return std::move(*this);}
        poly_t substr(size_t l, size_t k) const & {
            l = std::min(l, a.size());
            k = std::min(k, a.size() - l);
            return Vector(begin(a) + l, begin(a) + l + k);
        }
        poly_t substr(size_t l, size_t k) && {substr_inplace(l, k); return std::move(*this);}

        poly_t& operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
        friend poly_t operator * (poly_t p, const poly_t &t) {p *= t; return p;}

        poly_t& operator /= (const poly_t &t) {
            assert(!t.is_zero());
            if(this == &t) {return *this = T(1);}
            auto [q, r] = divmod(std::move(*this), t);
            return *this = std::move(q);
        }
        poly_t& operator %= (const poly_t &t) {
            assert(!t.is_zero());
            if(this == &t) {a.clear(); return *this;}
            auto [q, r] = divmod(std::move(*this), t);
            return *this = std::move(r);
        }
        friend poly_t operator / (poly_t p, poly_t const& t) {p /= t; return p;}
        friend poly_t operator % (poly_t p, poly_t const& t) {p %= t; return p;}

        poly_t& operator *= (T const& x) {
            for(auto &it: a) {
                it *= x;
            }
            return normalize();
        }
        poly_t& operator /= (T const& x) {return *this *= x.inv();}
        friend poly_t operator * (poly_t p, T const& x) {p *= x; return p;}
        friend poly_t operator / (poly_t p, T const& x) {p /= x; return p;}

        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 & {auto p = *this; p.reverse(n); return p;}
        poly_t reversed(size_t n) && {reverse(n); return std::move(*this);}
        poly_t reversed() const & {return reversed(a.size());}
        poly_t reversed() && {reverse(); return std::move(*this);}

        friend poly_t operator * (T const& x, poly_t p) {p *= x; return p;}

        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;}

        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;
        }

        poly_t& mul_truncate(poly_t const& t, size_t k) {
            fft::mul_truncate(a, t.a, k);
            return normalize();
        }

        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);
        }

        poly_t x2() const { // 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^2) + xP1(x^2)
        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 {std::move(res[0]), std::move(res[1])};
        }
        std::array<poly_t, 2> bisect() const {
            return bisect(size(a));
        }

    };
}
#pragma GCC pop_options

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


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


#line 5 "cp-algo/util/bump_alloc.hpp"
namespace cp_algo {
    template<class T, size_t max_len>
    struct bump_alloc {
        static char* buf;
        static size_t buf_ind;
        using value_type = T;
        template <class U> struct rebind { using other = bump_alloc<U, max_len>; };
        constexpr bool operator==(const bump_alloc&) const = default;
        constexpr bool operator!=(const bump_alloc&) const = default;
        bump_alloc() = default;
        template<class U> bump_alloc(const U&) {}
        T* allocate(size_t n) {
            buf_ind -= n * sizeof(T);
            buf_ind &= 0 - alignof(T);
            return (T*)(buf + buf_ind);
        }
        void deallocate(T*, size_t) {}
    };
    template<class T, size_t max_len>
    char* bump_alloc<T, max_len>::buf = big_alloc<char>().allocate(max_len * sizeof(T));
    template<class T, size_t max_len>
    size_t bump_alloc<T, max_len>::buf_ind = max_len * sizeof(T);
}

#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 9 "cp-algo/math/factorials.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
    template<bool use_bump_alloc = false, int maxn = -1>
    auto facts(auto const& args) {
        static_assert(!use_bump_alloc || maxn > 0, "maxn must be set if use_bump_alloc is true");
        constexpr int max_mod = 1'000'000'000;
        constexpr int accum = 4;
        constexpr int simd_size = 8;
        constexpr int block = 1 << 18;
        constexpr int subblock = block / simd_size;
        using base = std::decay_t<decltype(args[0])>;
        static_assert(modint_type<base>, "Base type must be a modint type");
        using T = std::array<int, 2>;
        using alloc = std::conditional_t<use_bump_alloc,
            bump_alloc<T, 30 * maxn>,
            big_alloc<T>>;
        std::basic_string<T, std::char_traits<T>, alloc> odd_args_per_block[max_mod / subblock];
        std::basic_string<T, std::char_traits<T>, alloc> reg_args_per_block[max_mod / subblock];
        constexpr int limit_reg = max_mod / 64;
        int limit_odd = 0;

