CP-Algorithms Library

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:warning: cp-algo/math/laurent.hpp

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Code

#ifndef CP_ALGO_MATH_LAURENT_HPP
#define CP_ALGO_MATH_LAURENT_HPP
#include "../util/big_alloc.hpp"
#include <memory>
#include <optional>
#include <vector>
#include <cassert>
#include <algorithm>
#include <type_traits>
#include <bit>

#include "cvector.hpp"
#include "convolution.hpp"

namespace cp_algo::math {
    // Base provider interface for lazy coefficient evaluation
    template<typename T>
    struct provider {
        mutable big_vector<T> cache;
        mutable int cache_offset = 0; // Index of first cached coefficient
        mutable bool initialized = false;
        mutable bool all_cached = false; // True if all non-zero coeffs are cached
        
        virtual ~provider() = default;
        virtual int offset() const { return 0; }
        
        // Returns true if this provider requires lazy evaluation (coefficients must be
        // computed in order). False means dependencies can be bulk-cached for FFT.
        // Examples: multiply needs lazy eval, add/subtract/negate/scale don't.
        virtual bool needs_lazy_eval() const { return false; }
        
        // Compute k-th coefficient lazily without caching
        virtual T coeff_lazy(int k) const = 0;
        
        // Double the number of known coefficients (or cache all for finite series)
        // Default: use coeff_lazy, but can be overridden for efficiency (e.g., FFT)
        virtual void double_up() const {
            int old_size = cache.size();
            int new_size = old_size == 0 ? 1 : 2 * old_size;
            
            cache.resize(new_size);
            for(int i = old_size; i < new_size; i++) {
                cache[i] = coeff_lazy(cache_offset + i);
            }
        }
        
        // Get coefficient with caching and doubling (default implementation)
        virtual T coeff(int k) const {
            if(!initialized) {
                cache_offset = offset();
                initialized = true;
            }
            
            int idx = k - cache_offset;
            if(idx < 0) {
                return T(0); // Below cached range
            }
            
            // If all coeffs are cached and we're beyond cache, return 0
            if(all_cached && idx >= (int)cache.size()) {
                return T(0);
            }
            
            if(needs_lazy_eval()) {
                // Sequentially extend cache to the requested index
                while(idx >= (int)cache.size() && !all_cached) {
                    int next_k = cache_offset + (int)cache.size();
                    cache.push_back(coeff_lazy(next_k));
                }
            } else {
                // Extend cache by doubling until we have enough
                while(idx >= (int)cache.size() && !all_cached) {
                    double_up();
                }
            }
            
            if(idx < (int)cache.size()) {
                return cache[idx];
            }
            
            return T(0);
        }

        // Alias for backwards compatibility
        T get(int k) const {
            return coeff(k);
        }
    };
    
    // Constant provider - returns a single coefficient at position offset
    template<typename T>
    struct constant_provider : provider<T> {
        T value;
        int offset;
        
        constant_provider(T value, int offset = 0) : value(value), offset(offset) {}
        
        int offset() const override {
            return offset;
        }
        
        T coeff_lazy(int k) const override {
            return k == offset ? value : T(0);
        }
        
        T coeff(int k) const override {
            return coeff_lazy(k);
        }
    };
    
    // Polynomial provider - wraps a vector of coefficients
    template<typename T>
    struct polynomial_provider : provider<T> {
        polynomial_provider(big_vector<T> coeffs, int offset = 0) {
            // Find first and last non-zero coefficients
            auto non_zero = [](const T& x) { return x != T(0); };
            auto first = std::ranges::find_if(coeffs, non_zero);
            auto last = std::ranges::find_if(coeffs | std::views::reverse, non_zero);
            
            // Extract non-zero range
            if(first != coeffs.end()) {
                int leading_zeros = first - coeffs.begin();
                int trailing_zeros = last - coeffs.rbegin();
                coeffs = big_vector<T>(first, coeffs.end() - trailing_zeros);
                offset += leading_zeros;
            } else {
                // All zeros
                coeffs.clear();
            }
            
            // Initialize cache directly with the coefficients
            this->cache = std::move(coeffs);
            this->cache_offset = offset;
            this->initialized = true;
            this->all_cached = true;
        }
        
        int offset() const override {
            return this->cache_offset;
        }
        
        T coeff_lazy(int k) const override {
            int idx = k - this->cache_offset;
            if(idx < 0 || idx >= (int)this->cache.size()) {
                return T(0);
            }
            return this->cache[idx];
        }
        
        T coeff(int k) const override {
            return coeff_lazy(k);
        }
    };
    
    // Base class for unary operations
    template<typename T>
    struct unary_provider : provider<T> {
        std::shared_ptr<provider<T>> operand;
        
        unary_provider(std::shared_ptr<provider<T>> operand)
            : operand(std::move(operand)) {}
        
        virtual T transform(T const& a) const = 0;
        
        int offset() const override {
            return operand->offset();
        }
        
        T coeff_lazy(int k) const override {
            return transform(operand->coeff_lazy(k));
        }
        
        T coeff(int k) const {
            return transform(operand->coeff(k));
        }
    };
    
    // Base class for binary operations
    template<typename T>
    struct binary_provider : provider<T> {
        std::shared_ptr<provider<T>> lhs, rhs;
        
        binary_provider(std::shared_ptr<provider<T>> lhs, std::shared_ptr<provider<T>> rhs)
            : lhs(std::move(lhs)), rhs(std::move(rhs)) {}
        
        virtual T combine(T const& a, T const& b) const = 0;
        
        int offset() const override {
            return std::min(lhs->offset(), rhs->offset());
        }
        
        T coeff_lazy(int k) const override {
            return combine(lhs->coeff_lazy(k), rhs->coeff_lazy(k));
        }
        
        T coeff(int k) const {
            return combine(lhs->coeff(k), rhs->coeff(k));
        }
    };
    
    // Addition provider
    template<typename T>
    struct add_provider : binary_provider<T> {
        using binary_provider<T>::binary_provider;
        
        T combine(T const& a, T const& b) const override {
            return a + b;
        }
    };
    
    // Subtraction provider
    template<typename T>
    struct subtract_provider : binary_provider<T> {
        using binary_provider<T>::binary_provider;
        
        T combine(T const& a, T const& b) const override {
            return a - b;
        }
    };
    
    // Negation provider
    template<typename T>
    struct negate_provider : unary_provider<T> {
        using unary_provider<T>::unary_provider;
        
        T transform(T const& a) const override {
            return -a;
        }
    };
    
    // Scalar multiplication provider
    template<typename T>
    struct scale_provider : unary_provider<T> {
        T scalar;
        
        scale_provider(std::shared_ptr<provider<T>> operand, T scalar)
            : unary_provider<T>(std::move(operand)), scalar(scalar) {}
        
        T transform(T const& a) const override {
            return a * scalar;
        }
    };
    
    // Multiplication provider (Cauchy product)
    template<typename T>
    struct multiply_provider : provider<T> {
        std::shared_ptr<provider<T>> lhs, rhs;
        
        multiply_provider(std::shared_ptr<provider<T>> lhs, std::shared_ptr<provider<T>> rhs)
            : lhs(std::move(lhs)), rhs(std::move(rhs)) {}
        
        int offset() const override {
            return lhs->offset() + rhs->offset();
        }
        
        bool needs_lazy_eval() const override {
            return lhs->needs_lazy_eval() || rhs->needs_lazy_eval();
        }
        
        T coeff_lazy(int k) const override {
            int n = k - offset();
            if(n < 0) return T(0);
            T result = T(0);
            bool lazy_lhs = lhs->needs_lazy_eval();
            bool lazy_rhs = rhs->needs_lazy_eval();
            for(int j = 0; j <= n; j++) {
                int i_l = lhs->offset() + j;
                int i_r = rhs->offset() + (n - j);
                auto a = lazy_lhs ? lhs->coeff(i_l) : lhs->coeff_lazy(i_l);
                auto b = lazy_rhs ? rhs->coeff(i_r) : rhs->coeff_lazy(i_r);
                result += a * b;
            }
            return result;
        }

        void double_up() const override {
            int old_size = this->cache.size();
            int new_size = old_size == 0 ? 1 : 2 * old_size;

            // Lazy path: compute the next coefficient sequentially
            if(needs_lazy_eval()) {
                int k = this->cache_offset + old_size;
                this->cache.push_back(coeff_lazy(k));
                return;
            }

            // Ensure operands have enough cached coefficients for the prefix we need
            int lhs_need = lhs->offset() + new_size - 1;
            int rhs_need = rhs->offset() + new_size - 1;
            lhs->coeff(lhs_need);
            rhs->coeff(rhs_need);

