#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-algorithms-aux/cp-algo/algebra/fft.hpp"


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


#include <functional>
#include <cstdint>
namespace cp_algo::algebra {
    const int maxn = 1 << 12;
    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 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 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 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];
    }
}

#line 1 "cp-algorithms-aux/cp-algo/algebra/modular.hpp"


#line 1 "cp-algorithms-aux/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 6 "cp-algorithms-aux/cp-algo/algebra/modular.hpp"
#include <algorithm>
#include <iostream>
#line 9 "cp-algorithms-aux/cp-algo/algebra/modular.hpp"
namespace cp_algo::algebra {
    template<int m>
    struct modular {
        // https://en.wikipedia.org/wiki/Berlekamp-Rabin_algorithm
        std::optional<modular> sqrt() const {
            if(r == 0) {
                return 0;
            } else if(bpow(*this, (m - 1) / 2) != modular(1)) {
                return std::nullopt;
            } else {
                while(true) {
                    modular z = random::rng();
                    if(z * z == *this) {
                        return z;
                    }
                    lin<modular> x(1, z, *this); // x + z (mod x^2 - b)
                    x = bpow(x, (m - 1) / 2, lin<modular>(0, 1, *this));
                    if(x.a != modular(0)) {
                        return x.a.inv();
                    }
                }
            }
        }
        
        uint64_t r;
        constexpr modular(): r(0) {}
        constexpr modular(int64_t rr): r(rr % m) {r = std::min<uint64_t>(r, r + m);}
        modular inv() const {return bpow(*this, m - 2);}
        modular operator - () const {return std::min(-r, m - r);}
        modular operator * (const modular &t) const {return r * t.r;}
        modular operator / (const modular &t) const {return *this * t.inv();}
        modular& operator += (const modular &t) {r += t.r; r = std::min(r, r - m); return *this;}
        modular& operator -= (const modular &t) {r -= t.r; r = std::min(r, r + m); return *this;}
        modular operator + (const modular &t) const {return modular(*this) += t;}
        modular operator - (const modular &t) const {return modular(*this) -= t;}
        modular& operator *= (const modular &t) {return *this = *this * t;}
        modular& operator /= (const modular &t) {return *this = *this / t;}
        
        auto operator <=> (const modular &t) const = default;
        
        explicit operator int() const {return r;}
        int64_t rem() const {return 2 * r > m ? r - m : r;}

        // Only use if you really know what you're doing!
        static constexpr uint64_t mm = 8LL * m * m;
        void add_unsafe(uint64_t t) {r += t;}
        void pseudonormalize() {r = std::min(r, r - mm);}
        modular& normalize() {if(r >= m) r %= m; return *this;}
    };
    
    template<int m>
    std::istream& operator >> (std::istream &in, modular<m> &x) {
        return in >> x.r;
    }
    
    template<int m>
    std::ostream& operator << (std::ostream &out, modular<m> const& x) {
        return out << x.r % m;
    }
}

#line 7 "cp-algorithms-aux/cp-algo/algebra/fft.hpp"
#include <vector>
namespace cp_algo::algebra::fft {
    using ftype = double;
    struct point {
        ftype x, y;
        
        ftype real() {return x;}
        ftype imag() {return y;}
        
        point(): x(0), y(0){}
        point(ftype x, ftype y = 0): x(x), y(y){}
        
        static point polar(ftype rho, ftype ang) {
            return point{rho * cos(ang), rho * sin(ang)};
        }
        
        point conj() const {
            return {x, -y};
        }
        
        point operator +=(const point &t) {x += t.x, y += t.y; return *this;}
        point operator +(const point &t) const {return point(*this) += t;}
        point operator -(const point &t) const {return {x - t.x, y - t.y};}
        point operator *(const point &t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
    };

    point w[maxn]; // w[2^n + k] = exp(pi * k / (2^n))
    int bitr[maxn];// b[2^n + k] = bitreverse(k)
    const ftype pi = acos(-1);
    bool initiated = 0;
    void init() {
        if(!initiated) {
            for(int i = 1; i < maxn; i *= 2) {
                int ti = i / 2;
                for(int j = 0; j < i; j++) {
                    w[i + j] = point::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;
                }
            }
        }
    }
    