        big_vector<base> res(size(args), 1);
        const int mod = base::mod();
        const int imod = -math::inv2(mod);
        for(auto [i, xy]: std::views::zip(args, res) | std::views::enumerate) {
            auto [x, y] = xy;
            int t = x.getr();
            if(t >= mod / 2) {
                t = mod - t - 1;
                y = t % 2 ? 1 : mod-1;
            }
            auto pw = 32ull * (t + 1);
            while(t > limit_reg) {
                limit_odd = std::max(limit_odd, (t - 1) / 2);
                odd_args_per_block[(t - 1) / 2 / subblock].push_back({int(i), (t - 1) / 2});
                t /= 2;
                pw += t;
            }
            reg_args_per_block[t / subblock].push_back({int(i), t});
            y *= pow_fixed<base, 2>(int(pw % (mod - 1)));
        }
        checkpoint("init");
        base bi2x32 = pow_fixed<base, 2>(32).inv();
        auto process = [&](int limit, auto &args_per_block, auto step, auto &&proj) {
            base fact = 1;
            for(int b = 0; b <= limit; b += accum * block) {
                u32x8 cur[accum];
                static std::array<u32x8, subblock> prods[accum];
                for(int z = 0; z < accum; z++) {
                    for(int j = 0; j < simd_size; j++) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
                        cur[z][j] = uint32_t(b + z * block + j * subblock);
                        cur[z][j] = proj(cur[z][j]);
                        prods[z][0][j] = cur[z][j] + !cur[z][j];
                        prods[z][0][j] = uint32_t(uint64_t(prods[z][0][j]) * bi2x32.getr() % mod);
#pragma GCC diagnostic pop
                    }
                }
                for(int i = 1; i < block / simd_size; i++) {
                    for(int z = 0; z < accum; z++) {
                        cur[z] += step;
                        prods[z][i] = montgomery_mul(prods[z][i - 1], cur[z], mod, imod);
                    }
                }
                checkpoint("inner loop");
                for(int z = 0; z < accum; z++) {
                    for(int j = 0; j < simd_size; j++) {
                        int bl = b + z * block + j * subblock;
                        for(auto [i, x]: args_per_block[bl / subblock]) {
                            res[i] *= fact * prods[z][x - bl][j];
                        }
                        fact *= base(prods[z].back()[j]);
                    }
                }
                checkpoint("mul ans");
            }
        };
        process(limit_reg, reg_args_per_block, 1, std::identity{});
        process(limit_odd, odd_args_per_block, 2, [](uint32_t x) {return 2 * x + 1;});
        auto invs = bulk_invs<base>(res);
        for(auto [i, x]: res | std::views::enumerate) {
            if (args[i] >= mod / 2) {
                x = invs[i];
            }
        }
        checkpoint("inv ans");
        return res;
    }
}
#pragma GCC pop_options

#line 5 "cp-algo/math/poly/calculus.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
    // k-th derivative; consumes temporaries and moved polynomials.
    template<typename T>
    poly_t<T> deriv(poly_t<T> p, int k = 1) {
        assert(k >= 0);
        if(k > p.deg()) {return k == 0 ? p : poly_t<T>{};}
        if(k == 0) {return p;}
        if(k == 1) {
            for(int i = 1; i <= p.deg(); i++) {p.a[i - 1] = T(i) * p.a[i];}
            p.a.pop_back();
            p.normalize();
            return p;
        }
        for(int i = k; i <= p.deg(); i++) {
            p.a[i - k] = fact<T>(i) * rfact<T>(i - k) * p.a[i];
        }
        p.a.resize(p.a.size() - k);
        p.normalize();
        return p;
    }
    // Antiderivative with zero constant coefficient.
    template<typename T>
    poly_t<T> integr(poly_t<T> p) {
        if(p.is_zero()) {return p;}
        p.a.push_back(0);
        for(int i = p.deg() - 1; i >= 0; i--) {
            p.a[i + 1] = p.a[i] * small_inv<T>(i + 1);
        }
        p.a[0] = 0;
        return p;
    }
}
#pragma GCC pop_options

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


#line 4 "cp-algo/math/poly/series/inv.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::poly::impl {
    template<typename poly>
    poly& inv_inplace(poly& p, size_t n) {
        using base = poly::base;
        if(n == 0) {
            p.a.clear();
            return p;
        }
        assert(p[0] != base(0));
        if(n < magic) {
            typename poly::Vector q(n);
            q[0] = base(1) / p[0];
            for(size_t i = 1; i < n; i++) {
                for(size_t j = 1; j <= std::min(i, p.a.size() - 1); j++) {
                    q[i] -= p.a[j] * q[i - j];
                }
                q[i] *= q[0];
            }
            return p = std::move(q);
        }
        size_t m = std::bit_floor(size_t(magic - 1));
        auto q = p.mod_xk(m);
        inv_inplace(q, m);
        for(; m < n; m *= 2) {
            size_t k = std::min(2 * m, n);
            typename poly::Vector error((k + fft::flen - 1) / fft::flen * fft::flen);
            auto Q = fft::dft<base>(q.a, m);
            {
                auto P = fft::dft<base>(p.a | std::views::take(k), m);
                // Wrapping modulo x^(2m) + factor^(2m) only changes the discarded low half.
                P.mul(Q, error, k);
            }
            auto E = fft::dft<base>(error | std::views::drop(m) | std::views::take(k - m), m);
            Q.mul_inplace(E, error, k - m);
            q.a.resize(k);
            for(size_t i = m; i < k; i++) {q.a[i] = -error[i - m];}
        }
        p = std::move(q);
        p.normalize();
        return p;
    }
}
namespace cp_algo::math {
    // Inverse modulo x^n; the constant coefficient must be invertible.
    template<typename T>
    poly_t<T> inv(poly_t<T> p, size_t n) {
        poly::impl::inv_inplace(p, n);
        return p;
    }
}
#pragma GCC pop_options