            // Build aligned prefixes starting at operand offsets
            big_vector<T> la(new_size), rb(new_size);
            for(int i = 0; i < new_size; i++) {
                la[i] = lhs->coeff(lhs->offset() + i);
                rb[i] = rhs->coeff(rhs->offset() + i);
            }

            this->cache.resize(new_size);
            convolution_prefix(la, rb, new_size);
            for(int i = old_size; i < new_size && i < (int)la.size(); i++) {
                this->cache[i] = la[i];
            }

            // If both operands are fully cached and we reached their total length, mark as done
            if(lhs->all_cached && rhs->all_cached) {
                size_t total_len = lhs->cache.size() + rhs->cache.size() - 1;
                if((size_t)new_size >= total_len) {
                    this->cache.resize(total_len);
                    this->all_cached = true;
                }
            }
        }
    };
    
    // Main Laurent series class
    template<typename T>
    struct laurent {
        std::shared_ptr<provider<T>> impl;
        
        laurent() : impl(std::make_shared<constant_provider<T>>(T(0), 0)) {}
        
        laurent(T value, int offset = 0) 
            : impl(std::make_shared<constant_provider<T>>(value, offset)) {}
        
        laurent(big_vector<T> coeffs, int offset = 0)
            : impl(std::make_shared<polynomial_provider<T>>(std::move(coeffs), offset)) {}
        
        laurent(std::shared_ptr<provider<T>> impl) : impl(std::move(impl)) {}
        
        // Get k-th coefficient (delegates to provider's caching)
        T operator[](int k) const {
            return impl->get(k);
        }
        
        // Arithmetic operations
        laurent operator-() const {
            return std::make_shared<negate_provider<T>>(impl);
        }
        
        laurent operator+(const laurent& other) const {
            return std::make_shared<add_provider<T>>(impl, other.impl);
        }
        
        laurent operator-(const laurent& other) const {
            return std::make_shared<subtract_provider<T>>(impl, other.impl);
        }
        
        laurent operator*(const laurent& other) const {
            return std::make_shared<multiply_provider<T>>(impl, other.impl);
        }
        
        laurent& operator+=(const laurent& other) {
            return *this = *this + other;
        }
        
        laurent& operator-=(const laurent& other) {
            return *this = *this - other;
        }
        
        laurent& operator*=(const laurent& other) {
            return *this = *this * other;
        }
        
        // Scalar multiplication
        laurent operator*(T const& scalar) const {
            return std::make_shared<scale_provider<T>>(impl, scalar);
        }
        
        laurent& operator*=(T const& scalar) {
            return *this = *this * scalar;
        }
    };
    
    template<typename T>
    laurent<T> operator*(T const& scalar, laurent<T> const& series) {
        return series * scalar;
    }
}

#endif // CP_ALGO_MATH_LAURENT_HPP

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


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



#include <map>
#include <deque>
#include <vector>
#include <string>
#include <cstddef>
#include <iostream>

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

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

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

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

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

    template<typename T>
    using big_vector = std::vector<T, big_alloc<T>>;
    template<typename T>
    using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
    template<typename T>
    using big_deque = std::deque<T, big_alloc<T>>;
    template<typename Key, typename Value, typename Compare = std::less<Key>>
    using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
    using big_string = big_basic_string<char>;
}

#line 4 "cp-algo/math/laurent.hpp"
#include <memory>
#include <optional>
#line 7 "cp-algo/math/laurent.hpp"
#include <cassert>
#include <algorithm>
#include <type_traits>
#include <bit>

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


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


#include <experimental/simd>
#include <cstdint>
#line 7 "cp-algo/util/simd.hpp"
#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") \
    _Pragma("GCC optimize(\"O3,unroll-loops\")") \
    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 i64x4 = simd<int64_t, 4>;
    using u64x4 = simd<uint64_t, 4>;
    using u32x8 = simd<uint32_t, 8>;
    using i32x4 = simd<int32_t, 4>;
    using u32x4 = simd<uint32_t, 4>;
    using i16x4 = simd<int16_t, 4>;
    using u8x32 = simd<uint8_t, 32>;
    using dx4 = simd<double, 4>;

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


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

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

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

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

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

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

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

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

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


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


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


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


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

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

#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {

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

        // Only use these if you really know what you're doing!
        static UInt modmod8() {return UInt(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/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 9 "cp-algo/math/fft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft {
    template<modint_type base>
    struct dft {
        cvector A, B;
        static base factor, ifactor;
        using Int2 = base::Int2;
        static bool _init;
        static int split() {
            static const int splt = int(std::sqrt(base::mod())) + 1;
            return splt;
        }
        static uint32_t mod, imod;

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

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

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

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

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

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

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

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

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

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

#line 10 "cp-algo/math/convolution.hpp"

namespace cp_algo::math {

// Convolution limited to the first `need` coefficients.
// Writes the result into `a`; performs in-place when possible (modint path).
template<class VecA, class VecB>
void convolution_prefix(VecA& a, VecB const& b, size_t need) {
    using T = typename std::decay_t<VecA>::value_type;
    size_t na = std::min(need, std::size(a));
    size_t nb = std::min(need, std::size(b));
    a.resize(na);
    auto bv = b | std::views::take(nb);

    if(na == 0 || nb == 0) {
        a.clear();
        return;
    }

    if constexpr (modint_type<T>) {
        // Use NTT-based truncated multiplication. Works in-place on `a`.
        fft::mul_truncate(a, bv, need);
    } else if constexpr (std::is_same_v<T, fft::point>) {
        size_t conv_len = na + nb - 1;
        size_t n = std::bit_ceil(conv_len);
        n = std::max(n, (size_t)fft::flen);
        fft::cvector A(n), B(n);
        for(size_t i = 0; i < na; i++) {
            A.set(i, a[i]);
        }
        for(size_t i = 0; i < nb; i++) {
            B.set(i, bv[i]);
        }
        A.fft();
        B.fft();
        A.dot(B);
        A.ifft();
        a.assign(need, T(0));
        for(size_t i = 0; i < std::min(need, conv_len); i++) {
            a[i] = A.template get<fft::point>(i);
        }
    } else if constexpr (std::is_same_v<T, fft::ftype>) {
        // Imaginary-cyclic convolution modulo x^n-i to compute acyclic convolution
        // Represents polynomials as point(a[i], a[i+n]) to work in x^n-i basis
        size_t conv_len = na + nb - 1;
        size_t n = std::bit_ceil(conv_len) / 2;
        n = std::max(n, (size_t)fft::flen);
        
        fft::cvector A(n), B(n);
        // Pack as modulo x^n-i: A[i] = point(a[i], a[i+n])
        for(size_t i = 0; i < std::min(n, na); i++) {
            fft::ftype re = a[i], im = 0;
            if(i + n < na) im = a[i + n];
            A.set(i, fft::point(re, im));
        }
        for(size_t i = 0; i < std::min(n, nb); i++) {
            fft::ftype re = bv[i], im = 0;
            if(i + n < nb) im = bv[i + n];
            B.set(i, fft::point(re, im));
        }
        A.fft();
        B.fft();
        A.dot(B);
        A.ifft();
        a.assign(2 * n, T(0));
        for(size_t i = 0; i < n; i++) {
            auto v = A.template get<fft::point>(i);
            a[i] = v.real();
            a[i + n] = v.imag();
        }
        a.resize(need);
    } else {
        // Generic fallback: use simple O(n^2) convolution from fft utilities.
        fft::mul_slow(a, bv, need);
    }
}

} // namespace cp_algo::math


#line 14 "cp-algo/math/laurent.hpp"

namespace cp_algo::math {
    // Base provider interface for lazy coefficient evaluation
    template<typename T>
    struct provider {
        mutable big_vector<T> cache;
        mutable int cache_offset = 0; // Index of first cached coefficient
        mutable bool initialized = false;
        mutable bool all_cached = false; // True if all non-zero coeffs are cached
        
        virtual ~provider() = default;
        virtual int offset() const { return 0; }
        
        // Returns true if this provider requires lazy evaluation (coefficients must be
        // computed in order). False means dependencies can be bulk-cached for FFT.
        // Examples: multiply needs lazy eval, add/subtract/negate/scale don't.
        virtual bool needs_lazy_eval() const { return false; }
        
        // Compute k-th coefficient lazily without caching
        virtual T coeff_lazy(int k) const = 0;
        
        // Double the number of known coefficients (or cache all for finite series)
        // Default: use coeff_lazy, but can be overridden for efficiency (e.g., FFT)
        virtual void double_up() const {
            int old_size = cache.size();
            int new_size = old_size == 0 ? 1 : 2 * old_size;
            
            cache.resize(new_size);
            for(int i = old_size; i < new_size; i++) {
                cache[i] = coeff_lazy(cache_offset + i);
            }
        }
        
