CP-Algorithms Library

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:heavy_check_mark: cp-algo/algebra/poly.hpp

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Code

#ifndef CP_ALGO_ALGEBRA_POLY_HPP
#define CP_ALGO_ALGEBRA_POLY_HPP
#include "poly/impl/euclid.hpp"
#include "poly/impl/base.hpp"
#include "poly/impl/div.hpp"
#include "number_theory.hpp"
#include "fft.hpp"
#include <functional>
#include <algorithm>
#include <iostream>
#include <optional>
#include <utility>
#include <vector>
#include <list>
namespace cp_algo::algebra {
    template<typename T>
    struct poly_t {
        using base = T;
        std::vector<T> a;
        
        void normalize() {poly::impl::normalize(*this);}
        
        poly_t(){}
        poly_t(T a0): a{a0} {normalize();}
        poly_t(std::vector<T> const& t): a(t) {normalize();}
        
        poly_t operator -() const {return poly::impl::neg(*this);}
        poly_t& operator += (poly_t const& t) {return poly::impl::add(*this, t);}
        poly_t& operator -= (poly_t const& t) {return poly::impl::sub(*this, t);}
        poly_t operator + (poly_t const& t) const {return poly_t(*this) += t;}
        poly_t operator - (poly_t const& t) const {return poly_t(*this) -= t;}
        
        poly_t mod_xk(size_t k) const {return poly::impl::mod_xk(*this, k);} // %= x^k
        poly_t mul_xk(size_t k) const {return poly::impl::mul_xk(*this, k);} // *= x^k
        poly_t div_xk(size_t k) const {return poly::impl::div_xk(*this, k);} // /= x^k
        poly_t substr(size_t l, size_t k) const {return poly::impl::substr(*this, l, k);}
        
        poly_t operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
        poly_t operator * (const poly_t &t) const {return poly_t(*this) *= t;}

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

        poly_t& operator *= (T const& x) {return *this = poly::impl::scale(*this, x);}
        poly_t& operator /= (T const& x) {return *this *= x.inv();}
        poly_t operator * (T const& x) const {return poly_t(*this) *= x;}
        poly_t operator / (T const& x) const {return poly_t(*this) /= x;}
        
        poly_t reverse(size_t n) const {return poly::impl::reverse(*this, n);}
        poly_t reverse() const {return reverse(size(a));}
        
        std::array<poly_t, 2> divmod(poly_t const& b) const {
            return poly::impl::divmod(*this, b);
        }
        
        // reduces A/B to A'/B' such that
        // deg B' < deg A / 2
        static std::pair<std::list<poly_t>, linfrac<poly_t>> half_gcd(auto &&A, auto &&B) {
            return poly::impl::half_gcd(A, B);
        }
        // reduces A / B to gcd(A, B) / 0
        static std::pair<std::list<poly_t>, linfrac<poly_t>> full_gcd(auto &&A, auto &&B) {
            return poly::impl::full_gcd(A, B);
        }
        static poly_t gcd(poly_t &&A, poly_t &&B) {
            full_gcd(A, B);
            return A;
        }
        
        // Returns a (non-monic) characteristic polynomial
        // of the minimum linear recurrence for the sequence
        poly_t min_rec(size_t d) const {
            return poly::impl::min_rec(*this, d);
        }
        
        // calculate inv to *this modulo t
        std::optional<poly_t> inv_mod(poly_t const& t) const {
            return poly::impl::inv_mod(*this, t);
        };
        
        poly_t negx() const { // A(x) -> A(-x)
            auto res = *this;
            for(int i = 1; i <= deg(); i += 2) {
                res.a[i] = -res[i];
            }
            return res;
        }
        
        void print(int n) const {
            for(int i = 0; i < n; i++) {
                std::cout << (*this)[i] << ' ';
            }
            std::cout << "\n";
        }
        
        void print() const {
            print(deg() + 1);
        }
        
        T eval(T x) const { // evaluates in single point x
            T res(0);
            for(int i = deg(); i >= 0; i--) {
                res *= x;
                res += a[i];
            }
            return res;
        }
        