#line 5 "cp-algo/math/poly/transform.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
    // Multiply coefficient k by c^k, i.e. substitute cx for x.
    template<typename T>
    poly_t<T> mulx(poly_t<T> p, std::type_identity_t<T> c) {
        T cur = 1;
        for(auto &x: p.a) {x *= cur; cur *= c;}
        p.normalize();
        return p;
    }
    // Multiply coefficient k by c^{k(k+1)/2} (chirp factors).
    template<typename T>
    poly_t<T> mulx_sq(poly_t<T> p, std::type_identity_t<T> c) {
        T cur = 1, total = 1;
        for(auto &x: p.a) {x *= total; cur *= c; total *= cur;}
        p.normalize();
        return p;
    }
    template<typename T>
    poly_t<T> invborel(poly_t<T> p) { // a[k] *= k!
        for(int i = 0; i <= p.deg(); i++) {p.a[i] *= fact<T>(i);}
        return p;
    }
    template<typename T>
    poly_t<T> borel(poly_t<T> p) { // a[k] /= k!
        for(int i = 0; i <= p.deg(); i++) {p.a[i] *= rfact<T>(i);}
        return p;
    }
    template<typename T>
    poly_t<T> expx(size_t n) {return borel(poly_t<T>::ones(n));}

    template<typename T>
    poly_t<T> log1px(size_t n) {
        typename poly_t<T>::Vector a(n);
        for(size_t i = 1; i < n; i++) {a[i] = (i & 1 ? small_inv<T>(i) : -small_inv<T>(i));}
        return a;
    }
    template<typename T>
    poly_t<T> log1mx(size_t n) {
        return n ? -integr(poly_t<T>::ones(n - 1)) : poly_t<T>{};
    }
    // Cross-correlation, with the reversed second argument.
    template<typename T>
    poly_t<T> corr(poly_t<T> a, poly_t<T> b) {
        b.reverse();
        a *= b;
        return a;
    }
    // Coefficient k is sum_i a[i+k] b[i].
    template<typename T>
    poly_t<T> forward_corr(poly_t<T> a, poly_t<T> b) {
        if(b.is_zero()) {return {};}
        int d = b.deg();
        return corr(std::move(a), std::move(b)).div_xk(d);
    }
    // Dot product of b with every full-length substring of a.
    template<typename T>
    poly_t<T> inner_corr(poly_t<T> a, poly_t<T> b) {
        if(b.is_zero() || a.deg() < b.deg()) {return {};}
        int d = b.deg(), n = a.deg() - d + 1;
        size_t N = std::max(fft::flen, std::bit_ceil(a.a.size()));
        std::ranges::reverse(b.a);
        a.a.resize(N);
        b.a.resize(N);
        fft::cyclic_mul(a.a, b.a, N);
        a.substr_inplace(d, n);
        return a;
    }
    template<typename T>
    poly_t<T> apply_diff(poly_t<T> p, poly_t<T> g) { // g(D) p(x)
        return borel(forward_corr(invborel(std::move(p)), std::move(g)));
    }
    template<typename T>
    poly_t<T> shift(poly_t<T> p, std::type_identity_t<T> c) { // p(x+c)
        auto g = mulx(expx<T>(p.deg() + 1), c);
        return apply_diff(std::move(p), std::move(g));
    }
    // q(0) = 0 and q(x+1) - q(x) = p(x).
    template<typename T>
    poly_t<T> prefix_sum(poly_t<T> p) {
        auto g = inv(expx<T>(p.deg() + 2).div_xk(1), p.deg() + 1);
        return integr(apply_diff(std::move(p), std::move(g)));
    }
}
#pragma GCC pop_options

#ifndef CP_ALGO_MATH_POLY_TRANSFORM_HPP
#define CP_ALGO_MATH_POLY_TRANSFORM_HPP
#include "calculus.hpp"
#include "series/inv.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<typename T>poly_t<T>mulx(poly_t<T>p,std::type_identity_t<T>c){T cur=1;for(auto&x:p.a){x*=cur;cur*=c;}p.normalize();return p;}template<typename T>poly_t<T>mulx_sq(poly_t<T>p,std::type_identity_t<T>c){T cur=1,total=1;for(auto&x:p.a){x*=total;cur*=c;total*=cur;}p.normalize();return p;}template<typename T>poly_t<T>invborel(poly_t<T>p){for(int i=0;i<=p.deg();i++){p.a[i]*=fact<T>(i);}return p;}template<typename T>poly_t<T>borel(poly_t<T>p){for(int i=0;i<=p.deg();i++){p.a[i]*=rfact<T>(i);}return p;}template<typename T>poly_t<T>expx(size_t n){return borel(poly_t<T>::ones(n));}template<typename T>poly_t<T>log1px(size_t n){typename poly_t<T>::Vector a(n);for(size_t i=1;i<n;i++){a[i]=(i&1?small_inv<T>(i):-small_inv<T>(i));}return a;}template<typename T>poly_t<T>log1mx(size_t n){return n?-integr(poly_t<T>::ones(n-1)):poly_t<T>{};}template<typename T>poly_t<T>corr(poly_t<T>a,poly_t<T>b){b.reverse();a*=b;return a;}template<typename T>poly_t<T>forward_corr(poly_t<T>a,poly_t<T>b){if(b.is_zero()){return{};}int d=b.deg();return corr(std::move(a),std::move(b)).div_xk(d);}template<typename T>poly_t<T>inner_corr(poly_t<T>a,poly_t<T>b){if(b.is_zero()||a.deg()<b.deg()){return{};}int d=b.deg(),n=a.deg()-d+1;size_t N=std::max(fft::flen,std::bit_ceil(a.a.size()));std::ranges::reverse(b.a);a.a.resize(N);b.a.resize(N);fft::cyclic_mul(a.a,b.a,N);a.substr_inplace(d,n);return a;}template<typename T>poly_t<T>apply_diff(poly_t<T>p,poly_t<T>g){return borel(forward_corr(invborel(std::move(p)),std::move(g)));}template<typename T>poly_t<T>shift(poly_t<T>p,std::type_identity_t<T>c){auto g=mulx(expx<T>(p.deg()+1),c);return apply_diff(std::move(p),std::move(g));}template<typename T>poly_t<T>prefix_sum(poly_t<T>p){auto g=inv(expx<T>(p.deg()+2).div_xk(1),p.deg()+1);return integr(apply_diff(std::move(p),std::move(g)));}}
#pragma GCC pop_options
#endif