        // Get coefficient with caching and doubling (default implementation)
        virtual T coeff(int k) const {
            if(!initialized) {
                cache_offset = offset();
                initialized = true;
            }
            
            int idx = k - cache_offset;
            if(idx < 0) {
                return T(0); // Below cached range
            }
            
            // If all coeffs are cached and we're beyond cache, return 0
            if(all_cached && idx >= (int)cache.size()) {
                return T(0);
            }
            
            if(needs_lazy_eval()) {
                // Sequentially extend cache to the requested index
                while(idx >= (int)cache.size() && !all_cached) {
                    int next_k = cache_offset + (int)cache.size();
                    cache.push_back(coeff_lazy(next_k));
                }
            } else {
                // Extend cache by doubling until we have enough
                while(idx >= (int)cache.size() && !all_cached) {
                    double_up();
                }
            }
            
            if(idx < (int)cache.size()) {
                return cache[idx];
            }
            
            return T(0);
        }

        // Alias for backwards compatibility
        T get(int k) const {
            return coeff(k);
        }
    };
    
    // Constant provider - returns a single coefficient at position offset
    template<typename T>
    struct constant_provider : provider<T> {
        T value;
        int offset;
        
        constant_provider(T value, int offset = 0) : value(value), offset(offset) {}
        
        int offset() const override {
            return offset;
        }
        
        T coeff_lazy(int k) const override {
            return k == offset ? value : T(0);
        }
        
        T coeff(int k) const override {
            return coeff_lazy(k);
        }
    };
    
    // Polynomial provider - wraps a vector of coefficients
    template<typename T>
    struct polynomial_provider : provider<T> {
        polynomial_provider(big_vector<T> coeffs, int offset = 0) {
            // Find first and last non-zero coefficients
            auto non_zero = [](const T& x) { return x != T(0); };
            auto first = std::ranges::find_if(coeffs, non_zero);
            auto last = std::ranges::find_if(coeffs | std::views::reverse, non_zero);
            
            // Extract non-zero range
            if(first != coeffs.end()) {
                int leading_zeros = first - coeffs.begin();
                int trailing_zeros = last - coeffs.rbegin();
                coeffs = big_vector<T>(first, coeffs.end() - trailing_zeros);
                offset += leading_zeros;
            } else {
                // All zeros
                coeffs.clear();
            }
            
            // Initialize cache directly with the coefficients
            this->cache = std::move(coeffs);
            this->cache_offset = offset;
            this->initialized = true;
            this->all_cached = true;
        }
        
        int offset() const override {
            return this->cache_offset;
        }
        
        T coeff_lazy(int k) const override {
            int idx = k - this->cache_offset;
            if(idx < 0 || idx >= (int)this->cache.size()) {
                return T(0);
            }
            return this->cache[idx];
        }
        
        T coeff(int k) const override {
            return coeff_lazy(k);
        }
    };
    
    // Base class for unary operations
    template<typename T>
    struct unary_provider : provider<T> {
        std::shared_ptr<provider<T>> operand;
        
        unary_provider(std::shared_ptr<provider<T>> operand)
            : operand(std::move(operand)) {}
        
        virtual T transform(T const& a) const = 0;
        
        int offset() const override {
            return operand->offset();
        }
        
        T coeff_lazy(int k) const override {
            return transform(operand->coeff_lazy(k));
        }
        
        T coeff(int k) const {
            return transform(operand->coeff(k));
        }
    };
    
    // Base class for binary operations
    template<typename T>
    struct binary_provider : provider<T> {
        std::shared_ptr<provider<T>> lhs, rhs;
        
        binary_provider(std::shared_ptr<provider<T>> lhs, std::shared_ptr<provider<T>> rhs)
            : lhs(std::move(lhs)), rhs(std::move(rhs)) {}
        
        virtual T combine(T const& a, T const& b) const = 0;
        
        int offset() const override {
            return std::min(lhs->offset(), rhs->offset());
        }
        
        T coeff_lazy(int k) const override {
            return combine(lhs->coeff_lazy(k), rhs->coeff_lazy(k));
        }
        
        T coeff(int k) const {
            return combine(lhs->coeff(k), rhs->coeff(k));
        }
    };
    
    // Addition provider
    template<typename T>
    struct add_provider : binary_provider<T> {
        using binary_provider<T>::binary_provider;
        
        T combine(T const& a, T const& b) const override {
            return a + b;
        }
    };
    
    // Subtraction provider
    template<typename T>
    struct subtract_provider : binary_provider<T> {
        using binary_provider<T>::binary_provider;
        
        T combine(T const& a, T const& b) const override {
            return a - b;
        }
    };
    
    // Negation provider
    template<typename T>
    struct negate_provider : unary_provider<T> {
        using unary_provider<T>::unary_provider;
        
        T transform(T const& a) const override {
            return -a;
        }
    };
    
    // Scalar multiplication provider
    template<typename T>
    struct scale_provider : unary_provider<T> {
        T scalar;
        
        scale_provider(std::shared_ptr<provider<T>> operand, T scalar)
            : unary_provider<T>(std::move(operand)), scalar(scalar) {}
        
        T transform(T const& a) const override {
            return a * scalar;
        }
    };
    
    // Multiplication provider (Cauchy product)
    template<typename T>
    struct multiply_provider : provider<T> {
        std::shared_ptr<provider<T>> lhs, rhs;
        
        multiply_provider(std::shared_ptr<provider<T>> lhs, std::shared_ptr<provider<T>> rhs)
            : lhs(std::move(lhs)), rhs(std::move(rhs)) {}
        
        int offset() const override {
            return lhs->offset() + rhs->offset();
        }
        
        bool needs_lazy_eval() const override {
            return lhs->needs_lazy_eval() || rhs->needs_lazy_eval();
        }
        
        T coeff_lazy(int k) const override {
            int n = k - offset();
            if(n < 0) return T(0);
            T result = T(0);
            bool lazy_lhs = lhs->needs_lazy_eval();
            bool lazy_rhs = rhs->needs_lazy_eval();
            for(int j = 0; j <= n; j++) {
                int i_l = lhs->offset() + j;
                int i_r = rhs->offset() + (n - j);
                auto a = lazy_lhs ? lhs->coeff(i_l) : lhs->coeff_lazy(i_l);
                auto b = lazy_rhs ? rhs->coeff(i_r) : rhs->coeff_lazy(i_r);
                result += a * b;
            }
            return result;
        }

        void double_up() const override {
            int old_size = this->cache.size();
            int new_size = old_size == 0 ? 1 : 2 * old_size;

            // Lazy path: compute the next coefficient sequentially
            if(needs_lazy_eval()) {
                int k = this->cache_offset + old_size;
                this->cache.push_back(coeff_lazy(k));
                return;
            }

            // Ensure operands have enough cached coefficients for the prefix we need
            int lhs_need = lhs->offset() + new_size - 1;
            int rhs_need = rhs->offset() + new_size - 1;
            lhs->coeff(lhs_need);
            rhs->coeff(rhs_need);

            // Build aligned prefixes starting at operand offsets
            big_vector<T> la(new_size), rb(new_size);
            for(int i = 0; i < new_size; i++) {
                la[i] = lhs->coeff(lhs->offset() + i);
                rb[i] = rhs->coeff(rhs->offset() + i);
            }

            this->cache.resize(new_size);
            convolution_prefix(la, rb, new_size);
            for(int i = old_size; i < new_size && i < (int)la.size(); i++) {
                this->cache[i] = la[i];
            }

            // If both operands are fully cached and we reached their total length, mark as done
            if(lhs->all_cached && rhs->all_cached) {
                size_t total_len = lhs->cache.size() + rhs->cache.size() - 1;
                if((size_t)new_size >= total_len) {
                    this->cache.resize(total_len);
                    this->all_cached = true;
                }
            }
        }
    };
    
    // Main Laurent series class
    template<typename T>
    struct laurent {
        std::shared_ptr<provider<T>> impl;
        
        laurent() : impl(std::make_shared<constant_provider<T>>(T(0), 0)) {}
        
        laurent(T value, int offset = 0) 
            : impl(std::make_shared<constant_provider<T>>(value, offset)) {}
        
        laurent(big_vector<T> coeffs, int offset = 0)
            : impl(std::make_shared<polynomial_provider<T>>(std::move(coeffs), offset)) {}
        
        laurent(std::shared_ptr<provider<T>> impl) : impl(std::move(impl)) {}
        
        // Get k-th coefficient (delegates to provider's caching)
        T operator[](int k) const {
            return impl->get(k);
        }
        