        T lead() const { // leading coefficient
            assert(!is_zero());
            return a.back();
        }
        
        int deg() const { // degree, -1 for P(x) = 0
            return (int)a.size() - 1;
        }
        
        bool is_zero() const {
            return a.empty();
        }
        
        T operator [](int idx) const {
            return idx < 0 || idx > deg() ? T(0) : a[idx];
        }
        
        T& coef(size_t idx) { // mutable reference at coefficient
            return a[idx];
        }
        
        bool operator == (const poly_t &t) const {return a == t.a;}
        bool operator != (const poly_t &t) const {return a != t.a;}
        
        poly_t deriv(int k = 1) const { // calculate derivative
            if(deg() + 1 < k) {
                return poly_t(T(0));
            }
            std::vector<T> res(deg() + 1 - k);
            for(int i = k; i <= deg(); i++) {
                res[i - k] = fact<T>(i) * rfact<T>(i - k) * a[i];
            }
            return res;
        }
        
        poly_t integr() const { // calculate integral with C = 0
            std::vector<T> res(deg() + 2);
            for(int i = 0; i <= deg(); i++) {
                res[i + 1] = a[i] * small_inv<T>(i + 1);
            }
            return res;
        }
        
        size_t trailing_xk() const { // Let p(x) = x^k * t(x), return k
            if(is_zero()) {
                return -1;
            }
            int res = 0;
            while(a[res] == T(0)) {
                res++;
            }
            return res;
        }
        
        poly_t log(size_t n) const { // calculate log p(x) mod x^n
            assert(a[0] == T(1));
            return (deriv().mod_xk(n) * inv(n)).integr().mod_xk(n);
        }
        
        poly_t exp(size_t n) const { // calculate exp p(x) mod x^n
            if(is_zero()) {
                return T(1);
            }
            assert(a[0] == T(0));
            poly_t ans = T(1);
            size_t a = 1;
            while(a < n) {
                poly_t C = ans.log(2 * a).div_xk(a) - substr(a, 2 * a);
                ans -= (ans * C).mod_xk(a).mul_xk(a);
                a *= 2;
            }
            return ans.mod_xk(n);
        }
        
        poly_t pow_bin(int64_t k, size_t n) const { // O(n log n log k)
            if(k == 0) {
                return poly_t(1).mod_xk(n);
            } else {
                auto t = pow(k / 2, n);
                t = (t * t).mod_xk(n);
                return (k % 2 ? *this * t : t).mod_xk(n);
            }
        }

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

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

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

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

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

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

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

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

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

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

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

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

        poly_t to_newton(std::vector<T> p) {
            if(is_zero()) {
                return *this;
            }
            int n = p.size();
            std::vector<poly_t> tree(4 * n);
            build(tree, 1, begin(p), end(p));
            return to_newton(tree, 1, begin(p), end(p));
        }

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

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

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

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

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

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

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


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


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


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

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

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

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

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

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

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

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

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

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

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


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


#include <functional>
#include <cstdint>
namespace cp_algo::algebra {
#ifdef CP_ALGO_MAXN
    const int maxn = CP_ALGO_MAXN;
#else
    const int maxn = 1 << 20;
#endif
    const int magic = 250; // threshold for sizes to run the naive algo

    auto bpow(auto const& x, int64_t 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, int64_t n, auto ans) {
        return bpow(x, n, ans, std::multiplies{});
    }
    template<typename T>
    T bpow(T const& x, int64_t n) {
        return bpow(x, n, T(1));
    }

    template<typename T>
    T fact(int n) {
        static std::vector<T> F(maxn);
        static bool init = false;
        if(!init) {
            F[0] = T(1);
            for(int i = 1; i < maxn; i++) {
                F[i] = F[i - 1] * T(i);
            }
            init = true;
        }
        return F[n];
    }
    template<typename T>
    T rfact(int n) {
        static std::vector<T> F(maxn);
        static bool init = false;
        if(!init) {
            F[maxn - 1] = T(1) / fact<T>(maxn - 1);
            for(int i = maxn - 2; i >= 0; i--) {
                F[i] = F[i + 1] * T(i + 1);
            }
            init = true;
        }
        return F[n];
    }
    template<typename T>
    T small_inv(int n) {
        static std::vector<T> F(maxn);
        static bool init = false;
        if(!init) {
            for(int i = 1; i < maxn; i++) {
                F[i] = rfact<T>(i) * fact<T>(i - 1);
            }
            init = true;
        }
        return F[n];
    }

