#include <bits/stdc++.h> #include <algorithm> #include <array> #ifdef _MSC_VER #include <intrin.h> #endif namespace atcoder { namespace internal { // @param n `0 <= n` // @return minimum non-negative `x` s.t. `n <= 2**x` int ceil_pow2(int n) { int x = 0; while ((1U << x) < (unsigned int)(n)) x++; return x; } // @param n `1 <= n` // @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0` int bsf(unsigned int n) { #ifdef _MSC_VER unsigned long index; _BitScanForward(&index, n); return index; #else return __builtin_ctz(n); #endif } } // namespace internal } // namespace atcoder #include <utility> #ifdef _MSC_VER #include <intrin.h> #endif namespace atcoder { namespace internal { // @param m `1 <= m` // @return x mod m constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } // Fast modular multiplication by barrett reduction // Reference: https://en.wikipedia.org/wiki/Barrett_reduction // NOTE: reconsider after Ice Lake struct barrett { unsigned int _m; unsigned long long im; // @param m `1 <= m < 2^31` barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {} // @return m unsigned int umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` unsigned int mul(unsigned int a, unsigned int b) const { // [1] m = 1 // a = b = im = 0, so okay // [2] m >= 2 // im = ceil(2^64 / m) // -> im * m = 2^64 + r (0 <= r < m) // let z = a*b = c*m + d (0 <= c, d < m) // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2 // ((ab * im) >> 64) == c or c + 1 unsigned long long z = a; z *= b; #ifdef _MSC_VER unsigned long long x; _umul128(z, im, &x); #else unsigned long long x = (unsigned long long)(((unsigned __int128)(z)*im) >> 64); #endif unsigned int v = (unsigned int)(z - x * _m); if (_m <= v) v += _m; return v; } }; // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` constexpr long long pow_mod_constexpr(long long x, long long n, int m) { if (m == 1) return 0; unsigned int _m = (unsigned int)(m); unsigned long long r = 1; unsigned long long y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; long long d = n - 1; while (d % 2 == 0) d /= 2; constexpr long long bases[3] = {2, 7, 61}; for (long long a : bases) { long long t = d; long long y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template <int n> constexpr bool is_prime = is_prime_constexpr(n); // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; // Contracts: // [1] s - m0 * a = 0 (mod b) // [2] t - m1 * a = 0 (mod b) // [3] s * |m1| + t * |m0| <= b long long s = b, t = a; long long m0 = 0, m1 = 1; while (t) { long long u = s / t; s -= t * u; m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b // [3]: // (s - t * u) * |m1| + t * |m0 - m1 * u| // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u) // = s * |m1| + t * |m0| <= b auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } // by [3]: |m0| <= b/g // by g != b: |m0| < b/g if (m0 < 0) m0 += b / s; return {s, m0}; } // Compile time primitive root // @param m must be prime // @return primitive root (and minimum in now) constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (long long)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template <int m> constexpr int primitive_root = primitive_root_constexpr(m); } // namespace internal } // namespace atcoder #include <cassert> #include <numeric> #include <type_traits> namespace atcoder { namespace internal { #ifndef _MSC_VER template <class T> using is_signed_int128 = typename std::conditional<std::is_same<T, __int128_t>::value || std::is_same<T, __int128>::value, std::true_type, std::false_type>::type; template <class T> using is_unsigned_int128 = typename std::conditional<std::is_same<T, __uint128_t>::value || std::is_same<T, unsigned __int128>::value, std::true_type, std::false_type>::type; template <class T> using make_unsigned_int128 = typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t, unsigned __int128>; template <class T> using is_integral = typename std::conditional<std::is_integral<T>::value || is_signed_int128<T>::value || is_unsigned_int128<T>::value, std::true_type, std::false_type>::type; template <class T> using is_signed_int = typename std::conditional<(is_integral<T>::value && std::is_signed<T>::value) || is_signed_int128<T>::value, std::true_type, std::false_type>::type; template <class T> using is_unsigned_int = typename std::conditional<(is_integral<T>::value && std::is_unsigned<T>::value) || is_unsigned_int128<T>::value, std::true_type, std::false_type>::type; template <class T> using to_unsigned = typename std::conditional< is_signed_int128<T>::value, make_unsigned_int128<T>, typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>, std::common_type<T>>::type>::type; #else template <class T> using is_integral = typename std::is_integral<T>; template <class T> using is_signed_int = typename std::conditional<is_integral<T>::value && std::is_signed<T>::value, std::true_type, std::false_type>::type; template <class T> using is_unsigned_int = typename std::conditional<is_integral<T>::value && std::is_unsigned<T>::value, std::true_type, std::false_type>::type; template <class T> using to_unsigned = typename std::conditional<is_signed_int<T>::value, std::make_unsigned<T>, std::common_type<T>>::type; #endif template <class T> using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>; template <class T> using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>; template <class T> using to_unsigned_t = typename to_unsigned<T>::type; } // namespace internal } // namespace atcoder #include <cassert> #include <numeric> #include <type_traits> #ifdef _MSC_VER #include <intrin.h> #endif namespace atcoder { namespace internal { struct modint_base {}; struct static_modint_base : modint_base {}; template <class T> using is_modint = std::is_base_of<modint_base, T>; template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>; } // namespace internal template <int m, std::enable_if_t<(1 <= m)>* = nullptr> struct static_modint : internal::static_modint_base { using mint = static_modint; public: static constexpr int mod() { return m; } static mint raw(int v) { mint x; x._v = v; return x; } static_modint() : _v(0) {} template <class T, internal::is_signed_int_t<T>* = nullptr> static_modint(T v) { long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template <class T, internal::is_unsigned_int_t<T>* = nullptr> static_modint(T v) { _v = (unsigned int)(v % umod()); } static_modint(bool v) { _v = ((unsigned int)(v) % umod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } mint& operator*=(const mint& rhs) { unsigned long long z = _v; z *= rhs._v; _v = (unsigned int)(z % umod()); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { if (prime) { assert(_v); return pow(umod() - 2); } else { auto eg = internal::inv_gcd(_v, m); assert(eg.first == 1); return eg.second; } } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static constexpr unsigned int umod() { return m; } static constexpr bool prime = internal::is_prime<m>; }; template <int id> struct dynamic_modint : internal::modint_base { using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = internal::barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template <class T, internal::is_signed_int_t<T>* = nullptr> dynamic_modint(T v) { long long x = (long long)(v % (long long)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } template <class T, internal::is_unsigned_int_t<T>* = nullptr> dynamic_modint(T