        // Arithmetic operations
        laurent operator-() const {
            return std::make_shared<negate_provider<T>>(impl);
        }
        
        laurent operator+(const laurent& other) const {
            return std::make_shared<add_provider<T>>(impl, other.impl);
        }
        
        laurent operator-(const laurent& other) const {
            return std::make_shared<subtract_provider<T>>(impl, other.impl);
        }
        
        laurent operator*(const laurent& other) const {
            return std::make_shared<multiply_provider<T>>(impl, other.impl);
        }
        
        laurent& operator+=(const laurent& other) {
            return *this = *this + other;
        }
        
        laurent& operator-=(const laurent& other) {
            return *this = *this - other;
        }
        
        laurent& operator*=(const laurent& other) {
            return *this = *this * other;
        }
        
        // Scalar multiplication
        laurent operator*(T const& scalar) const {
            return std::make_shared<scale_provider<T>>(impl, scalar);
        }
        
        laurent& operator*=(T const& scalar) {
            return *this = *this * scalar;
        }
    };
    
    template<typename T>
    laurent<T> operator*(T const& scalar, laurent<T> const& series) {
        return series * scalar;
    }
}



#ifndef CP_ALGO_MATH_LAURENT_HPP
#define CP_ALGO_MATH_LAURENT_HPP
#include "../util/big_alloc.hpp"
#include <memory>
#include <optional>
#include <vector>
#include <cassert>
#include <algorithm>
#include <type_traits>
#include <bit>
#include "cvector.hpp"
#include "convolution.hpp"
namespace cp_algo::math{template<typename T>struct provider{mutable big_vector<T>cache;mutable int cache_offset=0;mutable bool initialized=false;mutable bool all_cached=false;virtual~provider()=default;virtual int offset()const{return 0;}virtual bool needs_lazy_eval()const{return false;}virtual T coeff_lazy(int k)const=0;virtual void double_up()const{int old_size=cache.size();int new_size=old_size==0?1:2*old_size;cache.resize(new_size);for(int i=old_size;i<new_size;i++){cache[i]=coeff_lazy(cache_offset+i);}}virtual T coeff(int k)const{if(!initialized){cache_offset=offset();initialized=true;}int idx=k-cache_offset;if(idx<0){return T(0);}if(all_cached&&idx>=(int)cache.size()){return T(0);}if(needs_lazy_eval()){while(idx>=(int)cache.size()&&!all_cached){int next_k=cache_offset+(int)cache.size();cache.push_back(coeff_lazy(next_k));}}else{while(idx>=(int)cache.size()&&!all_cached){double_up();}}if(idx<(int)cache.size()){return cache[idx];}return T(0);}T get(int k)const{return coeff(k);}};template<typename T>struct constant_provider:provider<T>{T value;int offset;constant_provider(T value,int offset=0):value(value),offset(offset){}int offset()const override{return offset;}T coeff_lazy(int k)const override{return k==offset?value:T(0);}T coeff(int k)const override{return coeff_lazy(k);}};template<typename T>struct polynomial_provider:provider<T>{polynomial_provider(big_vector<T>coeffs,int offset=0){auto non_zero=[](const T&x){return x!=T(0);};auto first=std::ranges::find_if(coeffs,non_zero);auto last=std::ranges::find_if(coeffs|std::views::reverse,non_zero);if(first!=coeffs.end()){int leading_zeros=first-coeffs.begin();int trailing_zeros=last-coeffs.rbegin();coeffs=big_vector<T>(first,coeffs.end()-trailing_zeros);offset+=leading_zeros;}else{coeffs.clear();}this->cache=std::move(coeffs);this->cache_offset=offset;this->initialized=true;this->all_cached=true;}int offset()const override{return this->cache_offset;}T coeff_lazy(int k)const override{int idx=k-this->cache_offset;if(idx<0||idx>=(int)this->cache.size()){return T(0);}return this->cache[idx];}T coeff(int k)const override{return coeff_lazy(k);}};template<typename T>struct unary_provider:provider<T>{std::shared_ptr<provider<T>>operand;unary_provider(std::shared_ptr<provider<T>>operand):operand(std::move(operand)){}virtual T transform(T const&a)const=0;int offset()const override{return operand->offset();}T coeff_lazy(int k)const override{return transform(operand->coeff_lazy(k));}T coeff(int k)const{return transform(operand->coeff(k));}};template<typename T>struct binary_provider:provider<T>{std::shared_ptr<provider<T>>lhs,rhs;binary_provider(std::shared_ptr<provider<T>>lhs,std::shared_ptr<provider<T>>rhs):lhs(std::move(lhs)),rhs(std::move(rhs)){}virtual T combine(T const&a,T const&b)const=0;int offset()const override{return std::min(lhs->offset(),rhs->offset());}T coeff_lazy(int k)const override{return combine(lhs->coeff_lazy(k),rhs->coeff_lazy(k));}T coeff(int k)const{return combine(lhs->coeff(k),rhs->coeff(k));}};template<typename T>struct add_provider:binary_provider<T>{using binary_provider<T>::binary_provider;T combine(T const&a,T const&b)const override{return a+b;}};template<typename T>struct subtract_provider:binary_provider<T>{using binary_provider<T>::binary_provider;T combine(T const&a,T const&b)const override{return a-b;}};template<typename T>struct negate_provider:unary_provider<T>{using unary_provider<T>::unary_provider;T transform(T const&a)const override{return-a;}};template<typename T>struct scale_provider:unary_provider<T>{T scalar;scale_provider(std::shared_ptr<provider<T>>operand,T scalar):unary_provider<T>(std::move(operand)),scalar(scalar){}T transform(T const&a)const override{return a*scalar;}};template<typename T>struct multiply_provider:provider<T>{std::shared_ptr<provider<T>>lhs,rhs;multiply_provider(std::shared_ptr<provider<T>>lhs,std::shared_ptr<provider<T>>rhs):lhs(std::move(lhs)),rhs(std::move(rhs)){}int offset()const override{return lhs->offset()+rhs->offset();}bool needs_lazy_eval()const override{return lhs->needs_lazy_eval()||rhs->needs_lazy_eval();}T coeff_lazy(int k)const override{int n=k-offset();if(n<0)return T(0);T result=T(0);bool lazy_lhs=lhs->needs_lazy_eval();bool lazy_rhs=rhs->needs_lazy_eval();for(int j=0;j<=n;j++){int i_l=lhs->offset()+j;int i_r=rhs->offset()+(n-j);auto a=lazy_lhs?lhs->coeff(i_l):lhs->coeff_lazy(i_l);auto b=lazy_rhs?rhs->coeff(i_r):rhs->coeff_lazy(i_r);result+=a*b;}return result;}void double_up()const override{int old_size=this->cache.size();int new_size=old_size==0?1:2*old_size;if(needs_lazy_eval()){int k=this->cache_offset+old_size;this->cache.push_back(coeff_lazy(k));return;}int lhs_need=lhs->offset()+new_size-1;int rhs_need=rhs->offset()+new_size-1;lhs->coeff(lhs_need);rhs->coeff(rhs_need);big_vector<T>la(new_size),rb(new_size);for(int i=0;i<new_size;i++){la[i]=lhs->coeff(lhs->offset()+i);rb[i]=rhs->coeff(rhs->offset()+i);}this->cache.resize(new_size);convolution_prefix(la,rb,new_size);for(int i=old_size;i<new_size&&i<(int)la.size();i++){this->cache[i]=la[i];}if(lhs->all_cached&&rhs->all_cached){size_t total_len=lhs->cache.size()+rhs->cache.size()-1;if((size_t)new_size>=total_len){this->cache.resize(total_len);this->all_cached=true;}}}};template<typename T>struct laurent{std::shared_ptr<provider<T>>impl;laurent():impl(std::make_shared<constant_provider<T>>(T(0),0)){}laurent(T value,int offset=0):impl(std::make_shared<constant_provider<T>>(value,offset)){}laurent(big_vector<T>coeffs,int offset=0):impl(std::make_shared<polynomial_provider<T>>(std::move(coeffs),offset)){}laurent(std::shared_ptr<provider<T>>impl):impl(std::move(impl)){}T operator[](int k)const{return impl->get(k);}laurent operator-()const{return std::make_shared<negate_provider<T>>(impl);}laurent operator+(const laurent&other)const{return std::make_shared<add_provider<T>>(impl,other.impl);}laurent operator-(const laurent&other)const{return std::make_shared<subtract_provider<T>>(impl,other.impl);}laurent operator*(const laurent&other)const{return std::make_shared<multiply_provider<T>>(impl,other.impl);}laurent&operator+=(const laurent&other){return*this=*this+other;}laurent&operator-=(const laurent&other){return*this=*this-other;}laurent&operator*=(const laurent&other){return*this=*this*other;}laurent operator*(T const&scalar)const{return std::make_shared<scale_provider<T>>(impl,scalar);}laurent&operator*=(T const&scalar){return*this=*this*scalar;}};template<typename T>laurent<T>operator*(T const&scalar,laurent<T>const&series){return series*scalar;}}
#endif