    template<typename T>
    T nCr(int n, int r) {
        if(r < 0 || r > n) {
            return T(0);
        } else {
            return fact<T>(n) * rfact<T>(r) * rfact<T>(n-r);
        }
    }
}

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


#line 4 "cp-algo/algebra/modint.hpp"
#include <iostream>
namespace cp_algo::algebra {
    template<typename modint>
    struct modint_base {
        static int mod() {
            return modint::mod();
        }
        modint_base(): r(0) {}
        modint_base(int64_t rr): r(rr % mod()) {
            r = std::min(r, r + mod());
        }
        modint inv() const {
            return bpow(to_modint(), mod() - 2);
        }
        modint operator - () const {return std::min(-r, mod() - r);}
        modint& operator /= (const modint &t) {
            return to_modint() *= t.inv();
        }
        modint& operator *= (const modint &t) {
            r *= t.r; if(mod()) {r %= mod();}
            return to_modint();
        }
        modint& operator += (const modint &t) {
            r += t.r; r = std::min(r, r - mod());
            return to_modint();
        }
        modint& operator -= (const modint &t) {
            r -= t.r; r = std::min(r, r + mod());
            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_base &t) const = default;
        explicit operator int() const {return r;}
        int64_t rem() const {return 2 * r > (uint64_t)mod() ? r - mod() : r;}

        // Only use if you really know what you're doing!
        uint64_t modmod() const {return 8LL * mod() * mod();};
        void add_unsafe(uint64_t t) {r += t;}
        void pseudonormalize() {r = std::min(r, r - modmod());}
        modint const& normalize() {
            if(r >= (uint64_t)mod()) {
                r %= mod();
            }
            return to_modint();
        }
        uint64_t& setr() {return r;}
        uint64_t getr() const {return r;}
    private:
        uint64_t r;
        modint& to_modint() {return static_cast<modint&>(*this);}
        modint const& to_modint() const {return static_cast<modint const&>(*this);}
    };
    template<typename modint>
    std::istream& operator >> (std::istream &in, modint_base<modint> &x) {
        return in >> x.setr();
    }
    template<typename modint>
    std::ostream& operator << (std::ostream &out, modint_base<modint> const& x) {
        return out << x.getr();
    }

    template<int m>
    struct modint: modint_base<modint<m>> {
        static constexpr int mod() {return m;}
        using Base = modint_base<modint<m>>;
        using Base::Base;
    };

    struct dynamic_modint: modint_base<dynamic_modint> {
        static int mod() {return m;}
        static void switch_mod(int nm) {m = nm;}
        using Base = modint_base<dynamic_modint>;
        using Base::Base;
    private:
        static int m;
    };
    int dynamic_modint::m = 0;
}

#line 5 "cp-algo/algebra/fft.hpp"
#include <algorithm>
#include <complex>
#line 8 "cp-algo/algebra/fft.hpp"
#include <vector>
namespace cp_algo::algebra::fft {
    using ftype = double;
    using point = std::complex<ftype>;

    std::vector<point> w; // w[2^n + k] = exp(pi * k / (2^n))
    std::vector<int> bitr;// b[2^n + k] = bitreverse(k)
    const ftype pi = acos(-1);
    bool initiated = 0;
    void init() {
        if(!initiated) {
            w.resize(maxn);
            bitr.resize(maxn);
            for(int i = 1; i < maxn; i *= 2) {
                int ti = i / 2;
                for(int j = 0; j < i; j++) {
                    w[i + j] = std::polar(ftype(1), pi * j / i);
                    if(ti) {
                        bitr[i + j] = 2 * bitr[ti + j % ti] + (j >= ti);
                    }
                }
            }
            initiated = 1;
        }
    }
    