#line 1 "cp-algo/math/poly/transform.hpp"
#line 1 "cp-algo/math/poly/calculus.hpp"
#line 1 "cp-algo/math/poly/base.hpp"
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/math/dft.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
#include <bit>
#include <vector>
#include <algorithm>
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;template<int window=1>auto bpow(auto const&x,auto n,auto const&one,auto op){static_assert(window>=1&&window<=6);if constexpr(window>1){if(n==0){return one;}int bits=std::bit_width(uint64_t(n));auto low_bit=[&](int high){int low=std::max(0,high-window+1);while(!((n>>low)&1)){low++;}return low;};int first=low_bit(bits-1);int cost=(1<<(window-1))+first;for(int j=first-1;j>=0;){if(!((n>>j)&1)){j--;}else{cost++;j=low_bit(j)-1;}}if(cost>=bits+std::popcount(uint64_t(n))-2){return bpow<1>(x,n,one,op);}using T=std::decay_t<decltype(x)>;std::vector<T>odd;odd.reserve(1<<(window-1));odd.push_back(x);auto square=op(x,x);while(odd.size()<size_t(1<<(window-1))){odd.push_back(op(odd.back(),square));}auto ans=odd[(n>>first)/2];for(int j=first-1;j>=0;){if(!((n>>j)&1)){ans=op(ans,ans);j--;}else{int low=low_bit(j),length=j-low+1;auto digit=(n>>low)&((1u<<length)-1);for(int i=0;i<length;i++){ans=op(ans,ans);}ans=op(ans,odd[digit/2]);j=low-1;}}return ans;}else{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;}}template<int window=1>auto bpow(auto x,auto n,auto ans){return bpow<window>(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()));}constexpr 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>
#line 10 "cp-algo/util/big_alloc.hpp"
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#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);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>>;}
#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 u8x16=simd<uint8_t,16>;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);prepare_roots(n/16);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 if(n/4-pre_evals<extra.size()){return extra[n/4-pre_evals];}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,bool normalize=true>void ifft(){size_t n=size();if constexpr(!partial){prepare_roots(n/4);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;});}transform<true>(n,parity);checkpoint("ifft");if constexpr(normalize){auto scale=vz+ftype(partial?flen:1)/ftype(n);for(size_t k=0;k<n;k+=flen){set(k,get<vpoint>(k)*scale);}}}template<bool partial=true>void fft(){size_t n=size();prepare_roots(n/(partial?16:4));bool parity=std::countr_zero(n)%2;transform<false>(n,parity);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){prepare_roots(n/4);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;private:template<bool inverse>void transform(size_t n,bool parity){auto butterfly=[&](size_t offset,size_t length,size_t begin,size_t end)__attribute__((always_inline)){size_t i=length/4;point rt=root(16*n/length)*eval_point(4*offset/length);vpoint v1={vz+real(rt),inverse?vz-imag(rt):vz+imag(rt)};vpoint v2=v1*v1,v3=v1*v2;for(size_t j=offset+begin;j<offset+end;j+=flen){auto A=at(j),B=at(j+i),C=at(j+2*i),D=at(j+3*i);if constexpr(inverse){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;}else{B=B*v1;C=C*v2;D=D*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);}}};auto recurse=[&](auto&&self,size_t offset,size_t length)->void{if(length<4*flen){return;}if(length>=(1<<15)){size_t step=length/16;if constexpr(inverse){for(size_t t=0;t<16;t++){self(self,offset+t*step,step);}}for(size_t j=0;j<step;j+=256){size_t end=std::min(step,j+256);if constexpr(inverse){for(size_t t=0;t<4;t++){butterfly(offset+t*length/4,length/4,j,end);}for(size_t t=0;t<4;t++){butterfly(offset,length,j+t*step,end+t*step);}}else{for(size_t t=0;t<4;t++){butterfly(offset,length,j+t*step,end+t*step);}for(size_t t=0;t<4;t++){butterfly(offset+t*length/4,length/4,j,end);}}}if constexpr(!inverse){for(size_t t=0;t<16;t++){self(self,offset+t*step,step);}}}else{if constexpr(inverse){for(size_t leaf=offset+3*flen;leaf<offset+length;leaf+=4*flen){size_t level=std::min<size_t>(std::countr_one(leaf+3),std::countr_zero(length));for(size_t lvl=4+parity;lvl<=level;lvl+=2){size_t len=size_t(1)<<lvl;butterfly(leaf/len*len,len,0,len/4);}}}else{for(size_t leaf=offset;leaf<offset+length;leaf+=4*flen){size_t level=std::min<size_t>(std::countr_zero(n+leaf),std::countr_zero(length));level-=level%2!