#line 1 "cp-algo/math/laurent.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <map>
#include <deque>
#include <vector>
#include <string>
#include <cstddef>
#include <iostream>
#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,std::size_t Align=32>class big_alloc{static_assert(Align>=alignof(void*),"Align must be at least pointer-size");static_assert(std::popcount(Align)==1,"Align must be a power of two");public:using value_type=T;template<class U>struct rebind{using other=big_alloc<U,Align>;};constexpr bool operator==(const big_alloc&)const=default;constexpr bool operator!=(const big_alloc&)const=default;big_alloc()noexcept=default;template<typename U,std::size_t A>big_alloc(const big_alloc<U,A>&)noexcept{}[[nodiscard]]T*allocate(std::size_t n){std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0);madvise(raw,padded,MADV_HUGEPAGE);madvise(raw,padded,MADV_POPULATE_WRITE);return static_cast<T*>(raw);}
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align)));}void deallocate(T*p,std::size_t n)noexcept{if(!p)return;std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){munmap(p,padded);return;}
#endif
::operator delete(p,padded,std::align_val_t(align));}private:static constexpr std::size_t MEGABYTE=1<<20;static constexpr std::size_t round_up(std::size_t x)noexcept{return(x+Align-1)/Align*Align;}};template<typename T>using big_vector=std::vector<T,big_alloc<T>>;template<typename T>using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>;template<typename T>using big_deque=std::deque<T,big_alloc<T>>;template<typename Key,typename Value,typename Compare=std::less<Key>>using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>;using big_string=big_basic_string<char>;}
#line 4 "cp-algo/math/laurent.hpp"
#include <memory>
#include <optional>
#line 7 "cp-algo/math/laurent.hpp"
#include <cassert>
#include <algorithm>
#include <type_traits>
#include <bit>
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#include <cstdint>
#line 7 "cp-algo/util/simd.hpp"
#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")\_Pragma("GCC optimize(\"O3,unroll-loops\")")\CP_ALGO_SIMD_AVX2_TARGETCP_ALGO_SIMD_PRAGMA_PUSHnamespace cp_algo{template<typename T,size_t len>using simd[[gnu::vector_size(len*sizeof(T))]]=T;using i64x4=simd<int64_t,4>;using u64x4=simd<uint64_t,4>;using u32x8=simd<uint32_t,8>;using i32x4=simd<int32_t,4>;using u32x4=simd<uint32_t,4>;using i16x4=simd<int16_t,4>;using u8x32=simd<uint8_t,32>;using dx4=simd<double,4>;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>
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSHnamespace 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 1 "cp-algo/util/checkpoint.hpp"
#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 7 "cp-algo/math/cvector.hpp"
#include <ranges>
#line 9 "cp-algo/math/cvector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSHnamespace stdx=std::experimental;namespace cp_algo::math::fft{static constexpr size_t flen=4;using ftype=double;using vftype=dx4;using point=complex<ftype>;using vpoint=complex<vftype>;static constexpr vftype vz={};vpoint vi(vpoint const&r){return{-imag(r),real(r)};}struct cvector{big_vector<vpoint>r;cvector(size_t n){n=std::max(flen,std::bit_ceil(n));r.resize(n/flen);checkpoint("cvector create");}vpoint&at(size_t k){return r[k/flen];}vpoint at(size_t k)const{return r[k/flen];}template<class pt=point>inline void set(size_t k,pt const&t){if constexpr(std::is_same_v<pt,point>){real(r[k/flen])[k%flen]=real(t);imag(r[k/flen])[k%flen]=imag(t);}else{at(k)=t;}}template<class pt=point>inline pt get(size_t k)const{if constexpr(std::is_same_v<pt,point>){return{real(r[k/flen])[k%flen],imag(r[k/flen])[k%flen]};}else{return at(k);}}size_t size()const{return flen*r.size();}static constexpr size_t eval_arg(size_t n){if(n<pre_evals){return eval_args[n];}else{return eval_arg(n/2)|(n&1)<<(std::bit_width(n)-1);}}static constexpr point eval_point(size_t n){if(n%2){return-eval_point(n-1);}else if(n%4){return eval_point(n-2)*point(0,1);}else if(n/4<pre_evals){return evalp[n/4];}else{return polar<ftype>(1.,std::numbers::pi/(ftype)std::bit_floor(n)*(ftype)eval_arg(n));}}static constexpr std::array<point,32>roots=[](){std::array<point,32>res;for(size_t i=2;i<32;i++){res[i]=polar<ftype>(1.,std::numbers::pi/(1ull<<(i-2)));}return res;}();static constexpr point root(size_t n){return roots[std::bit_width(n)];}template<int step>static void exec_on_eval(size_t n,size_t k,auto&&callback){callback(k,root(4*step*n)*eval_point(step*k));}template<int step>static void exec_on_evals(size_t n,auto&&callback){point factor=root(4*step*n);for(size_t i=0;i<n;i++){callback(i,factor*eval_point(step*i));}}static void do_dot_iter(point rt,vpoint&Bv,vpoint const&Av,vpoint&res){res+=Av*Bv;real(Bv)=rotate_right(real(Bv));imag(Bv)=rotate_right(imag(Bv));auto x=real(Bv)[0],y=imag(Bv)[0];real(Bv)[0]=x*real(rt)-y*imag(rt);imag(Bv)[0]=x*imag(rt)+y*real(rt);}void dot(cvector const&t){size_t n=this->size();exec_on_evals<1>(n/flen,[&](size_t k,point rt)__attribute__((always_inline)){k*=flen;auto[Ax,Ay]=at(k);auto Bv=t.at(k);vpoint res=vz;for(size_t i=0;i<flen;i++){vpoint Av=vpoint(vz+Ax[i],vz+Ay[i]);do_dot_iter(rt,Bv,Av,res);}set(k,res);});checkpoint("dot");}template<bool partial=true>void ifft(){size_t n=size();if constexpr(!partial){point pi(0,1);exec_on_evals<4>(n/4,[&](size_t k,point rt)__attribute__((always_inline)){k*=4;point v1=conj(rt);point v2=v1*v1;point v3=v1*v2;auto A=get(k);auto B=get(k+1);auto C=get(k+2);auto D=get(k+3);set(k,(A+B)+(C+D));set(k+2,((A+B)-(C+D))*v2);set(k+1,((A-B)-pi*(C-D))*v1);set(k+3,((A-B)+pi*(C-D))*v3);});}bool parity=std::countr_zero(n)%2;if(parity){exec_on_evals<2>(n/(2*flen),[&](size_t k,point rt)__attribute__((always_inline)){k*=2*flen;vpoint cvrt={vz+real(rt),vz-imag(rt)};auto B=at(k)-at(k+flen);at(k)+=at(k+flen);at(k+flen)=B*cvrt;});}for(size_t leaf=3*flen;leaf<n;leaf+=4*flen){size_t level=std::countr_one(leaf+3);for(size_t lvl=4+parity;lvl<=level;lvl+=2){size_t i=(1<<lvl)/4;exec_on_eval<4>(n>>lvl,leaf>>lvl,[&](size_t k,point rt)__attribute__((always_inline)){k<<=lvl;vpoint v1={vz+real(rt),vz-imag(rt)};vpoint v2=v1*v1;vpoint v3=v1*v2;for(size_t j=k;j<k+i;j+=flen){auto A=at(j);auto B=at(j+i);auto C=at(j+2*i);auto D=at(j+3*i);at(j)=((A+B)+(C+D));at(j+2*i)=((A+B)-(C+D))*v2;at(j+i)=((A-B)-vi(C-D))*v1;at(j+3*i)=((A-B)+vi(C-D))*v3;}});}}checkpoint("ifft");for(size_t k=0;k<n;k+=flen){if constexpr(partial){set(k,get<vpoint>(k)/=vz+ftype(n/flen));}else{set(k,get<vpoint>(k)/=vz+ftype(n));}}}template<bool partial=true>void fft(){size_t n=size();bool parity=std::countr_zero(n)%2;for(size_t leaf=0;leaf<n;leaf+=4*flen){size_t level=std::countr_zero(n+leaf);level-=level%2!