    void fft(auto &a, int n) {
        init();
        if(n == 1) {
            return;
        }
        int hn = n / 2;
        for(int i = 0; i < n; i++) {
            int ti = 2 * bitr[hn + i % hn] + (i > hn);
            if(i < ti) {
                std::swap(a[i], a[ti]);
            }
        }
        for(int i = 1; i < n; i *= 2) {
            for(int j = 0; j < n; j += 2 * i) {
                for(int k = j; k < j + i; k++) {
                    point t = a[k + i] * w[i + k - j];
                    a[k + i] = a[k] - t;
                    a[k] += t;
                }
            }
        }
    }
    template<typename base>
    void mul_slow(std::vector<base> &a, const std::vector<base> &b) {
        if(a.empty() || b.empty()) {
            a.clear();
        } else {
            int n = a.size();
            int m = b.size();
            a.resize(n + m - 1);
            for(int k = n + m - 2; k >= 0; k--) {
                a[k] *= b[0];
                for(int j = std::max(k - n + 1, 1); j < std::min(k + 1, m); j++) {
                    a[k] += a[k - j] * b[j];
                }
            }
        }
    }
    
    template<int m>
    struct dft {
        static constexpr int split = 1 << 15;
        std::vector<point> A;
        
        dft(std::vector<modint<m>> const& a, size_t n): A(n) {
            for(size_t i = 0; i < std::min(n, a.size()); i++) {
                A[i] = point(
                    a[i].rem() % split,
                    a[i].rem() / split
                );
            }
            if(n) {
                fft(A, n);
            }
        }
    
        auto operator * (dft const& B) {
            assert(A.size() == B.A.size());
            size_t n = A.size();
            if(!n) {
                return std::vector<modint<m>>();
            }
            std::vector<point> C(n), D(n);
            for(size_t i = 0; i < n; i++) {
                C[i] = A[i] * (B[i] + conj(B[(n - i) % n]));
                D[i] = A[i] * (B[i] - conj(B[(n - i) % n]));
            }
            fft(C, n);
            fft(D, n);
            reverse(begin(C) + 1, end(C));
            reverse(begin(D) + 1, end(D));
            int t = 2 * n;
            std::vector<modint<m>> res(n);
            for(size_t i = 0; i < n; i++) {
                modint<m> A0 = llround(C[i].real() / t);
                modint<m> A1 = llround(C[i].imag() / t + D[i].imag() / t);
                modint<m> A2 = llround(D[i].real() / t);
                res[i] = A0 + A1 * split - A2 * split * split;
            }
            return res;
        }
        
        point& operator [](int i) {return A[i];}
        point operator [](int i) const {return A[i];}
    };
    
    size_t com_size(size_t as, size_t bs) {
        if(!as || !bs) {
            return 0;
        }
        size_t n = as + bs - 1;
        while(__builtin_popcount(n) != 1) {
            n++;
        }
        return n;
    }
    
    template<int m>
    void mul(std::vector<modint<m>> &a, std::vector<modint<m>> b) {
        if(std::min(a.size(), b.size()) < magic) {
            mul_slow(a, b);
            return;
        }
        auto n = com_size(a.size(), b.size());
        auto A = dft<m>(a, n);
        if(a == b) {
            a = A * A;
        } else {
            a = A * dft<m>(b, n);
        }
    }
}

#line 7 "cp-algo/algebra/poly/impl/euclid.hpp"
#include <numeric>
#line 11 "cp-algo/algebra/poly/impl/euclid.hpp"
#include <list>
// operations related to gcd and Euclidean algo
namespace cp_algo::algebra::poly::impl {
    template<typename poly>
    using gcd_result = std::pair<
        std::list<std::decay_t<poly>>,
        linfrac<std::decay_t<poly>>>;