v) { _v = (unsigned int)(v % mod()); } dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v += mod() - rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator*=(const mint& rhs) { _v = bt.mul(_v, rhs._v); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { auto eg = internal::inv_gcd(_v, mod()); assert(eg.first == 1); return eg.second; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static internal::barrett bt; static unsigned int umod() { return bt.umod(); } }; template <int id> internal::barrett dynamic_modint<id>::bt = 998244353; using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; using modint = dynamic_modint<-1>; namespace internal { template <class T> using is_static_modint = std::is_base_of<internal::static_modint_base, T>; template <class T> using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>; template <class> struct is_dynamic_modint : public std::false_type {}; template <int id> struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {}; template <class T> using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>; } // namespace internal } // namespace atcoder #include <cassert> #include <type_traits> #include <vector> namespace atcoder { namespace internal { template <class mint, internal::is_static_modint_t<mint>* = nullptr> void butterfly(std::vector<mint>& a) { static constexpr int g = internal::primitive_root<mint::mod()>; int n = int(a.size()); int h = internal::ceil_pow2(n); static bool first = true; static mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i] if (first) { first = false; mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1 int cnt2 = bsf(mint::mod() - 1); mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv(); for (int i = cnt2; i >= 2; i--) { // e^(2^i) == 1 es[i - 2] = e; ies[i - 2] = ie; e *= e; ie *= ie; } mint now = 1; for (int i = 0; i <= cnt2 - 2; i++) { sum_e[i] = es[i] * now; now *= ies[i]; } } for (int ph = 1; ph <= h; ph++) { int w = 1 << (ph - 1), p = 1 << (h - ph); mint now = 1; for (int s = 0; s < w; s++) { int offset = s << (h - ph + 1); for (int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p] * now; a[i + offset] = l + r; a[i + offset + p] = l - r; } now *= sum_e[bsf(~(unsigned int)(s))]; } } } template <class mint, internal::is_static_modint_t<mint>* = nullptr> void butterfly_inv(std::vector<mint>& a) { static constexpr int g = internal::primitive_root<mint::mod()>; int n = int(a.size()); int h = internal::ceil_pow2(n); static bool first = true; static mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i] if (first) { first = false; mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1 int cnt2 = bsf(mint::mod() - 1); mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv(); for (int i = cnt2; i >= 2; i--) { // e^(2^i) == 1 es[i - 2] = e; ies[i - 2] = ie; e *= e; ie *= ie; } mint now = 1; for (int i = 0; i <= cnt2 - 2; i++) { sum_ie[i] = ies[i] * now; now *= es[i]; } } for (int ph = h; ph >= 1; ph--) { int w = 1 << (ph - 1), p = 1 << (h - ph); mint inow = 1; for (int s = 0; s < w; s++) { int offset = s << (h - ph + 1); for (int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p]; a[i + offset] = l + r; a[i + offset + p] = (unsigned long long)(mint::mod() + l.val() - r.val()) * inow.val(); } inow *= sum_ie[bsf(~(unsigned int)(s))]; } } } } // namespace internal template <class mint, internal::is_static_modint_t<mint>* = nullptr> std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; if (std::min(n, m) <= 60) { if (n < m) { std::swap(n, m); std::swap(a, b); } std::vector<mint> ans(n + m - 1); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { ans[i + j] += a[i] * b[j]; } } return ans; } int z = 1 << internal::ceil_pow2(n + m - 1); a.resize(z); internal::butterfly(a); b.resize(z); internal::butterfly(b); for (int i = 0; i < z; i++) { a[i] *= b[i]; } internal::butterfly_inv(a); a.resize(n + m - 1); mint iz = mint(z).inv(); for (int i = 0; i < n + m - 1; i++) a[i] *= iz; return a; } template <unsigned int mod = 998244353, class T, std::enable_if_t<internal::is_integral<T>::value>* = nullptr> std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; using mint = static_modint<mod>; std::vector<mint> a2(n), b2(m); for (int i = 0; i < n; i++) { a2[i] = mint(a[i]); } for (int i = 0; i < m; i++) { b2[i] = mint(b[i]); } auto c2 = convolution(move(a2), move(b2)); std::vector<T> c(n + m - 1); for (int i = 0; i < n + m - 1; i++) { c[i] = c2[i].val(); } return c; } std::vector<long long> convolution_ll(const std::vector<long long>& a, const std::vector<long long>& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; static constexpr unsigned long long MOD1 = 754974721; // 2^24 static constexpr unsigned long long MOD2 = 167772161; // 2^25 static constexpr unsigned long long MOD3 = 469762049; // 2^26 static constexpr unsigned long long M2M3 = MOD2 * MOD3; static constexpr unsigned long long M1M3 = MOD1 * MOD3; static constexpr unsigned long long M1M2 = MOD1 * MOD2; static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3; static constexpr unsigned long long i1 = internal::inv_gcd(MOD2 * MOD3, MOD1).second; static constexpr unsigned long long i2 = internal::inv_gcd(MOD1 * MOD3, MOD2).second; static constexpr unsigned long long i3 = internal::inv_gcd(MOD1 * MOD2, MOD3).second; auto c1 = convolution<MOD1>(a, b); auto c2 = convolution<MOD2>(a, b); auto c3 = convolution<MOD3>(a, b); std::vector<long long> c(n + m - 1); for (int i = 0; i < n + m - 1; i++) { unsigned long long x = 0; x += (c1[i] * i1) % MOD1 * M2M3; x += (c2[i] * i2) % MOD2 * M1M3; x += (c3[i] * i3) % MOD3 * M1M2; // B = 2^63, -B <= x, r(real value) < B // (x, x - M, x - 2M, or x - 3M) = r (mod 2B) // r = c1[i] (mod MOD1) // focus on MOD1 // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B) // r = x, // x - M' + (0 or 2B), // x - 2M' + (0, 2B or 4B), // x - 3M' + (0, 2B, 4B or 6B) (without mod!) // (r - x) = 0, (0) // - M' + (0 or 2B), (1) // -2M' + (0 or 2B or 4B), (2) // -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1) // we checked that // ((1) mod MOD1) mod 5 = 2 // ((2) mod MOD1) mod 5 = 3 // ((3) mod MOD1) mod 5 = 4 long long diff = c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1)); if (diff < 0) diff += MOD1; static constexpr unsigned long long offset[5] = { 0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3}; x -= offset[diff % 5]; c[i] = x; } return c; } } // namespace atcoder using i64 = long long; using u64 = unsigned long long; #define REP(i, n) for (int i = 0, REP_N_ = int(n); i < REP_N_; ++i) #define ALL(x) std::begin(x), std::end(x) template <class T> inline bool chmax(T &a, T b) { return a < b and ((a = std::move(b)), true); } template <class T> inline bool chmin(T &a, T b) { return a > b and ((a = std::move(b)), true); } template <typename T> using V = std::vector<T>; template <typename T> std::istream &operator>>(std::istream &is, std::vector<T> &a) { for (auto &x : a) is >> x; return is; } template <typename Container> std::ostream &pprint(const Container &a, std::string_view sep = " ", std::string_view ends = "\n", std::ostream *os = nullptr) { if (os == nullptr) os = &std::cout; auto b = std::begin(a), e = std::end(a); for (auto it = std::begin(a); it != e; ++it) { if (it != b) *os << sep; *os << *it; } return *os << ends; } template <typename T, typename = void> struct is_iterable : std::false_type {}; template <typename