=parity;for(size_t lvl=level;lvl>=4;lvl-=2){size_t len=size_t(1)<<lvl;butterfly(leaf/len*len,len,0,len/4);}}}}};recurse(recurse,0,n);}static big_vector<point>extra;static void prepare_roots(size_t n){if(n<=pre_evals+extra.size()){return;}size_t old=extra.size();extra.resize(std::bit_ceil(n)-pre_evals);for(size_t i=old;i<extra.size();i++){size_t j=4*(i+pre_evals);extra[i]=polar<ftype>(1.,std::numbers::pi/(ftype)std::bit_floor(j)*(ftype)eval_arg(j));}}};big_vector<point>cvector::extra;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/dft.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*(1.0/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(0),B(0){A.r.clear();B.r.clear();size_t blocks=std::max(flen,std::bit_ceil(n))/flen;A.r.reserve(blocks);B.r.reserve(blocks);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.r.emplace_back(rai,rani);B.r.emplace_back(qai,qani);cur=montgomery_mul(cur,step4,mod,imod);}A.r.resize(blocks);B.r.resize(blocks);checkpoint("dft init");if(n){if(partial){A.fft();B.fft();}else{A.template fft<false>();B.template fft<false>();}}}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);vpoint vrt={vz+real(rt),vz+imag(rt)};auto Cr=Cv*vrt,Dr=Dv*vrt;auto iter=[&]<int i>()__attribute__((always_inline)){auto wrap=[&](vftype original,vftype rotated){if constexpr(i==0){return original;}else{return __builtin_shufflevector(rotated,original,4-i,5-i,6-i,7-i);}};vpoint Cw={wrap(real(Cv),real(Cr)),wrap(imag(Cv),imag(Cr))};vpoint Dw={wrap(real(Dv),real(Dr)),wrap(imag(Dv),imag(Dr))};vpoint Av={vz+Ax[i],vz+Ay[i]},Bv={vz+Bx[i],vz+By[i]};AC+=Av*Cw;AD+=Av*Dw;BC+=Bv*Cw;BD+=Bv*Dw;};iter.template operator()<0>();iter.template operator()<1>();iter.template operator()<2>();iter.template operator()<3>();}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::mod())<<31);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]));}}template<bool normalized=true>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 scale=vz+ftype(flen)/ftype(n);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 get=[&](auto const&x){if constexpr(normalized){return x.at(i);}else{return x.at(i)*scale;}};auto[Ax,Ay]=get(A);auto[Bx,By]=get(B);auto[Cx,Cy]=get(C);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.template ifft<true,false>();B.template ifft<true,false>();C.template ifft<true,false>();recover_mod<false>(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={};}
#pragma GCC pop_options
#line 4 "cp-algo/math/fft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft{void mul_slow(auto&a,auto const&b,size_t k){if(!std::empty(a)&&std::data(a)==std::data(b)){using base=std::decay_t<decltype(a[0])>;size_t n=std::min(k,std::size(a)),m=std::min(k,std::size(b));if(!m){a.clear();return;}a.resize(k);for(size_t j=k;j-->0;){base sum=0;size_t lo=j>=n?j+1-n:0,hi=std::min(j+1,m);for(size_t i=lo;i<hi;i++){if(n==m&&i>j-i){break;}auto term=a[i]*a[j-i];sum+=n==m&&i!=j-i?term+term:term;}a[j]=sum;}return;}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);size_t as=std::min(k,std::size(a)),bs=std::min(k,std::size(b));size_t tail=as+bs-1-n;if(tail<=32&&as<=n&&bs<=n){std::array<base,32>high{};for(size_t i=0;i<tail;i++){for(size_t j=n+i-bs+1;j<as;j++){high[i]+=a[j]*b[n+i-j];}}auto A=dft<base>(a|std::views::take(k),n/2);if(as==bs&&std::data(a)==std::data(b)){a.resize((k+flen-1)/flen*flen);A.mul(A,a,std::min(k,n));}else{auto B=dft<base>(b|std::views::take(k),n/2);a.resize((k+flen-1)/flen*flen);A.mul_inplace(B,a,std::min(k,n));}auto wrap=bpow(dft<base>::factor,n);for(size_t i=0;i<tail;i++){a[i]+=wrap*high[i];if(n+i<k){a[n+i]=high[i];}}a.resize(k);return;}auto A=dft<base>(a|std::views::take(k),n);if(as==bs&&std::data(a)==std::data(b)){a.resize((k+flen-1)/flen*flen);A.mul(A,a,k);}else{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);}template<bool inverse=false>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()};if constexpr(!inverse){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;if constexpr(inverse){xl=(xl+(xl&1)*base::mod())>>1;xr=montgomery_mul(xr,cur,dft<base>::mod,dft<base>::imod);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(inverse?"mod join":"mod split");}void cyclic_mul(auto&a,auto&&b,size_t