    void mul_slow(std::vector<auto> &a, const std::vector<auto> &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<modular<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<modular<m>>();
            }
            std::vector<point> C(n), D(n);
            for(size_t i = 0; i < n; i++) {
                C[i] = A[i] * (B[i] + B[(n - i) % n].conj());
                D[i] = A[i] * (B[i] - B[(n - i) % n].conj());
            }
            fft(C, n);
            fft(D, n);
            reverse(begin(C) + 1, end(C));
            reverse(begin(D) + 1, end(D));
            int t = 2 * n;
            std::vector<modular<m>> res(n);
            for(size_t i = 0; i < n; i++) {
                modular<m> A0 = llround(C[i].real() / t);
                modular<m> A1 = llround(C[i].imag() / t + D[i].imag() / t);
                modular<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<modular<m>> &a, std::vector<modular<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-algorithms-aux/cp-algo/algebra/poly/impl/euclid.hpp"
#include <numeric>
#line 11 "cp-algorithms-aux/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-algorithms-aux/cp-algo/algebra/poly/impl/base.hpp"


#line 6 "cp-algorithms-aux/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 r) {
        return std::vector(
            begin(p.a) + std::min(l, p.a.size()),
            begin(p.a) + std::min(r, 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-algorithms-aux/cp-algo/algebra/poly/impl/div.hpp"


#line 6 "cp-algorithms-aux/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 15 "cp-algorithms-aux/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 r) const {return poly::impl::substr(*this, l, r);}
        
        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 = (*this)[0].sqrt();
            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;
        }
    };
    
    static auto operator * (const auto& a, const poly_t<auto>& b) {
        return b * a;
    }
};

#line 1 "cp-algorithms-aux/cp-algo/linalg/matrix.hpp"


#line 1 "cp-algorithms-aux/cp-algo/linalg/vector.hpp"


#line 7 "cp-algorithms-aux/cp-algo/linalg/vector.hpp"
#include <valarray>
#line 9 "cp-algorithms-aux/cp-algo/linalg/vector.hpp"
#include <iterator>
namespace cp_algo::linalg {
    template<class vec, typename base>
    struct valarray_base: std::valarray<base> {
        using Base = std::valarray<base>;
        using Base::Base;

        valarray_base(base const& t): Base(t, 1) {}

        auto begin() {return std::begin(*static_cast<Base*>(this));}
        auto end() {return std::end(*static_cast<Base*>(this));}
        auto begin() const {return std::begin(*static_cast<Base const*>(this));}
        auto end() const {return std::end(*static_cast<Base const*>(this));}

        bool operator == (vec const& t) const {return std::ranges::equal(*this, t);}
        bool operator != (vec const& t) const {return !(*this == t);}

        vec operator-() const {return Base::operator-();}
        vec operator-(vec const& t) const {return Base::operator-(t);}
        vec operator+(vec const& t) const {return Base::operator+(t);}

        static vec from_range(auto const& R) {
            vec res(std::ranges::distance(R));
            std::ranges::copy(R, res.begin());
            return res;
        }
    };

    template<class vec, typename base>
    struct vec_base: valarray_base<vec, base> {
        using Base = valarray_base<vec, base>;
        using Base::Base;

        static vec ei(size_t n, size_t i) {
            vec res(n);
            res[i] = 1;
            return res;
        }

        // Make sure the result is vec, not Base
        vec operator*(base t) const {return Base::operator*(t);}

        void add_scaled(vec const& b, base scale, size_t i = 0) {
            assert(false);
            for(; i < size(*this); i++) {
                (*this)[i] += scale * b[i];
            }
        }
        auto& normalize() {
            return *this;
        }
        auto& normalize(size_t i) {
            return (*this)[i];
        }
        void read() {
            for(auto &it: *this) {
                std::cin >> it;
            }
        }
        void print() const {
            std::ranges::copy(*this, std::ostream_iterator<base>(std::cout, " "));
            std::cout << "\n";
        }
        static vec random(size_t n) {
            vec res(n);
            std::ranges::generate(res, random::rng);
            return res;
        }
        // Concatenate vectors
        vec operator |(vec const& t) const {
            vec res(size(*this) + size(t));
            res[std::slice(0, size(*this), 1)] = *this;
            res[std::slice(size(*this), size(t), 1)] = t;
            return res;
        }