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

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

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

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

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

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


#line 6 "cp-algo/algebra/poly/impl/base.hpp"
// really basic operations, typically taking O(n)
namespace cp_algo::algebra::poly::impl {
    void normalize(auto& p) {
        while(p.deg() >= 0 && p.lead() == 0) {
            p.a.pop_back();
        }
    }
    auto neg(auto p) {
        std::ranges::transform(p.a, begin(p.a), std::negate{});
        return p;
    }
    auto& scale(auto &p, auto x) {
        for(auto &it: p.a) {
            it *= x;
        }
        p.normalize();
        return p;
    }
    auto& add(auto &p, auto q) {
        p.a.resize(std::max(p.a.size(), q.a.size()));
        std::ranges::transform(p.a, q.a, begin(p.a), std::plus{});
        normalize(p);
        return p;
    }
    auto& sub(auto &p, auto q) {
        p.a.resize(std::max(p.a.size(), q.a.size()));
        std::ranges::transform(p.a, q.a, begin(p.a), std::minus{});
        normalize(p);
        return p;
    }
    auto mod_xk(auto const& p, size_t k) {
        return std::vector(begin(p.a), begin(p.a) + std::min(k, p.a.size()));
    }
    auto mul_xk(auto p, int k) {
        if(k < 0) {
            return p.div_xk(-k);
        }
        p.a.insert(begin(p.a), k, 0);
        normalize(p);
        return p;
    }
    template<typename poly>
    poly div_xk(poly const& p, int k) {
        if(k < 0) {
            return p.mul_xk(-k);
        }
        return std::vector(begin(p.a) + std::min<size_t>(k, p.a.size()), end(p.a));
    }
    auto substr(auto const& p, size_t l, size_t k) {
        return std::vector(
            begin(p.a) + std::min(l, p.a.size()),
            begin(p.a) + std::min(l + k, p.a.size())
        );
    }
    auto reverse(auto p, size_t n) {
        p.a.resize(n);
        std::ranges::reverse(p.a);
        normalize(p);
        return p;
    }
}

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


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

    template<typename poly>
    poly powmod_hint(poly const& p, int64_t k, poly const& md, poly const& mdri) {
        return bpow(p, k, poly(1), [&](auto const& p, auto const& q){
            return divmod_hint(p * q, md, mdri)[1];
        });
    }
    template<typename poly>
    auto powmod(poly const& p, int64_t k, poly const& md) {
        int d = md.deg();
        if(p == poly::xk(1) && false) { // does it actually speed anything up?..
            if(k < md.deg()) {
                return poly::xk(k);
            } else {
                auto mdr = md.reverse();
                return (mdr.inv(k - md.deg() + 1, md.deg()) * mdr).reverse(md.deg());
            }
        }
        if(md == poly::xk(d)) {
            return p.pow(k, d);
        }
        if(md == poly::xk(d) - poly(1)) {
            return p.powmod_circular(k, d);
        }
        return powmod_hint(p, k, md, md.reverse().inv(md.deg() + 1));
    }

    auto interleave(auto const& p) {
        auto [p0, p1] = p.bisect();
        return p0 * p0 - (p1 * p1).mul_xk(1);
    }
    template<typename poly>
    poly inv(poly const& q, int64_t k, size_t n) {
        if(k <= std::max<int64_t>(n, size(q.a))) {
            return q.inv(k + n).div_xk(k);
        }
        if(k % 2) {
            return inv(q, k - 1, n + 1).div_xk(1);
        }
        
        auto qq = inv(interleave(q), k / 2 - q.deg() / 2, (n + 1) / 2 + q.deg() / 2);
        auto [q0, q1] = q.negx().bisect();
        return (
            (q0 * qq).x2() + (q1 * qq).x2().mul_xk(1)
        ).div_xk(2*q0.deg()).mod_xk(n);
    }
    template<typename poly>
    poly inv(poly const& p, size_t n) {
        auto q = p.mod_xk(n);
        if(n == 1) {
            return poly(1) / q[0];
        }
        // Q(-x) = P0(x^2) + xP1(x^2)
        auto [q0, q1] = q.negx().bisect();
        
        int N = fft::com_size((n + 1) / 2, (n + 1) / 2);
        
        auto q0f = fft::dft(q0.a, N);
        auto q1f = fft::dft(q1.a, N);