k,bool zero_upper=false){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();bool square=std::data(a)==std::data(b);if(k<=(1<<16)){big_vector<base>ap(begin(a),end(a));if(square){mul_truncate(ap,ap,2*k);}else{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);if(zero_upper){std::ranges::copy(std::span(a).first(k),begin(a)+k);if(!square){std::ranges::copy(std::span(b).first(k),begin(b)+k);}}else{mod_split(a,k,factor);if(!square){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);if(square){A.mul(A,ra,k);}else{auto B=dft<base>(rb,k/2);A.mul_inplace(B,ra,k);}base i2=base(2).inv();factor=factor.inv()*i2;mod_split<true>(a,k,factor);}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);}namespace impl{void mul_unbalanced(auto&a,auto const&b){using base=std::decay_t<decltype(a[0])>;auto x=std::span<base const>(a),y=std::span<base const>(b);if(x.size()<y.size()){std::swap(x,y);}constexpr size_t length=1<<15;size_t step=length-y.size()+1;auto fixed=dft<base>(y,length/2);std::decay_t<decltype(a)>result(x.size()+y.size()-1);big_vector<base>work(length);for(size_t start=0;start<x.size();start+=step){size_t count=std::min(step,x.size()-start);auto block=dft<base>(x.subspan(start,count),length/2);size_t need=count+y.size()-1;block.mul(fixed,work,need);for(size_t i=0;i<need;i++){result[start+i]+=work[i];}}a=std::move(result);}}void mul(auto&a,auto&&b){if(std::empty(a)||std::empty(b)){a.clear();return;}bool square=std::data(a)==std::data(b)&&std::size(a)==std::size(b);if(!square&&std::data(a)==std::data(b)){auto copy=make_copy(b);return mul(a,copy);}size_t small=std::min(size(a),size(b)),large=std::max(size(a),size(b));if(small>=magic&&small<=4096&&large>=(1<<20)&&large/small>=64){return impl::mul_unbalanced(a,b);}using base=std::decay_t<decltype(a[0])>;size_t N=size(a)+size(b);if(N>(1<<20)){N--;size_t NN=std::bit_ceil(N);bool zero_upper=std::max(size(a),size(b))<=NN/2;a.resize(NN);if(zero_upper&&!square){size_t half=NN/2;b.resize(half);auto lo=std::span(a).first(half),hi=std::span(a).last(half);{auto A=dft<base>(lo,half/2);auto B=dft<base>(b,half/2);A.mul_inplace(B,hi,half);}cyclic_mul(lo,b,half);mod_split<true>(a,half,(base(2)*bpow(dft<base>::factor,half)).inv());}else{if(!square){b.resize(NN);}cyclic_mul(a,b,NN,zero_upper);}a.resize(N);}else{mul_truncate(a,b,N-1);}}void mul(auto&a,auto const&b){if(std::empty(a)||std::empty(b)){a.clear();return;}size_t small=std::min(size(a),size(b)),large=std::max(size(a),size(b));if(small>=magic&&small<=4096&&large>=(1<<20)&&large/small>=64){return impl::mul_unbalanced(a,b);}size_t N=size(a)+size(b);if(N>(1<<20)){if(std::data(a)==std::data(b)&&std::size(a)==std::size(b)){mul(a,a);}else{mul(a,make_copy(b));}}else{mul_truncate(a,b,N-1);}}}
#pragma GCC pop_options
#line 4 "cp-algo/math/poly/base.hpp"
#include <array>
#line 6 "cp-algo/math/poly/base.hpp"
#include <utility>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<typename T>struct poly_t;template<typename T>std::array<poly_t<T>,2>divmod(poly_t<T>p,poly_t<T>const&q);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()=default;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;}friend poly_t operator-(poly_t p){p.negate_inplace();return p;}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();}friend poly_t operator+(poly_t p,poly_t const&t){p+=t;return p;}friend poly_t operator-(poly_t p,poly_t const&t){p-=t;return p;}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){if(is_zero()){return*this;}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 substr(0,k);}poly_t mod_xk(size_t k)&&{mod_xk_inplace(k);return std::move(*this);}poly_t mul_xk(size_t k)const&{auto p=*this;p.mul_xk_inplace(k);return p;}poly_t mul_xk(size_t k)&&{mul_xk_inplace(k);return std::move(*this);}poly_t div_xk(int64_t k)const&{return k<0?mul_xk(-k):substr(k,a.size());}poly_t div_xk(int64_t k)&&{div_xk_inplace(k);return std::move(*this);}poly_t substr(size_t l,size_t k)const&{l=std::min(l,a.size());k=std::min(k,a.size()-l);return Vector(begin(a)+l,begin(a)+l+k);}poly_t substr(size_t l,size_t k)&&{substr_inplace(l,k);return std::move(*this);}poly_t&operator*=(const poly_t&t){fft::mul(a,t.a);normalize();return*this;}friend poly_t operator*(poly_t p,const poly_t&t){p*=t;return p;}poly_t&operator/=(const