=parity;for(size_t lvl=level;lvl>=4;lvl-=2){size_t i=(1<<lvl)/4;exec_on_eval<4>(n>>lvl,leaf>>lvl,[&](size_t k,point rt)__attribute__((always_inline)){k<<=lvl;vpoint v1={vz+real(rt),vz+imag(rt)};vpoint v2=v1*v1;vpoint v3=v1*v2;for(size_t j=k;j<k+i;j+=flen){auto A=at(j);auto B=at(j+i)*v1;auto C=at(j+2*i)*v2;auto D=at(j+3*i)*v3;at(j)=(A+C)+(B+D);at(j+i)=(A+C)-(B+D);at(j+2*i)=(A-C)+vi(B-D);at(j+3*i)=(A-C)-vi(B-D);}});}}if(parity){exec_on_evals<2>(n/(2*flen),[&](size_t k,point rt)__attribute__((always_inline)){k*=2*flen;vpoint vrt={vz+real(rt),vz+imag(rt)};auto t=at(k+flen)*vrt;at(k+flen)=at(k)-t;at(k)+=t;});}if constexpr(!partial){point pi(0,1);exec_on_evals<4>(n/4,[&](size_t k,point rt)__attribute__((always_inline)){k*=4;point v1=rt;point v2=v1*v1;point v3=v1*v2;auto A=get(k);auto B=get(k+1)*v1;auto C=get(k+2)*v2;auto D=get(k+3)*v3;set(k,(A+C)+(B+D));set(k+1,(A+C)-(B+D));set(k+2,(A-C)+pi*(B-D));set(k+3,(A-C)-pi*(B-D));});}checkpoint("fft");}static constexpr size_t pre_evals=1<<16;static const std::array<size_t,pre_evals>eval_args;static const std::array<point,pre_evals>evalp;};const std::array<size_t,cvector::pre_evals>cvector::eval_args=[](){std::array<size_t,pre_evals>res={};for(size_t i=1;i<pre_evals;i++){res[i]=res[i>>1]|(i&1)<<(std::bit_width(i)-1);}return res;}();const std::array<point,cvector::pre_evals>cvector::evalp=[](){std::array<point,pre_evals>res={};res[0]=1;for(size_t n=1;n<pre_evals;n++){res[n]=polar<ftype>(1.,std::numbers::pi*ftype(eval_args[n])/ftype(4*std::bit_floor(n)));}return res;}();}
#pragma GCC pop_options
#line 1 "cp-algo/math/convolution.hpp"
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#line 6 "cp-algo/math/common.hpp"
namespace cp_algo::math{
#ifdef CP_ALGO_MAXN
const int maxn=CP_ALGO_MAXN;
#else
const int maxn=1<<19;
#endif
const int magic=64;auto bpow(auto const&x,auto n,auto const&one,auto op){if(n==0){return one;}else{auto t=bpow(x,n/2,one,op);t=op(t,t);if(n%2){t=op(t,x);}return t;}}auto bpow(auto x,auto n,auto ans){return bpow(x,n,ans,std::multiplies{});}template<typename T>T bpow(T const&x,auto n){return bpow(x,n,T(1));}inline constexpr auto inv2(auto x){assert(x%2);std::make_unsigned_t<decltype(x)>y=1;while(y*x!=1){y*=2-x*y;}return y;}}
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math{template<typename modint,typename _Int>struct modint_base{using Int=_Int;using UInt=std::make_unsigned_t<Int>;static constexpr size_t bits=sizeof(Int)*8;using Int2=std::conditional_t<bits<=32,int64_t,__int128_t>;using UInt2=std::conditional_t<bits<=32,uint64_t,__uint128_t>;constexpr static Int mod(){return modint::mod();}constexpr static Int remod(){return modint::remod();}constexpr static UInt2 modmod(){return UInt2(mod())*mod();}constexpr modint_base()=default;constexpr modint_base(Int2 rr){to_modint().setr(UInt((rr+modmod())%mod()));}modint inv()const{return bpow(to_modint(),mod()-2);}modint operator-()const{modint neg;neg.r=std::min(-r,remod()-r);return neg;}modint&operator/=(const modint&t){return to_modint()*=t.inv();}modint&operator*=(const modint&t){r=UInt(UInt2(r)*t.r%mod());return to_modint();}modint&operator+=(const modint&t){r+=t.r;r=std::min(r,r-remod());return to_modint();}modint&operator-=(const modint&t){r-=t.r;r=std::min(r,r+remod());return to_modint();}modint operator+(const modint&t)const{return modint(to_modint())+=t;}modint operator-(const modint&t)const{return modint(to_modint())-=t;}modint operator*(const modint&t)const{return modint(to_modint())*=t;}modint operator/(const modint&t)const{return modint(to_modint())/=t;}auto operator==(const modint&t)const{return to_modint().getr()==t.getr();}auto operator!=(const modint&t)const{return to_modint().getr()!=t.getr();}auto operator<=(const modint&t)const{return to_modint().getr()<=t.getr();}auto operator>=(const modint&t)const{return to_modint().getr()>=t.getr();}auto operator<(const modint&t)const{return to_modint().getr()<t.getr();}auto operator>(const modint&t)const{return to_modint().getr()>t.getr();}Int rem()const{UInt R=to_modint().getr();return R-(R>(UInt)mod()/2)*mod();}constexpr void setr(UInt rr){r=rr;}constexpr UInt getr()const{return r;}static UInt modmod8(){return UInt(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/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 9 "cp-algo/math/fft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSHnamespace cp_algo::math::fft{template<modint_type base>struct dft{cvector A,B;static base factor,ifactor;using Int2=base::Int2;static bool _init;static int split(){static const int splt=int(std::sqrt(base::mod()))+1;return splt;}static uint32_t mod,imod;static void init(){if(!_init){factor=1+random::rng()%(base::mod()-1);ifactor=base(1)/factor;mod=base::mod();imod=-inv2<uint32_t>(base::mod());_init=true;}}static std::pair<vftype,vftype>do_split(auto const&a,size_t idx,u64x4 mul){if(idx>=std::size(a)){return std::pair{vftype(),vftype()};}u64x4 au={idx<std::size(a)?a[idx].getr():0,idx+1<std::size(a)?a[idx+1].getr():0,idx+2<std::size(a)?a[idx+2].getr():0,idx+3<std::size(a)?a[idx+3].getr():0};au=montgomery_mul(au,mul,mod,imod);au=au>=base::mod()?au-base::mod():au;auto ai=to_double(i64x4(au>=base::mod()/2?au-base::mod():au));auto quo=round(ai/split());return std::pair{ai-quo*split(),quo};}dft(size_t n):A(n),B(n){init();}dft(auto const&a,size_t n,bool partial=true):A(n),B(n){init();base b2x32=bpow(base(2),32);u64x4 cur={(bpow(factor,1)*b2x32).getr(),(bpow(factor,2)*b2x32).getr(),(bpow(factor,3)*b2x32).getr(),(bpow(factor,4)*b2x32).getr()};u64x4 step4=u64x4{}+(bpow(factor,4)*b2x32).getr();u64x4 stepn=u64x4{}+(bpow(factor,n)*b2x32).getr();for(size_t i=0;i<std::min(n,std::size(a));i+=flen){auto[rai,qai]=do_split(a,i,cur);auto[rani,qani]=do_split(a,n+i,montgomery_mul(cur,stepn,mod,imod));A.at(i)=vpoint(rai,rani);B.at(i)=vpoint(qai,qani);cur=montgomery_mul(cur,step4,mod,imod);}checkpoint("dft init");if(n){if(partial){A.fft();B.fft();}else{A.template fft<false>();B.template fft<false>();}}}static void do_dot_iter(point rt,vpoint&Cv,vpoint&Dv,vpoint const&Av,vpoint const&Bv,vpoint&AC,vpoint&AD,vpoint&BC,vpoint&BD){AC+=Av*Cv;AD+=Av*Dv;BC+=Bv*Cv;BD+=Bv*Dv;real(Cv)=rotate_right(real(Cv));imag(Cv)=rotate_right(imag(Cv));real(Dv)=rotate_right(real(Dv));imag(Dv)=rotate_right(imag(Dv));auto cx=real(Cv)[0],cy=imag(Cv)[0];auto dx=real(Dv)[0],dy=imag(Dv)[0];real(Cv)[0]=cx*real(rt)-cy*imag(rt);imag(Cv)[0]=cx*imag(rt)+cy*real(rt);real(Dv)[0]=dx*real(rt)-dy*imag(rt);imag(Dv)[0]=dx*imag(rt)+dy*real(rt);}template<bool overwrite=true,bool partial=true>void dot(auto const&C,auto const&D,auto&Aout,auto&Bout,auto&Cout)const{cvector::exec_on_evals<1>(A.size()/flen,[&](size_t k,point