        // Q(x)*Q(-x) = Q0(x^2)^2 - x^2 Q1(x^2)^2
        auto qqf = fft::dft(inv(
            poly(q0f * q0f) - poly(q1f * q1f).mul_xk(1)
        , (n + 1) / 2).a, N);
        
        return (
            poly(q0f * qqf).x2() + poly(q1f * qqf).x2().mul_xk(1)
        ).mod_xk(n);
    }
}

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


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


#include <chrono>
#include <random>
namespace cp_algo::random {
    uint64_t rng() {
        static std::mt19937_64 rng(std::chrono::steady_clock::now().time_since_epoch().count());
        return rng();
    }
}

#line 7 "cp-algo/algebra/number_theory.hpp"
namespace cp_algo::algebra {
    // https://en.wikipedia.org/wiki/Berlekamp-Rabin_algorithm
    template<typename base>
    requires(std::is_base_of_v<modint_base<base>, base>)
    std::optional<base> sqrt(base b) {
        if(b == base(0)) {
            return base(0);
        } else if(bpow(b, (b.mod() - 1) / 2) != base(1)) {
            return std::nullopt;
        } else {
            while(true) {
                base z = random::rng();
                if(z * z == b) {
                    return z;
                }
                lin<base> x(1, z, b); // x + z (mod x^2 - b)
                x = bpow(x, (b.mod() - 1) / 2, lin<base>(0, 1, b));
                if(x.a != base(0)) {
                    return x.a.inv();
                }
            }
        }
    }
}

#line 15 "cp-algo/algebra/poly.hpp"
namespace cp_algo::algebra {
    template<typename T>
    struct poly_t {
        using base = T;
        std::vector<T> a;
        
        void normalize() {poly::impl::normalize(*this);}
        
        poly_t(){}
        poly_t(T a0): a{a0} {normalize();}
        poly_t(std::vector<T> const& t): a(t) {normalize();}
        
        poly_t operator -() const {return poly::impl::neg(*this);}
        poly_t& operator += (poly_t const& t) {return poly::impl::add(*this, t);}
        poly_t& operator -= (poly_t const& t) {return poly::impl::sub(*this, t);}
        poly_t operator + (poly_t const& t) const {return poly_t(*this) += t;}
        poly_t operator - (poly_t const& t) const {return poly_t(*this) -= t;}
        
        poly_t mod_xk(size_t k) const {return poly::impl::mod_xk(*this, k);} // %= x^k
        poly_t mul_xk(size_t k) const {return poly::impl::mul_xk(*this, k);} // *= x^k
        poly_t div_xk(size_t k) const {return poly::impl::div_xk(*this, k);} // /= x^k
        poly_t substr(size_t l, size_t k) const {return poly::impl::substr(*this, l, k);}
        
        poly_t operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
        poly_t operator * (const poly_t &t) const {return poly_t(*this) *= t;}

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

        poly_t& operator *= (T const& x) {return *this = poly::impl::scale(*this, x);}
        poly_t& operator /= (T const& x) {return *this *= x.inv();}
        poly_t operator * (T const& x) const {return poly_t(*this) *= x;}
        poly_t operator / (T const& x) const {return poly_t(*this) /= x;}
        
        poly_t reverse(size_t n) const {return poly::impl::reverse(*this, n);}
        poly_t reverse() const {return reverse(size(a));}
        
        std::array<poly_t, 2> divmod(poly_t const& b) const {
            return poly::impl::divmod(*this, b);
        }
        
        // reduces A/B to A'/B' such that
        // deg B' < deg A / 2
        static std::pair<std::list<poly_t>, linfrac<poly_t>> half_gcd(auto &&A, auto &&B) {
            return poly::impl::half_gcd(A, B);
        }
        // reduces A / B to gcd(A, B) / 0
        static std::pair<std::list<poly_t>, linfrac<poly_t>> full_gcd(auto &&A, auto &&B) {
            return poly::impl::full_gcd(A, B);
        }
        static poly_t gcd(poly_t &&A, poly_t &&B) {
            full_gcd(A, B);
            return A;
        }
        
