#include #ifdef LOCAL #include #else #define debug(...) void(0) #endif #include #ifdef _MSC_VER #include #endif #ifdef _MSC_VER #include #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` explicit 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 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 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 constexpr int primitive_root = primitive_root_constexpr(m); // @param n `n < 2^32` // @param m `1 <= m < 2^32` // @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64) unsigned long long floor_sum_unsigned(unsigned long long n, unsigned long long m, unsigned long long a, unsigned long long b) { unsigned long long ans = 0; while (true) { if (a >= m) { ans += n * (n - 1) / 2 * (a / m); a %= m; } if (b >= m) { ans += n * (b / m); b %= m; } unsigned long long y_max = a * n + b; if (y_max < m) break; // y_max < m * (n + 1) // floor(y_max / m) <= n n = (unsigned long long)(y_max / m); b = (unsigned long long)(y_max % m); std::swap(m, a); } return ans; } } // namespace internal } // namespace atcoder namespace atcoder { namespace internal { #ifndef _MSC_VER template using is_signed_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using is_unsigned_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using make_unsigned_int128 = typename std::conditional::value, __uint128_t, unsigned __int128>; template using is_integral = typename std::conditional::value || is_signed_int128::value || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using is_signed_int = typename std::conditional<(is_integral::value && std::is_signed::value) || is_signed_int128::value, std::true_type, std::false_type>::type; template using is_unsigned_int = typename std::conditional<(is_integral::value && std::is_unsigned::value) || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using to_unsigned = typename std::conditional< is_signed_int128::value, make_unsigned_int128, typename std::conditional::value, std::make_unsigned, std::common_type>::type>::type; #else template using is_integral = typename std::is_integral; template using is_signed_int = typename std::conditional::value && std::is_signed::value, std::true_type, std::false_type>::type; template using is_unsigned_int = typename std::conditional::value && std::is_unsigned::value, std::true_type, std::false_type>::type; template using to_unsigned = typename std::conditional::value, std::make_unsigned, std::common_type>::type; #endif template using is_signed_int_t = std::enable_if_t::value>; template using is_unsigned_int_t = std::enable_if_t::value>; template using to_unsigned_t = typename to_unsigned::type; } // namespace internal } // namespace atcoder namespace atcoder { namespace internal { struct modint_base {}; struct static_modint_base : modint_base {}; template using is_modint = std::is_base_of; template using is_modint_t = std::enable_if_t::value>; } // namespace internal template * = 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 * = nullptr> static_modint(T v) { long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template * = nullptr> static_modint(T 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; }; template 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 * = nullptr> dynamic_modint(T v) { long long x = (long long)(v % (long long)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } template * = nullptr> dynamic_modint(T 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 internal::barrett dynamic_modint::bt(998244353); using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; using modint = dynamic_modint<-1>; namespace internal { template using is_static_modint = std::is_base_of; template using is_static_modint_t = std::enable_if_t::value>; template struct is_dynamic_modint : public std::false_type {}; template struct is_dynamic_modint> : public std::true_type {}; template using is_dynamic_modint_t = std::enable_if_t::value>; } // namespace internal } // namespace atcoder #ifdef _MSC_VER #include #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` constexpr int bsf_constexpr(unsigned int n) { int x = 0; while (!