rt)__attribute__((always_inline)){k*=flen;vpoint AC,AD,BC,BD;AC=AD=BC=BD=vz;auto Cv=C.at(k),Dv=D.at(k);if constexpr(partial){auto[Ax,Ay]=A.at(k);auto[Bx,By]=B.at(k);for(size_t i=0;i<flen;i++){vpoint Av={vz+Ax[i],vz+Ay[i]},Bv={vz+Bx[i],vz+By[i]};do_dot_iter(rt,Cv,Dv,Av,Bv,AC,AD,BC,BD);}}else{AC=A.at(k)*Cv;AD=A.at(k)*Dv;BC=B.at(k)*Cv;BD=B.at(k)*Dv;}if constexpr(overwrite){Aout.at(k)=AC;Cout.at(k)=AD+BC;Bout.at(k)=BD;}else{Aout.at(k)+=AC;Cout.at(k)+=AD+BC;Bout.at(k)+=BD;}});checkpoint("dot");}void dot(auto&&C,auto const&D){dot(C,D,A,B,C);}static void do_recover_iter(size_t idx,auto A,auto B,auto C,auto mul,uint64_t splitsplit,auto&res){auto A0=lround(A),A1=lround(C),A2=lround(B);auto Ai=A0+A1*split()+A2*splitsplit+uint64_t(base::modmod());auto Au=montgomery_reduce(u64x4(Ai),mod,imod);Au=montgomery_mul(Au,mul,mod,imod);Au=Au>=base::mod()?Au-base::mod():Au;for(size_t j=0;j<flen;j++){res[idx+j].setr(typename base::UInt(Au[j]));}}void recover_mod(auto&&C,auto&res,size_t k){size_t check=(k+flen-1)/flen*flen;assert(res.size()>=check);size_t n=A.size();auto const splitsplit=base(split()*split()).getr();base b2x32=bpow(base(2),32);base b2x64=bpow(base(2),64);u64x4 cur={(bpow(ifactor,2)*b2x64).getr(),(bpow(ifactor,3)*b2x64).getr(),(bpow(ifactor,4)*b2x64).getr(),(bpow(ifactor,5)*b2x64).getr()};u64x4 step4=u64x4{}+(bpow(ifactor,4)*b2x32).getr();u64x4 stepn=u64x4{}+(bpow(ifactor,n)*b2x32).getr();for(size_t i=0;i<std::min(n,k);i+=flen){auto[Ax,Ay]=A.at(i);auto[Bx,By]=B.at(i);auto[Cx,Cy]=C.at(i);do_recover_iter(i,Ax,Bx,Cx,cur,splitsplit,res);if(i+n<k){do_recover_iter(i+n,Ay,By,Cy,montgomery_mul(cur,stepn,mod,imod),splitsplit,res);}cur=montgomery_mul(cur,step4,mod,imod);}checkpoint("recover mod");}void mul(auto&&C,auto const&D,auto&res,size_t k){assert(A.size()==C.size());size_t n=A.size();if(!n){res={};return;}dot(C,D);A.ifft();B.ifft();C.ifft();recover_mod(C,res,k);}void mul_inplace(auto&&B,auto&res,size_t k){mul(B.A,B.B,res,k);}void mul(auto const&B,auto&res,size_t k){mul(cvector(B.A),B.B,res,k);}big_vector<base>operator*=(dft&B){big_vector<base>res(2*A.size());mul_inplace(B,res,2*A.size());return res;}big_vector<base>operator*=(dft const&B){big_vector<base>res(2*A.size());mul(B,res,2*A.size());return res;}auto operator*(dft const&B)const{return dft(*this)*=B;}point operator[](int i)const{return A.get(i);}};template<modint_type base>base dft<base>::factor=1;template<modint_type base>base dft<base>::ifactor=1;template<modint_type base>bool dft<base>::_init=false;template<modint_type base>uint32_t dft<base>::mod={};template<modint_type base>uint32_t dft<base>::imod={};void mul_slow(auto&a,auto const&b,size_t k){if(std::empty(a)||std::empty(b)){a.clear();}else{size_t n=std::min(k,std::size(a));size_t m=std::min(k,std::size(b));a.resize(k);for(int j=int(k-1);j>=0;j--){a[j]*=b[0];for(int i=std::max(j-(int)n,0)+1;i<std::min(j+1,(int)m);i++){a[j]+=a[j-i]*b[i];}}}}size_t com_size(size_t as,size_t bs){if(!as||!bs){return 0;}return std::max(flen,std::bit_ceil(as+bs-1)/2);}void mul_truncate(auto&a,auto const&b,size_t k){using base=std::decay_t<decltype(a[0])>;if(std::min({k,std::size(a),std::size(b)})<magic){mul_slow(a,b,k);return;}auto n=std::max(flen,std::bit_ceil(std::min(k,std::size(a))+std::min(k,std::size(b))-1)/2);auto A=dft<base>(a|std::views::take(k),n);auto B=dft<base>(b|std::views::take(k),n);a.resize((k+flen-1)/flen*flen);A.mul_inplace(B,a,k);a.resize(k);}void mod_split(auto&&x,size_t n,auto k){using base=std::decay_t<decltype(k)>;dft<base>::init();assert(std::size(x)==2*n);u64x4 cur=u64x4{}+(k*bpow(base(2),32)).getr();for(size_t i=0;i<n;i+=flen){u64x4 xl={x[i].getr(),x[i+1].getr(),x[i+2].getr(),x[i+3].getr()};u64x4 xr={x[n+i].getr(),x[n+i+1].getr(),x[n+i+2].getr(),x[n+i+3].getr()};xr=montgomery_mul(xr,cur,dft<base>::mod,dft<base>::imod);xr=xr>=base::mod()?xr-base::mod():xr;auto t=xr;xr=xl-t;xl+=t;xl=xl>=base::mod()?xl-base::mod():xl;xr=xr>=base::mod()?xr+base::mod():xr;for(size_t k=0;k<flen;k++){x[i+k].setr(typename base::UInt(xl[k]));x[n+i+k].setr(typename base::UInt(xr[k]));}}cp_algo::checkpoint("mod split");}void cyclic_mul(auto&a,auto&&b,size_t k){assert(std::popcount(k)==1);assert(std::size(a)==std::size(b)&&std::size(a)==k);using base=std::decay_t<decltype(a[0])>;dft<base>::init();if(k<=(1<<16)){big_vector<base>ap(begin(a),end(a));mul_truncate(ap,b,2*k);mod_split(ap,k,bpow(dft<base>::factor,k));std::ranges::copy(ap|std::views::take(k),begin(a));return;}k/=2;auto factor=bpow(dft<base>::factor,k);mod_split(a,k,factor);mod_split(b,k,factor);auto la=std::span(a).first(k);auto lb=std::span(b).first(k);auto ra=std::span(a).last(k);auto rb=std::span(b).last(k);cyclic_mul(la,lb,k);auto A=dft<base>(ra,k/2);auto B=dft<base>(rb,k/2);A.mul_inplace(B,ra,k);base i2=base(2).inv();factor=factor.inv()*i2;for(size_t i=0;i<k;i++){auto t=(a[i]+a[i+k])*i2;a[i+k]=(a[i]-a[i+k])*factor;a[i]=t;}cp_algo::checkpoint("mod join");}auto make_copy(auto&&x){return x;}void cyclic_mul(auto&a,auto const&b,size_t k){return cyclic_mul(a,make_copy(b),k);}void mul(auto&a,auto&&b){size_t N=size(a)+size(b);if(N>(1<<20)){N--;size_t NN=std::bit_ceil(N);a.resize(NN);b.resize(NN);cyclic_mul(a,b,NN);a.resize(N);}else{mul_truncate(a,b,N-1);}}void mul(auto&a,auto const&b){size_t N=size(a)+size(b);if(N>(1<<20)){mul(a,make_copy(b));}else{mul_truncate(a,b,N-1);}}}
#pragma GCC pop_options
#line 10 "cp-algo/math/convolution.hpp"
namespace cp_algo::math{template<class VecA,class VecB>void convolution_prefix(VecA&a,VecB const&b,size_t need){using T=typename std::decay_t<VecA>::value_type;size_t na=std::min(need,std::size(a));size_t nb=std::min(need,std::size(b));a.resize(na);auto bv=b|std::views::take(nb);if(na==0||nb==0){a.clear();return;}if constexpr(modint_type<T>){fft::mul_truncate(a,bv,need);}else if constexpr(std::is_same_v<T,fft::point>){size_t conv_len=na+nb-1;size_t n=std::bit_ceil(conv_len);n=std::max(n,(size_t)fft::flen);fft::cvector A(n),B(n);for(size_t i=0;i<na;i++){A.set(i,a[i]);}for(size_t i=0;i<nb;i++){B.set(i,bv[i]);}A.fft();B.fft();A.dot(B);A.ifft();a.assign(need,T(0));for(size_t i=0;i<std::min(need,conv_len);i++){a[i]=A.template get<fft::point>(i);}}else if constexpr(std::is_same_v<T,fft::ftype>){size_t conv_len=na+nb-1;size_t n=std::bit_ceil(conv_len)/2;n=std::max(n,(size_t)fft::flen);fft::cvector A(n),B(n);for(size_t i=0;i<std::min(n,na);i++){fft::ftype re=a[i],im=0;if(i+n<na)im=a[i+n];A.set(i,fft::point(re,im));}for(size_t i=0;i<std::min(n,nb);i++){fft::ftype re=bv[i],im=0;if(i+n<nb)im=bv[i+n];B.set(i,fft::point(re,im));}A.fft();B.fft();A.dot(B);A.ifft();a.assign(2*n,T(0));for(size_t i=0;i<n;i++){auto v=A.template get<fft::point>(i);a[i]=v.real();a[i+n]=v.imag();}a.resize(need);}else{fft::mul_slow(a,bv,need);}}}