        // Generally, vec shouldn't be modified
        // after it's pivot index is set
        std::pair<size_t, base> find_pivot() {
            auto true_this = static_cast<vec*>(this);
            if(pivot == size_t(-1)) {
                pivot = 0;
                while(pivot < size(*this) && true_this->normalize(pivot) == base(0)) {
                    pivot++;
                }
                if(pivot < size(*this)) {
                    pivot_inv = base(1) / (*this)[pivot];
                }
            }
            return {pivot, pivot_inv};
        }
        void reduce_by(vec &t) {
            auto true_this = static_cast<vec*>(this);
            auto [pivot, pinv] = t.find_pivot();
            if(pivot < size(*this)) {
                true_this->add_scaled(t, -true_this->normalize(pivot) * pinv, pivot);
            }
        }
    private:
        size_t pivot = -1;
        base pivot_inv;
    };

    template<typename base>
    struct vec: vec_base<vec<base>, base> {
        using Base = vec_base<vec<base>, base>;
        using Base::Base;
    };

    template<int mod>
    struct vec<algebra::modular<mod>>:
            vec_base<vec<algebra::modular<mod>>, algebra::modular<mod>> {
        using base = algebra::modular<mod>;
        using Base = vec_base<vec<base>, base>;
        using Base::Base;

        void add_scaled(vec const& b, base scale, size_t i = 0) {
            for(; i < size(*this); i++) {
                (*this)[i].add_unsafe(scale.r * b[i].r);
            }
            if(++counter == 8) {
                for(auto &it: *this) {
                    it.pseudonormalize();
                }
                counter = 0;
            }
        }
        auto& normalize() {
            for(auto &it: *this) {
                it.normalize();
            }
            return *this;
        }
        auto& normalize(size_t i) {
            return (*this)[i].normalize();
        }
    private:
        size_t counter = 0;
    };
}

#line 10 "cp-algorithms-aux/cp-algo/linalg/matrix.hpp"
#include <array>
namespace cp_algo::linalg {
    template<typename base>
    struct matrix: valarray_base<matrix<base>, vec<base>> {
        using Base = valarray_base<matrix<base>, vec<base>>;
        using Base::Base;

        matrix(size_t n): Base(vec<base>(n), n) {}
        matrix(size_t n, size_t m): Base(vec<base>(m), n) {}

        size_t n() const {return size(*this);}
        size_t m() const {return n() ? size(row(0)) : 0;}
        auto dim() const {return std::array{n(), m()};}

        auto& row(size_t i) {return (*this)[i];}
        auto const& row(size_t i) const {return (*this)[i];}

        matrix& operator *=(base t) {for(auto &it: *this) it *= t; return *this;}
        matrix operator *(base t) const {return matrix(*this) *= t;}

        // Make sure the result is matrix, not Base
        matrix& operator*=(matrix const& t) {return *this = *this * t;}

        void read() {
            for(auto &it: *this) {
                it.read();
            }
        }
        void print() const {
            for(auto const& it: *this) {
                it.print();
            }
        }

        static matrix eye(size_t n) {
            matrix res(n);
            for(size_t i = 0; i < n; i++) {
                res[i][i] = 1;
            }
            return res;
        }

        // Concatenate matrices
        matrix operator |(matrix const& b) const {
            assert(n() == b.n());
            matrix res(n(), m()+b.m());
            for(size_t i = 0; i < n(); i++) {
                res[i] = row(i) | b[i];
            }
            return res;
        }
        matrix submatrix(auto slicex, auto slicey) const {
            matrix res = (*this)[slicex];
            for(auto &row: res) {
                row = vec<base>(row[slicey]);
            }
            return res;
        }

        matrix T() const {
            matrix res(m(), n());
            for(size_t i = 0; i < n(); i++) {
                for(size_t j = 0; j < m(); j++) {
                    res[j][i] = row(i)[j];
                }
            }
            return res;
        }

        matrix operator *(matrix const& b) const {
            assert(m() == b.n());
            matrix res(n(), b.m());
            for(size_t i = 0; i < n(); i++) {
                for(size_t j = 0; j < m(); j++) {
                    res[i].add_scaled(b[j], row(i)[j]);
                }
            }
            return res.normalize();
        }

        vec<base> apply(vec<base> const& x) const {
            return (matrix(x) * *this)[0];
        }

        matrix pow(uint64_t k) const {
            assert(n() == m());
            return bpow(*this, k, eye(n()));
        }

        static matrix random(size_t n, size_t m) {
            matrix res(n, m);
            std::ranges::generate(res, std::bind(vec<base>::random, m));
            return res;
        }
        static matrix random(size_t n) {
            return random(n, n);
        }

        matrix& normalize() {
            for(auto &it: *this) {
                it.normalize();
            }
            return *this;
        }