poly_t&t){assert(!t.is_zero());if(this==&t){return*this=T(1);}auto[q,r]=divmod(std::move(*this),t);return*this=std::move(q);}poly_t&operator%=(const poly_t&t){assert(!t.is_zero());if(this==&t){a.clear();return*this;}auto[q,r]=divmod(std::move(*this),t);return*this=std::move(r);}friend poly_t operator/(poly_t p,poly_t const&t){p/=t;return p;}friend poly_t operator%(poly_t p,poly_t const&t){p%=t;return p;}poly_t&operator*=(T const&x){for(auto&it:a){it*=x;}return normalize();}poly_t&operator/=(T const&x){return*this*=x.inv();}friend poly_t operator*(poly_t p,T const&x){p*=x;return p;}friend poly_t operator/(poly_t p,T const&x){p/=x;return p;}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&{auto p=*this;p.reverse(n);return p;}poly_t reversed(size_t n)&&{reverse(n);return std::move(*this);}poly_t reversed()const&{return reversed(a.size());}poly_t reversed()&&{reverse();return std::move(*this);}friend poly_t operator*(T const&x,poly_t p){p*=x;return p;}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;}size_t trailing_xk()const{if(is_zero()){return-1;}int res=0;while(a[res]==T(0)){res++;}return res;}poly_t&mul_truncate(poly_t const&t,size_t k){fft::mul_truncate(a,t.a,k);return normalize();}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);}poly_t x2()const{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{std::move(res[0]),std::move(res[1])};}std::array<poly_t,2>bisect()const{return bisect(size(a));}};}
#pragma GCC pop_options
#line 1 "cp-algo/math/factorials.hpp"
#line 1 "cp-algo/util/bump_alloc.hpp"
#line 5 "cp-algo/util/bump_alloc.hpp"
namespace cp_algo{template<class T,size_t max_len>struct bump_alloc{static char*buf;static size_t buf_ind;using value_type=T;template<class U>struct rebind{using other=bump_alloc<U,max_len>;};constexpr bool operator==(const bump_alloc&)const=default;constexpr bool operator!=(const bump_alloc&)const=default;bump_alloc()=default;template<class U>bump_alloc(const U&){}T*allocate(size_t n){buf_ind-=n*sizeof(T);buf_ind&=0-alignof(T);return(T*)(buf+buf_ind);}void deallocate(T*,size_t){}};template<class T,size_t max_len>char*bump_alloc<T,max_len>::buf=big_alloc<char>().allocate(max_len*sizeof(T));template<class T,size_t max_len>size_t bump_alloc<T,max_len>::buf_ind=max_len*sizeof(T);}
#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 9 "cp-algo/math/factorials.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<bool use_bump_alloc=false,int maxn=-1>auto facts(auto const&args){static_assert(!use_bump_alloc||maxn>0,"maxn must be set if use_bump_alloc is true");constexpr int max_mod=1'000'000'000;constexpr int accum=4;constexpr int simd_size=8;constexpr int block=1<<18;constexpr int subblock=block/simd_size;using base=std::decay_t<decltype(args[0])>;static_assert(modint_type<base>,"Base type must be a modint type");using T=std::array<int,2>;using alloc=std::conditional_t<use_bump_alloc,bump_alloc<T,30*maxn>,big_alloc<T>>;std::basic_string<T,std::char_traits<T>,alloc>odd_args_per_block[max_mod/subblock];std::basic_string<T,std::char_traits<T>,alloc>reg_args_per_block[max_mod/subblock];constexpr int limit_reg=max_mod/64;int limit_odd=0;big_vector<base>res(size(args),1);const int mod=base::mod();const int imod=-math::inv2(mod);for(auto[i,xy]:std::views::zip(args,res)|std::views::enumerate){auto[x,y]=xy;int t=x.getr();if(t>=mod/2){t=mod-t-1;y=t%2?1:mod-1;}auto pw=32ull*(t+1);while(t>limit_reg){limit_odd=std::max(limit_odd,(t-1)/2);odd_args_per_block[(t-1)/2/subblock].push_back({int(i),(t-1)/2});t/=2;pw+=t;}reg_args_per_block[t/subblock].push_back({int(i),t});y*=pow_fixed<base,2>(int(pw%(mod-1)));}checkpoint("init");base bi2x32=pow_fixed<base,2>(32).inv();auto process=[&](int limit,auto&args_per_block,auto step,auto&&proj){base fact=1;for(int b=0;b<=limit;b+=accum*block){u32x8 cur[accum];static std::array<u32x8,subblock>prods[accum];for(int z=0;z<accum;z++){for(int j=0;j<simd_size;j++){
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
cur[z][j]=uint32_t(b+z*block+j*subblock);cur[z][j]=proj(cur[z][j]);prods[z][0][j]=cur[z][j]+!cur[z][j];prods[z][0][j]=uint32_t(uint64_t(prods[z][0][j])*bi2x32.getr()%mod);
#pragma GCC diagnostic pop