#line 14 "cp-algo/math/laurent.hpp"
namespace cp_algo::math{template<typename T>struct provider{mutable big_vector<T>cache;mutable int cache_offset=0;mutable bool initialized=false;mutable bool all_cached=false;virtual~provider()=default;virtual int offset()const{return 0;}virtual bool needs_lazy_eval()const{return false;}virtual T coeff_lazy(int k)const=0;virtual void double_up()const{int old_size=cache.size();int new_size=old_size==0?1:2*old_size;cache.resize(new_size);for(int i=old_size;i<new_size;i++){cache[i]=coeff_lazy(cache_offset+i);}}virtual T coeff(int k)const{if(!initialized){cache_offset=offset();initialized=true;}int idx=k-cache_offset;if(idx<0){return T(0);}if(all_cached&&idx>=(int)cache.size()){return T(0);}if(needs_lazy_eval()){while(idx>=(int)cache.size()&&!all_cached){int next_k=cache_offset+(int)cache.size();cache.push_back(coeff_lazy(next_k));}}else{while(idx>=(int)cache.size()&&!all_cached){double_up();}}if(idx<(int)cache.size()){return cache[idx];}return T(0);}T get(int k)const{return coeff(k);}};template<typename T>struct constant_provider:provider<T>{T value;int offset;constant_provider(T value,int offset=0):value(value),offset(offset){}int offset()const override{return offset;}T coeff_lazy(int k)const override{return k==offset?value:T(0);}T coeff(int k)const override{return coeff_lazy(k);}};template<typename T>struct polynomial_provider:provider<T>{polynomial_provider(big_vector<T>coeffs,int offset=0){auto non_zero=[](const T&x){return x!=T(0);};auto first=std::ranges::find_if(coeffs,non_zero);auto last=std::ranges::find_if(coeffs|std::views::reverse,non_zero);if(first!=coeffs.end()){int leading_zeros=first-coeffs.begin();int trailing_zeros=last-coeffs.rbegin();coeffs=big_vector<T>(first,coeffs.end()-trailing_zeros);offset+=leading_zeros;}else{coeffs.clear();}this->cache=std::move(coeffs);this->cache_offset=offset;this->initialized=true;this->all_cached=true;}int offset()const override{return this->cache_offset;}T coeff_lazy(int k)const override{int idx=k-this->cache_offset;if(idx<0||idx>=(int)this->cache.size()){return T(0);}return this->cache[idx];}T coeff(int k)const override{return coeff_lazy(k);}};template<typename T>struct unary_provider:provider<T>{std::shared_ptr<provider<T>>operand;unary_provider(std::shared_ptr<provider<T>>operand):operand(std::move(operand)){}virtual T transform(T const&a)const=0;int offset()const override{return operand->offset();}T coeff_lazy(int k)const override{return transform(operand->coeff_lazy(k));}T coeff(int k)const{return transform(operand->coeff(k));}};template<typename T>struct binary_provider:provider<T>{std::shared_ptr<provider<T>>lhs,rhs;binary_provider(std::shared_ptr<provider<T>>lhs,std::shared_ptr<provider<T>>rhs):lhs(std::move(lhs)),rhs(std::move(rhs)){}virtual T combine(T const&a,T const&b)const=0;int offset()const override{return std::min(lhs->offset(),rhs->offset());}T coeff_lazy(int k)const override{return combine(lhs->coeff_lazy(k),rhs->coeff_lazy(k));}T coeff(int k)const{return combine(lhs->coeff(k),rhs->coeff(k));}};template<typename T>struct add_provider:binary_provider<T>{using binary_provider<T>::binary_provider;T combine(T const&a,T const&b)const override{return a+b;}};template<typename T>struct subtract_provider:binary_provider<T>{using binary_provider<T>::binary_provider;T combine(T const&a,T const&b)const override{return a-b;}};template<typename T>struct negate_provider:unary_provider<T>{using unary_provider<T>::unary_provider;T transform(T const&a)const override{return-a;}};template<typename T>struct scale_provider:unary_provider<T>{T scalar;scale_provider(std::shared_ptr<provider<T>>operand,T scalar):unary_provider<T>(std::move(operand)),scalar(scalar){}T transform(T const&a)const override{return a*scalar;}};template<typename T>struct multiply_provider:provider<T>{std::shared_ptr<provider<T>>lhs,rhs;multiply_provider(std::shared_ptr<provider<T>>lhs,std::shared_ptr<provider<T>>rhs):lhs(std::move(lhs)),rhs(std::move(rhs)){}int offset()const override{return lhs->offset()+rhs->offset();}bool needs_lazy_eval()const override{return lhs->needs_lazy_eval()||rhs->needs_lazy_eval();}T coeff_lazy(int k)const override{int n=k-offset();if(n<0)return T(0);T result=T(0);bool lazy_lhs=lhs->needs_lazy_eval();bool lazy_rhs=rhs->needs_lazy_eval();for(int j=0;j<=n;j++){int i_l=lhs->offset()+j;int i_r=rhs->offset()+(n-j);auto a=lazy_lhs?lhs->coeff(i_l):lhs->coeff_lazy(i_l);auto b=lazy_rhs?rhs->coeff(i_r):rhs->coeff_lazy(i_r);result+=a*b;}return result;}void double_up()const override{int old_size=this->cache.size();int new_size=old_size==0?1:2*old_size;if(needs_lazy_eval()){int k=this->cache_offset+old_size;this->cache.push_back(coeff_lazy(k));return;}int lhs_need=lhs->offset()+new_size-1;int rhs_need=rhs->offset()+new_size-1;lhs->coeff(lhs_need);rhs->coeff(rhs_need);big_vector<T>la(new_size),rb(new_size);for(int i=0;i<new_size;i++){la[i]=lhs->coeff(lhs->offset()+i);rb[i]=rhs->coeff(rhs->offset()+i);}this->cache.resize(new_size);convolution_prefix(la,rb,new_size);for(int i=old_size;i<new_size&&i<(int)la.size();i++){this->cache[i]=la[i];}if(lhs->all_cached&&rhs->all_cached){size_t total_len=lhs->cache.size()+rhs->cache.size()-1;if((size_t)new_size>=total_len){this->cache.resize(total_len);this->all_cached=true;}}}};template<typename T>struct laurent{std::shared_ptr<provider<T>>impl;laurent():impl(std::make_shared<constant_provider<T>>(T(0),0)){}laurent(T value,int offset=0):impl(std::make_shared<constant_provider<T>>(value,offset)){}laurent(big_vector<T>coeffs,int offset=0):impl(std::make_shared<polynomial_provider<T>>(std::move(coeffs),offset)){}laurent(std::shared_ptr<provider<T>>impl):impl(std::move(impl)){}T operator[](int k)const{return impl->get(k);}laurent operator-()const{return std::make_shared<negate_provider<T>>(impl);}laurent operator+(const laurent&other)const{return std::make_shared<add_provider<T>>(impl,other.impl);}laurent operator-(const laurent&other)const{return std::make_shared<subtract_provider<T>>(impl,other.impl);}laurent operator*(const laurent&other)const{return std::make_shared<multiply_provider<T>>(impl,other.impl);}laurent&operator+=(const laurent&other){return*this=*this+other;}laurent&operator-=(const laurent&other){return*this=*this-other;}laurent&operator*=(const laurent&other){return*this=*this*other;}laurent operator*(T const&scalar)const{return std::make_shared<scale_provider<T>>(impl,scalar);}laurent&operator*=(T const&scalar){return*this=*this*scalar;}};template<typename T>laurent<T>operator*(T const&scalar,laurent<T>const&series){return series*scalar;}}
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