        // Returns a (non-monic) characteristic polynomial
        // of the minimum linear recurrence for the sequence
        poly_t min_rec(size_t d) const {
            return poly::impl::min_rec(*this, d);
        }
        
        // calculate inv to *this modulo t
        std::optional<poly_t> inv_mod(poly_t const& t) const {
            return poly::impl::inv_mod(*this, t);
        };
        
        poly_t negx() const { // A(x) -> A(-x)
            auto res = *this;
            for(int i = 1; i <= deg(); i += 2) {
                res.a[i] = -res[i];
            }
            return res;
        }
        
        void print(int n) const {
            for(int i = 0; i < n; i++) {
                std::cout << (*this)[i] << ' ';
            }
            std::cout << "\n";
        }
        
        void print() const {
            print(deg() + 1);
        }
        
        T eval(T x) const { // evaluates in single point x
            T res(0);
            for(int i = deg(); i >= 0; i--) {
                res *= x;
                res += a[i];
            }
            return res;
        }
        
        T lead() const { // leading coefficient
            assert(!is_zero());
            return a.back();
        }
        
        int deg() const { // degree, -1 for P(x) = 0
            return (int)a.size() - 1;
        }
        
        bool is_zero() const {
            return a.empty();
        }
        
        T operator [](int idx) const {
            return idx < 0 || idx > deg() ? T(0) : a[idx];
        }
        
        T& coef(size_t idx) { // mutable reference at coefficient
            return a[idx];
        }
        
        bool operator == (const poly_t &t) const {return a == t.a;}
        bool operator != (const poly_t &t) const {return a != t.a;}
        
        poly_t deriv(int k = 1) const { // calculate derivative
            if(deg() + 1 < k) {
                return poly_t(T(0));
            }
            std::vector<T> res(deg() + 1 - k);
            for(int i = k; i <= deg(); i++) {
                res[i - k] = fact<T>(i) * rfact<T>(i - k) * a[i];
            }
            return res;
        }
        
        poly_t integr() const { // calculate integral with C = 0
            std::vector<T> res(deg() + 2);
            for(int i = 0; i <= deg(); i++) {
                res[i + 1] = a[i] * small_inv<T>(i + 1);
            }
            return res;
        }
        
        size_t trailing_xk() const { // Let p(x) = x^k * t(x), return k
            if(is_zero()) {
                return -1;
            }
            int res = 0;
            while(a[res] == T(0)) {
                res++;
            }
            return res;
        }
        
        poly_t log(size_t n) const { // calculate log p(x) mod x^n
            assert(a[0] == T(1));
            return (deriv().mod_xk(n) * inv(n)).integr().mod_xk(n);
        }
        
        poly_t exp(size_t n) const { // calculate exp p(x) mod x^n
            if(is_zero()) {
                return T(1);
            }
            assert(a[0] == T(0));
            poly_t ans = T(1);
            size_t a = 1;
            while(a < n) {
                poly_t C = ans.log(2 * a).div_xk(a) - substr(a, 2 * a);
                ans -= (ans * C).mod_xk(a).mul_xk(a);
                a *= 2;
            }
            return ans.mod_xk(n);
        }
        
        poly_t pow_bin(int64_t k, size_t n) const { // O(n log n log k)
            if(k == 0) {
                return poly_t(1).mod_xk(n);
            } else {
                auto t = pow(k / 2, n);
                t = (t * t).mod_xk(n);
                return (k % 2 ? *this * t : t).mod_xk(n);
            }
        }

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

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

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

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

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

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

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

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

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

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

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

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

        poly_t to_newton(std::vector<T> p) {
            if(is_zero()) {
                return *this;
            }
            int n = p.size();
            std::vector<poly_t> tree(4 * n);
            build(tree, 1, begin(p), end(p));
            return to_newton(tree, 1, begin(p), end(p));
        }

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

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

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

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

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

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