}}for(int i=1;i<block/simd_size;i++){for(int z=0;z<accum;z++){cur[z]+=step;prods[z][i]=montgomery_mul(prods[z][i-1],cur[z],mod,imod);}}checkpoint("inner loop");for(int z=0;z<accum;z++){for(int j=0;j<simd_size;j++){int bl=b+z*block+j*subblock;for(auto[i,x]:args_per_block[bl/subblock]){res[i]*=fact*prods[z][x-bl][j];}fact*=base(prods[z].back()[j]);}}checkpoint("mul ans");}};process(limit_reg,reg_args_per_block,1,std::identity{});process(limit_odd,odd_args_per_block,2,[](uint32_t x){return 2*x+1;});auto invs=bulk_invs<base>(res);for(auto[i,x]:res|std::views::enumerate){if(args[i]>=mod/2){x=invs[i];}}checkpoint("inv ans");return res;}}
#pragma GCC pop_options
#line 5 "cp-algo/math/poly/calculus.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<typename T>poly_t<T>deriv(poly_t<T>p,int k=1){assert(k>=0);if(k>p.deg()){return k==0?p:poly_t<T>{};}if(k==0){return p;}if(k==1){for(int i=1;i<=p.deg();i++){p.a[i-1]=T(i)*p.a[i];}p.a.pop_back();p.normalize();return p;}for(int i=k;i<=p.deg();i++){p.a[i-k]=fact<T>(i)*rfact<T>(i-k)*p.a[i];}p.a.resize(p.a.size()-k);p.normalize();return p;}template<typename T>poly_t<T>integr(poly_t<T>p){if(p.is_zero()){return p;}p.a.push_back(0);for(int i=p.deg()-1;i>=0;i--){p.a[i+1]=p.a[i]*small_inv<T>(i+1);}p.a[0]=0;return p;}}
#pragma GCC pop_options
#line 1 "cp-algo/math/poly/series/inv.hpp"
#line 4 "cp-algo/math/poly/series/inv.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::poly::impl{template<typename poly>poly&inv_inplace(poly&p,size_t n){using base=poly::base;if(n==0){p.a.clear();return p;}assert(p[0]!=base(0));if(n<magic){typename poly::Vector q(n);q[0]=base(1)/p[0];for(size_t i=1;i<n;i++){for(size_t j=1;j<=std::min(i,p.a.size()-1);j++){q[i]-=p.a[j]*q[i-j];}q[i]*=q[0];}return p=std::move(q);}size_t m=std::bit_floor(size_t(magic-1));auto q=p.mod_xk(m);inv_inplace(q,m);for(;m<n;m*=2){size_t k=std::min(2*m,n);typename poly::Vector error((k+fft::flen-1)/fft::flen*fft::flen);auto Q=fft::dft<base>(q.a,m);{auto P=fft::dft<base>(p.a|std::views::take(k),m);P.mul(Q,error,k);}auto E=fft::dft<base>(error|std::views::drop(m)|std::views::take(k-m),m);Q.mul_inplace(E,error,k-m);q.a.resize(k);for(size_t i=m;i<k;i++){q.a[i]=-error[i-m];}}p=std::move(q);p.normalize();return p;}}namespace cp_algo::math{template<typename T>poly_t<T>inv(poly_t<T>p,size_t n){poly::impl::inv_inplace(p,n);return p;}}
#pragma GCC pop_options
#line 5 "cp-algo/math/poly/transform.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{template<typename T>poly_t<T>mulx(poly_t<T>p,std::type_identity_t<T>c){T cur=1;for(auto&x:p.a){x*=cur;cur*=c;}p.normalize();return p;}template<typename T>poly_t<T>mulx_sq(poly_t<T>p,std::type_identity_t<T>c){T cur=1,total=1;for(auto&x:p.a){x*=total;cur*=c;total*=cur;}p.normalize();return p;}template<typename T>poly_t<T>invborel(poly_t<T>p){for(int i=0;i<=p.deg();i++){p.a[i]*=fact<T>(i);}return p;}template<typename T>poly_t<T>borel(poly_t<T>p){for(int i=0;i<=p.deg();i++){p.a[i]*=rfact<T>(i);}return p;}template<typename T>poly_t<T>expx(size_t n){return borel(poly_t<T>::ones(n));}template<typename T>poly_t<T>log1px(size_t n){typename poly_t<T>::Vector a(n);for(size_t i=1;i<n;i++){a[i]=(i&1?small_inv<T>(i):-small_inv<T>(i));}return a;}template<typename T>poly_t<T>log1mx(size_t n){return n?-integr(poly_t<T>::ones(n-1)):poly_t<T>{};}template<typename T>poly_t<T>corr(poly_t<T>a,poly_t<T>b){b.reverse();a*=b;return a;}template<typename T>poly_t<T>forward_corr(poly_t<T>a,poly_t<T>b){if(b.is_zero()){return{};}int d=b.deg();return corr(std::move(a),std::move(b)).div_xk(d);}template<typename T>poly_t<T>inner_corr(poly_t<T>a,poly_t<T>b){if(b.is_zero()||a.deg()<b.deg()){return{};}int d=b.deg(),n=a.deg()-d+1;size_t N=std::max(fft::flen,std::bit_ceil(a.a.size()));std::ranges::reverse(b.a);a.a.resize(N);b.a.resize(N);fft::cyclic_mul(a.a,b.a,N);a.substr_inplace(d,n);return a;}template<typename T>poly_t<T>apply_diff(poly_t<T>p,poly_t<T>g){return borel(forward_corr(invborel(std::move(p)),std::move(g)));}template<typename T>poly_t<T>shift(poly_t<T>p,std::type_identity_t<T>c){auto g=mulx(expx<T>(p.deg()+1),c);return apply_diff(std::move(p),std::move(g));}template<typename T>poly_t<T>prefix_sum(poly_t<T>p){auto g=inv(expx<T>(p.deg()+2).div_xk(1),p.deg()+1);return integr(apply_diff(std::move(p),std::move(g)));}}
#pragma GCC pop_options

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