(n & (1 << x))) 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 namespace atcoder { namespace internal { template , internal::is_static_modint_t* = nullptr> struct fft_info { static constexpr int rank2 = bsf_constexpr(mint::mod() - 1); std::array root; // root[i]^(2^i) == 1 std::array iroot; // root[i] * iroot[i] == 1 std::array rate2; std::array irate2; std::array rate3; std::array irate3; fft_info() { root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2); iroot[rank2] = root[rank2].inv(); for (int i = rank2 - 1; i >= 0; i--) { root[i] = root[i + 1] * root[i + 1]; iroot[i] = iroot[i + 1] * iroot[i + 1]; } { mint prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 2; i++) { rate2[i] = root[i + 2] * prod; irate2[i] = iroot[i + 2] * iprod; prod *= iroot[i + 2]; iprod *= root[i + 2]; } } { mint prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 3; i++) { rate3[i] = root[i + 3] * prod; irate3[i] = iroot[i + 3] * iprod; prod *= iroot[i + 3]; iprod *= root[i + 3]; } } } }; template * = nullptr> void butterfly(std::vector& a) { int n = int(a.size()); int h = internal::ceil_pow2(n); static const fft_info info; int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed while (len < h) { if (h - len == 1) { int p = 1 << (h - len - 1); mint rot = 1; for (int s = 0; s < (1 << len); s++) { int offset = s << (h - len); for (int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p] * rot; a[i + offset] = l + r; a[i + offset + p] = l - r; } if (s + 1 != (1 << len)) rot *= info.rate2[bsf(~(unsigned int)(s))]; } len++; } else { // 4-base int p = 1 << (h - len - 2); mint rot = 1, imag = info.root[2]; for (int s = 0; s < (1 << len); s++) { mint rot2 = rot * rot; mint rot3 = rot2 * rot; int offset = s << (h - len); for (int i = 0; i < p; i++) { auto mod2 = 1ULL * mint::mod() * mint::mod(); auto a0 = 1ULL * a[i + offset].val(); auto a1 = 1ULL * a[i + offset + p].val() * rot.val(); auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val(); auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val(); auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val(); auto na2 = mod2 - a2; a[i + offset] = a0 + a2 + a1 + a3; a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3)); a[i + offset + 2 * p] = a0 + na2 + a1na3imag; a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag); } if (s + 1 != (1 << len)) rot *= info.rate3[bsf(~(unsigned int)(s))]; } len += 2; } } } template * = nullptr> void butterfly_inv(std::vector& a) { int n = int(a.size()); int h = internal::ceil_pow2(n); static const fft_info info; int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed while (len) { if (len == 1) { int p = 1 << (h - len); mint irot = 1; for (int s = 0; s < (1 << (len - 1)); s++) { int offset = s << (h - len + 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()) * irot.val(); ; } if (s + 1 != (1 << (len - 1))) irot *= info.irate2[bsf(~(unsigned int)(s))]; } len--; } else { // 4-base int p = 1 << (h - len); mint irot = 1, iimag = info.iroot[2]; for (int s = 0; s < (1 << (len - 2)); s++) { mint irot2 = irot * irot; mint irot3 = irot2 * irot; int offset = s << (h - len + 2); for (int i = 0; i < p; i++) { auto a0 = 1ULL * a[i + offset + 0 * p].val(); auto a1 = 1ULL * a[i + offset + 1 * p].val(); auto a2 = 1ULL * a[i + offset + 2 * p].val(); auto a3 = 1ULL * a[i + offset + 3 * p].val(); auto a2na3iimag = 1ULL * mint((mint::mod() + a2 - a3) * iimag.val()).val(); a[i + offset] = a0 + a1 + a2 + a3; a[i + offset + 1 * p] = (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val(); a[i + offset + 2 * p] = (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) * irot2.val(); a[i + offset + 3 * p] = (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) * irot3.val(); } if (s + 1 != (1 << (len - 2))) irot *= info.irate3[bsf(~(unsigned int)(s))]; } len -= 2; } } } template * = nullptr> std::vector convolution_naive(const std::vector& a, const std::vector& b) { int n = int(a.size()), m = int(b.size()); std::vector ans(n + m - 1); if (n < m) { for (int j = 0; j < m; j++) { for (int i = 0; i < n; i++) { ans[i + j] += a[i] * b[j]; } } } else { for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { ans[i + j] += a[i] * b[j]; } } } return ans; } template * = nullptr> std::vector convolution_fft(std::vector a, std::vector b) { int n = int(a.size()), m = int(b.size()); 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; } } // namespace internal template * = nullptr> std::vector convolution(std::vector&& a, std::vector&& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; if (std::min(n, m) <= 60) return convolution_naive(a, b); return internal::convolution_fft(a, b); } template * = nullptr> std::vector convolution(const std::vector& a, const std::vector& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; if (std::min(n, m) <= 60) return