        enum Mode {normal, reverse};
        template<Mode mode = normal>
        auto gauss(size_t lim) {
            size_t rk = 0;
            std::vector<size_t> free, pivots;
            for(size_t i = 0; i < lim; i++) {
                for(size_t j = rk; j < n() && row(rk).normalize(i) == base(0); j++) {
                    if(row(j).normalize(i) != 0) {
                        row(rk) += row(j);
                    }
                }
                if(rk == n() || row(rk).normalize()[i] == base(0)) {
                    free.push_back(i);
                } else {
                    pivots.push_back(i);
                    for(size_t j = (mode == normal) * rk; j < n(); j++) {
                        if(j != rk) {
                            row(j).reduce_by(row(rk));
                        }
                    }
                    rk += 1;
                }
            }
            normalize();
            return std::array{pivots, free};
        }
        template<Mode mode = normal>
        auto gauss() {
            return gauss<mode>(m());
        }

        size_t rank() const {
            if(n() < m()) {
                return T().rank();
            }
            return size(matrix(*this).gauss()[0]);
        }

        base det() const {
            assert(n() == m());
            matrix b = *this;
            b.gauss();
            base res = 1;
            for(size_t i = 0; i < n(); i++) {
                res *= b[i][i];
            }
            return res;
        }

        std::optional<matrix> inv() const {
            assert(n() == m());
            matrix b = *this | eye(n());
            if(size(b.gauss<reverse>(n())[0]) < n()) {
                return std::nullopt;
            }
            for(size_t i = 0; i < n(); i++) {
                b[i] *= base(1) / b[i][i];
            }
            return b.submatrix(std::slice(0, n(), 1), std::slice(n(), n(), 1));
        }

        // Can also just run gauss on T() | eye(m)
        // but it would be slower :(
        auto kernel() const {
            auto A = *this;
            auto [pivots, free] = A.template gauss<reverse>();
            matrix sols(size(free), m());
            for(size_t j = 0; j < size(pivots); j++) {
                base scale = base(1) / A[j][pivots[j]];
                for(size_t i = 0; i < size(free); i++) {
                    sols[i][pivots[j]] = A[j][free[i]] * scale;
                }
            }
            for(size_t i = 0; i < size(free); i++) {
                sols[i][free[i]] = -1;
            }
            return sols;
        }

        // [solution, basis], transposed
        std::optional<std::array<matrix, 2>> solve(matrix t) const {
            matrix sols = (*this | t).kernel();
            if(sols.n() < t.m() || sols.submatrix(
                std::slice(sols.n() - t.m(), t.m(), 1),
                std::slice(m(), t.m(), 1)
            ) != -eye(t.m())) {
                return std::nullopt;
            } else {
                return std::array{
                    sols.submatrix(std::slice(sols.n() - t.m(), t.m(), 1),
                                   std::slice(0, m(), 1)),
                    sols.submatrix(std::slice(0, sols.n() - t.m(), 1),
                                   std::slice(0, m(), 1))
                };
            }
        }
    };
}

#line 7 "main.cpp"
#include <bits/stdc++.h>

using namespace std;
using namespace cp_algo::algebra;
using namespace cp_algo::linalg;

const int mod = 998244353;
using base = modular<mod>;
using polyn = poly_t<base>;

void solve() {
    size_t n, q;
    cin >> n >> q;
    matrix<base> A(n);
    A.read();
    vector<vec<base>> basis;
    auto x = vec<base>::random(n);
    size_t degree = 0;
    polyn ans = base((n % 2) ? -1 : 1);
    while(size(basis) <= n) {
        vec<base> y = x | vec<base>::ei(n + 1, size(basis));
        for(auto &it: basis) {
            y.reduce_by(it);
        }
        y.normalize();
        if(vec<base>(y[slice(0, n, 1)]) == vec<base>(n)) {
            vector<base> cur(begin(y) + n + size(basis) - degree,
                             begin(y) + n + size(basis) + 1);
            ans *= polyn(cur);
            degree = 0;
            if(size(basis) < n) {
                x = vec<base>::random(n);
            } else {
                break;
            }
        } else {
            basis.push_back(y);
            x = A.apply(x);
            degree++;
        }
    }

    #define MXITER 2000
    vector<base> qs(MXITER);
    while (q > 0) {
        int v = (min((int) q, MXITER));
        qs.resize(v);
        for (int i = 0; i < v; i++) {
            cin >> qs[i];
        }
        for (auto x : ans.eval(qs)) {
            cout << x << " ";
        }
        q -= MXITER;
    }
    cout << endl;
}

signed main() {
    //freopen("input.txt", "r", stdin);
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t = 1;
    //cin >> t;
    while(t--) {
        solve();
    }
}