T> struct is_iterable<T, std::void_t<decltype(std::begin(std::declval<T>())), decltype(std::end(std::declval<T>()))>> : std::true_type {}; template <typename T, typename = std::enable_if_t<is_iterable<T>::value && !std::is_same<T, std::string>::value>> std::ostream &operator<<(std::ostream &os, const T &a) { return pprint(a, ", ", "", &(os << "{")) << "}"; } template <typename T, typename U> std::ostream &operator<<(std::ostream &os, const std::pair<T, U> &a) { return os << "(" << a.first << ", " << a.second << ")"; } #ifdef ENABLE_DEBUG template <typename T> void pdebug(const T &value) { std::cerr << value; } template <typename T, typename... Ts> void pdebug(const T &value, const Ts &... args) { pdebug(value); std::cerr << ", "; pdebug(args...); } #define DEBUG(...) \ do { \ std::cerr << " \033[33m (L" << __LINE__ << ") "; \ std::cerr << #__VA_ARGS__ << ":\033[0m "; \ pdebug(__VA_ARGS__); \ std::cerr << std::endl; \ } while (0) #else #define pdebug(...) #define DEBUG(...) #endif using namespace std; // Formal Power Series (dense format). template <typename T, int DMAX> struct DenseFPS { // Coefficients of terms from x^0 to x^DMAX. std::vector<T> coeff_; DenseFPS() : coeff_(DMAX + 1) {} // zero-initialized explicit DenseFPS(std::vector<T> c) : coeff_(std::move(c)) { assert((int)coeff_.size() == DMAX + 1); } DenseFPS(const DenseFPS &other) : coeff_(other.coeff_) {} DenseFPS(DenseFPS &&other) : coeff_(std::move(other.coeff_)) {} DenseFPS &operator=(const DenseFPS &other) { coeff_ = other.coeff_; return *this; } DenseFPS &operator=(DenseFPS &&other) { coeff_ = std::move(other.coeff_); return *this; } static constexpr int size() { return DMAX + 1; } // Returns the coefficient of x^d. const T &operator[](int d) const { return coeff_[d]; } DenseFPS &operator+=(const T &scalar) { coeff_[0] += scalar; return *this; } friend DenseFPS operator+(const DenseFPS &x, const T &scalar) { DenseFPS res = x; res += scalar; return res; } DenseFPS &operator+=(const DenseFPS &other) { for (int i = 0; i < size(); ++i) coeff_[i] += other[i]; return *this; } friend DenseFPS operator+(const DenseFPS &x, const DenseFPS &y) { DenseFPS res = x; res += y; return res; } DenseFPS &operator-=(const DenseFPS &other) { for (int i = 0; i < size(); ++i) coeff_[i] -= other[i]; return *this; } friend DenseFPS operator-(const DenseFPS &x, const DenseFPS &y) { DenseFPS res = x; res -= y; return res; } DenseFPS &operator*=(const T &scalar) { for (auto &x : coeff_) x *= scalar; return *this; } friend DenseFPS operator*(const DenseFPS &x, const T &scalar) { DenseFPS res = x; res *= scalar; return res; } DenseFPS &operator*=(const DenseFPS &other) { *this = this->mul_naive(other); return *this; } friend DenseFPS operator*(const DenseFPS &x, const DenseFPS &y) { return x.mul_naive(y); } private: // Naive multiplication. O(N^2) DenseFPS mul_naive(const DenseFPS &other) const { DenseFPS res; for (int i = 0; i < size(); ++i) { for (int j = 0; i + j < size(); ++j) { res.coeff_[i + j] += (*this)[i] * other[j]; } } return res; } }; namespace fps { // Fast polynomial multiplication by single NTT. template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> mul_ntt(const DenseFPS<ModInt, DMAX> &x, const DenseFPS<ModInt, DMAX> &y) { static_assert(ModInt::mod() != 1'000'000'007); // Must be a NTT-friendly MOD! auto z = atcoder::convolution(x.coeff_, y.coeff_); z.resize(DMAX + 1); // Maybe shrink. return {std::move(z)}; } // Polynomial multiplication by NTT + Garner (arbitrary mod). template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> mul_mod(const DenseFPS<ModInt, DMAX> &x, const DenseFPS<ModInt, DMAX> &y) { std::vector<i64> xll(x.size()), yll(y.size()); for (int i = 0; i < x.size(); ++i) { xll[i] = x[i].val(); } for (int i = 0; i < y.size(); ++i) { yll[i] = y[i].val(); } auto zll = atcoder::convolution_ll(xll, yll); DenseFPS<ModInt, DMAX> res; int n = std::min<int>(res.size(), zll.size()); for (int i = 0; i < n; ++i) { res.coeff_[i] = zll[i]; } return res; } // Polynomial multiplication by NTT + Garner (long long). template <int DMAX> DenseFPS<i64, DMAX> mul_ll(const DenseFPS<i64, DMAX> &x, const DenseFPS<i64, DMAX> &y) { auto z = atcoder::convolution_ll(x.coeff_, y.coeff_); z.resize(DMAX + 1); // Maybe shrink. return DenseFPS<i64, DMAX>(std::move(z)); } template <typename T, int DMAX, typename Func> DenseFPS<T, DMAX> pow_generic(const DenseFPS<T, DMAX> &x, u64 t, Func mulfunc) { DenseFPS<T, DMAX> base = x, res; res.coeff_[0] = 1; while (t) { if (t & 1) res = mulfunc(res, base); base = mulfunc(base, base); t >>= 1; } return res; } template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> pow_ntt(const DenseFPS<ModInt, DMAX> &x, u64 t) { return pow_generic(x, t, mul_ntt); } template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> pow_mod(const DenseFPS<ModInt, DMAX> &x, u64 t) { return pow_generic(x, t, mul_mod); } template <int DMAX> DenseFPS<i64, DMAX> pow_ll(const DenseFPS<i64, DMAX> &x, u64 t) { return pow_generic(x, t, mul_ll); } } // namespace fps // Formal Power Series (sparse format). template <typename T> struct SparseFPS { int size_; std::vector<int> degree_; std::vector<T> coeff_; SparseFPS() : size_(0) {} explicit SparseFPS(std::vector<std::pair<int, T>> terms) : size_(terms.size()), degree_(size_), coeff_(size_) { // Sort by degree_ in ascending order. sort(terms.begin(), terms.end()); for (int i = 0; i < size_; ++i) { degree_[i] = terms[i].first; coeff_[i] = terms[i].second; } } inline int size() const { return size_; } inline int degree(int i) const { return degree_[i]; } inline const T &coeff(int i) const { return coeff_[i]; } int DMAX() const { return (size_ == 0) ? 0 : degree_.back(); } void emplace_back(int d, T c) { if (not degree_.empty()) { assert(d > degree_.back()); } degree_.push_back(std::move(d)); coeff_.push_back(std::move(c)); ++size_; } // Returns the coefficient of x^d. T operator[](int d) const { auto it = std::lower_bound(degree_.begin(), degree_.end(), d); if (it == degree_.end() or *it != d) return (T)(0); int j = std::distance(degree_.begin(), it); return coeff_[j]; } SparseFPS &operator+=(const T &scalar) { for (auto &x : coeff_) x += scalar; return *this; } friend SparseFPS operator+(const SparseFPS &x, const T &scalar) { SparseFPS res = x; res += scalar; return res; } SparseFPS &operator+=(const SparseFPS &other) { *this = this->add(other); return *this; } friend SparseFPS operator+(const SparseFPS &x, const SparseFPS &y) { return x.add(y); } SparseFPS &operator*=(const T &scalar) { for (auto &x : coeff_) x *= scalar; return *this; } friend SparseFPS operator*(const SparseFPS &x, const T &scalar) { SparseFPS res = x; res *= scalar; return res; } SparseFPS &operator-=(const SparseFPS &other) { *this = this->add(other * -1); return *this; } friend SparseFPS operator-(const SparseFPS &x, const SparseFPS &y) { return x.add(y * -1); } SparseFPS mul(const SparseFPS &other, int max_degree) const { std::map<int, T> terms; for (int i = 0; i < size(); ++i) { int di = degree(i); T ci = coeff(i); for (int j = 0; j < other.size(); ++j) { int dj = other.degree(j); if (di + dj > max_degree) break; terms[di + dj] += ci * other.coeff(j); } } SparseFPS res; for (auto &[d, c] : terms) { res.emplace_back(d, c); } return res; } private: SparseFPS add(const SparseFPS &other) const { SparseFPS res; int j = 0; for (int i = 0; i < size();) { const int deg = this->degree(i); for (; j < other.size() and other.degree(j) < deg; ++j) { res.emplace_back(other.degree(j), other.coeff(j)); } T c = this->coeff(i); if (j < other.size() and other.degree(j) == deg) { c += other.coeff(j); ++j; } if (c != 0) { res.emplace_back(deg, c); } } for (; j < other.size(); ++j) { res.emplace_back(other.degree(j), other.coeff(j)); } return res; } }; // Polynomial addition (dense + sparse). template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> &operator+=(DenseFPS<ModInt, DMAX> &x, const SparseFPS<ModInt> &y) { for (int i = 0; i < y.size(); ++i) { if (y.degree(i) > DMAX) break; // ignore x.coeff_[y.degree(i)] += y.coeff(i); } return x; } template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> operator+(const DenseFPS<ModInt, DMAX> &x, const SparseFPS<ModInt> &y) { DenseFPS<ModInt, DMAX> res = x; res += y; return res; } template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> operator+(const SparseFPS<ModInt> &x, const DenseFPS<ModInt, DMAX> &y) { return y + x; // commutative } // Polynomial multiplication (dense * sparse). template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> &operator*=(DenseFPS<ModInt, DMAX> &x, const SparseFPS<ModInt> &y) { if (y.size() == 0) { return x *= 0; } ModInt c0 = 0; int j0 = 0; if (y.degree(0) == 0) { c0 = y.coeff(0); ++j0; } for (int i = DMAX; i >= 0; --i) { x.coeff_[i] *= c0; for (int j = j0; j < y.size(); ++j) { int d = y.degree(j); if (d > i) break; x.coeff_[i] += x[i - d] * y.coeff(j); } } return x; } template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> operator*(const DenseFPS<ModInt, DMAX> &x, const SparseFPS<ModInt> &y) { DenseFPS<ModInt, DMAX> res = x; res *= y; return res; } template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> operator*(const SparseFPS<ModInt> &x, const DenseFPS<ModInt, DMAX> &y) { return y * x; // commutative } // Polynomial division (dense / sparse). template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> &operator/=(DenseFPS<ModInt, DMAX> &x, const SparseFPS<ModInt> &y) { assert(y.size() > 0 and y.degree(0) == 0 and y.coeff(0) != 0); ModInt inv_c0 = y.coeff(0).inv(); for (int i = 0; i < x.size(); ++i) { for (int j = 1; j < y.size(); ++j) { int d = y.degree(j); if (d > i) break; x.coeff_[i] -= x.coeff_[i - d] * y.coeff[j]; } x.coeff_[i] *= inv_c0; } return x; } template <typename ModInt, int DMAX> DenseFPS<ModInt, DMAX> operator/(const DenseFPS<ModInt, DMAX> &x, const SparseFPS<ModInt> &y) { DenseFPS<ModInt, DMAX> res = x; res /= y; return res; } using mint = atcoder::modint998244353; const int MOD = 998244353; const int BMAX = 200'000; template <class T> struct Factorials { // factorials and inverse factorials. std::vector<T> fact, ifact; // n: max cached value. Factorials(size_t n) : fact(n + 1), ifact(n + 1) { assert(n > 0 and n < MOD); fact[0] = 1; for (size_t i = 1; i <= n; ++i) fact[i] = fact[i - 1] * i; ifact[n] = fact[n].inv(); for (size_t i = n; i >= 1; --i) ifact[i - 1] = ifact[i] * i; } // Combination (nCk) T C(int n, int k) { if (k < 0 || k > n) return 0; return fact[n] * ifact[k] * ifact[n - k]; } // Permutation (nPk) T P(int n, int k) { if (k < 0 || k > n) return 0; return fact[n] * ifact[n - k]; } }; int main() { ios::sync_with_stdio(false); cin.tie(nullptr); int N, Q; cin >> N >> Q; V<int> A(N); cin >> A; deque<SparseFPS<mint>> q; REP(i, N) { SparseFPS<mint> g; g.emplace_back(0, mint(A[i] - 1)); g.emplace_back(1, mint(1)); q.push_back(move(g)); } while (q.size() > 1) { SparseFPS p = q[0].mul(q[1], BMAX); q.pop_front(); q.pop_front(); q.push_back(move(p)); } const SparseFPS<mint> &f = q.front(); REP(i, Q) { int b; cin >> b; cout << f[b].val() << '\n'; assert(false); } }