convolution_naive(a, b); return internal::convolution_fft(a, b); } template ::value>* = nullptr> std::vector convolution(const std::vector& a, const std::vector& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; using mint = static_modint; std::vector 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 c(n + m - 1); for (int i = 0; i < n + m - 1; i++) { c[i] = c2[i].val(); } return c; } std::vector convolution_ll(const std::vector& a, const std::vector& 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(a, b); auto c2 = convolution(a, b); auto c3 = convolution(a, b); std::vector 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 template struct FormalPowerSeries : std::vector { private: using std::vector::vector; using FPS = FormalPowerSeries; void shrink() { while (this->size() and this->back() == T(0)) this->pop_back(); } FPS pre(size_t sz) const { return FPS(this->begin(), this->begin() + std::min(this->size(), sz)); } FPS rev() const { FPS ret(*this); std::reverse(ret.begin(), ret.end()); return ret; } FPS operator>>(size_t sz) const { if (this->size() <= sz) return {}; return FPS(this->begin() + sz, this->end()); } FPS operator<<(size_t sz) const { if (this->empty()) return {}; FPS ret(*this); ret.insert(ret.begin(), sz, T(0)); return ret; } public: FPS& operator+=(const FPS& r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < int(r.size()); i++) (*this)[i] += r[i]; shrink(); return *this; } FPS& operator+=(const T& v) { if (this->empty()) this->resize(1); (*this)[0] += v; shrink(); return *this; } FPS& operator-=(const FPS& r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < int(r.size()); i++) (*this)[i] -= r[i]; shrink(); return *this; } FPS& operator-=(const T& v) { if (this->empty()) this->resize(1); (*this)[0] -= v; shrink(); return *this; } FPS& operator*=(const FPS& r) { auto res = atcoder::convolution(*this, r); return *this = {res.begin(), res.end()}; } FPS& operator*=(const T& v) { for (auto& x : (*this)) x *= v; shrink(); return *this; } FPS& operator/=(const FPS& r) { if (this->size() < r.size()) { this->clear(); return *this; } int n = this->size() - r.size() + 1; return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(); } FPS& operator%=(const FPS& r) { *this -= *this / r * r; shrink(); return *this; } FPS operator+(const FPS& r) const { return FPS(*this) += r; } FPS operator+(const T& v) const { return FPS(*this) += v; } FPS operator-(const FPS& r) const { return FPS(*this) -= r; } FPS operator-(const T& v) const { return FPS(*this) -= v; } FPS operator*(const FPS& r) const { return FPS(*this) *= r; } FPS operator*(const T& v) const { return FPS(*this) *= v; } FPS operator/(const FPS& r) const { return FPS(*this) /= r; } FPS operator%(const FPS& r) const { return FPS(*this) %= r; } FPS operator-() const { FPS ret = *this; for (auto& v : ret) v = -v; return ret; } FPS differential() const { const int n = (int)this->size(); FPS ret(std::max(0, n - 1)); for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i); return ret; } FPS integral() const { const int n = (int)this->size(); FPS ret(n + 1); ret[0] = T(0); if (n > 0) ret[1] = T(1); auto mod = T::mod(); for (int i = 2; i <= n; i++) ret[i] = -ret[mod % i] * (mod / i); for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i]; return ret; } FPS inv(int deg = -1) const { assert((*this)[0] != T(0)); const int n = (int)this->size(); if (deg == -1) deg = n; FPS ret{(*this)[0].inv()}; ret.reserve(deg); for (int d = 1; d < deg; d <<= 1) { FPS f(d << 1), g(d << 1); std::copy(this->begin(), this->begin() + std::min(n, d << 1), f.begin()); std::copy(ret.begin(), ret.end(), g.begin()); atcoder::internal::butterfly(f); atcoder::internal::butterfly(g); for (int i = 0; i < (d << 1); i++) f[i] *= g[i]; atcoder::internal::butterfly_inv(f); std::fill(f.begin(), f.begin() + d, T(0)); atcoder::internal::butterfly(f); for (int i = 0; i < (d << 1); i++) f[i] *= g[i]; atcoder::internal::butterfly_inv(f); T iz = T(d << 1).inv(); iz *= -iz; for (int i = d; i < std::min(d << 1, deg); i++) ret.push_back(f[i] * iz); } return ret.pre(deg); } FPS log(int deg = -1) const { assert((*this)[0] == T(1)); if (deg == -1) deg = (int)this->size(); return (differential() * inv(deg)).pre(deg - 1).integral(); } FPS sqrt(const std::function& get_sqrt, int deg = -1) const { const int n = this->size(); if (deg == -1) deg = n; if (this->empty()) return FPS(deg, 0); if ((*this)[0] == T(0)) { for (int i = 1; i < n; i++) { if ((*this)[i] != T(0)) { if (i & 1) return {}; if (deg - i / 2 <= 0) break; auto ret = (*this >> i).sqrt(get_sqrt, deg - i / 2); if (ret.empty()) return {}; ret = ret << (i / 2); if ((int)ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return FPS(deg, T(0)); } auto sqrtf0 = T(get_sqrt((*this)[0])); if (sqrtf0 * sqrtf0 != (*this)[0]) return {}; FPS ret{sqrtf0}; T inv2 = T(2).inv(); for (int i = 1; i < deg; i <<= 1) ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2; return ret.pre(deg); } /** * @brief Exp of Formal Power Series * * @see https://arxiv.org/pdf/1301.5804.pdf */ FPS exp(int deg = -1) const { assert(this->empty() or (*this)[0] == T(0)); if (this->size() <= 1) return {T(1)}; if (deg == -1) deg = (int)this->size(); FPS inv; inv.reserve(deg + 1); inv.push_back(T(0)); inv.push_back(T(1)); auto inplace_integral = [&](FPS& F) -> void { const int n = (int)F.size(); auto mod = T::mod(); while ((int)inv.size() <= n) { int i = inv.size(); inv.push_back(-inv[mod % i] * (mod / i)); } F.insert(F.begin(), T(0)); for (int i = 1; i <= n; i++) F[i] *= inv[i]; }; auto inplace_differential = [](FPS& F) -> void { if (F.empty()) return; F.erase(F.begin()); for (size_t i = 0; i < F.size(); i++) F[i] *= T(i + 1); }; FPS f{1, (*this)[1]}, g{T(1)}, g_fft{T(1), T(1)}; for (int m = 2; m < deg; m <<= 1) { const T iz1 = T(m).inv(), iz2 = T(m << 1).inv(); auto f_fft = f; f_fft.resize(m << 1); atcoder::internal::butterfly(f_fft); { // Step 2.a' FPS _g(m); for (int i = 0; i < m; i++) _g[i] = f_fft[i] * g_fft[i]; atcoder::internal::butterfly_inv(_g); std::fill(_g.begin(), _g.begin() + (m >> 1), T(0)); atcoder::internal::butterfly(_g); for (int i = 0; i < m; i++) _g[i] *= -g_fft[i] * iz1 * iz1; atcoder::internal::butterfly_inv(_g); g.insert(g.end(), _g.begin() + (m >> 1), _g.end()); g_fft = g; g_fft.resize(m << 1); atcoder::internal::butterfly(g_fft); } FPS x(this->begin(), this->begin() + std::min((int)this->size(), m)); { // Step 2.b' x.resize(m); inplace_differential(x); x.push_back(T(0)); atcoder::internal::butterfly(x); } { // Step 2.c' for (int i = 0; i < m; i++) x[i] *= f_fft[i] * iz1; atcoder::internal::butterfly_inv(x); } { // Step 2.d' and 2.e' x -= f.differential(); x.resize(m << 1); for (int i = 0; i < m - 1; i++) x[m + i] = x[i], x[i] = T(0); atcoder::internal::butterfly(x); for (int i = 0; i < (m << 1); i++) x[i] *= g_fft[i] * iz2; atcoder::internal::butterfly_inv(x); } { // Step 2.f' x.pop_back(); inplace_integral(x); for (int i = m; i < std::min((int)this->size(), m << 1); i++) x[i] += (*this)[i]; std::fill(x.begin(), x.begin() + m, T(0)); } { // Step 2.g' and 2.h' atcoder::internal::butterfly(x); for (int i = 0; i < (m << 1); i++) x[i] *= f_fft[i] * iz2; atcoder::internal::butterfly_inv(x); f.insert(f.end(), x.begin() + m, x.end()); } } return FPS{f.begin(), f.begin() + deg}; } FPS pow(int64_t k, int deg = -1) const { const int n = (int)this->size(); if (deg == -1) deg = n; if (k == 0) { auto res = FPS(deg, T(0)); res[0] = T(1); return res; } for (int i = 0; i < n; i++) { if ((*this)[i] != T(0)) { if (i >= (deg + k - 1) / k) return FPS(deg, T(0)); T rev = (*this)[i].inv(); FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg) * ((*this)[i].pow(k)); ret = (ret << (i * k)).pre(deg); if ((int)ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return FPS(deg, T(0)); } T eval(T x) const { T ret = 0, w = 1; for (const auto& v : *this) ret += w * v, w *= x; return ret; } static FPS product_of_polynomial_sequence(const std::vector& fs) { if (fs.empty()) return {T(1)}; auto comp = [](const FPS& f, const FPS& g) { return f.size() > g.size(); }; std::priority_queue, decltype(comp)> pq{comp}; for (const auto& f : fs) pq.emplace(f); while (pq.size() > 1) { auto f = pq.top(); pq.pop(); auto g = pq.top(); pq.pop(); pq.emplace(f * g); } return pq.top(); } static FPS pow_sparse(const std::vector>& f, int64_t k, int n) { assert(k >= 0); int d = f.size(), offset = 0; while (offset < d and f[offset].second == 0) offset++; FPS res(n, 0); if (offset == d) { if (k == 0) res[0]++; return res; } if (f[offset].first > 0) { int deg = f[offset].first; if (k > (n - 1) / deg) return res; std::vector> g(f.begin() + offset, f.end()); for (auto& p : g) p.first -= deg; auto tmp = pow_sparse(g, k, n - k * deg); for (int i = 0; i < n - k * deg; i++) res[k * deg + i] = tmp[i]; return res; } std::vector invs(n + 1); invs[0] = T(0); invs[1] = T(1); auto mod = T::mod(); for (int i = 2; i <= n; i++) invs[i] = -invs[mod % i] * (mod / i); res[0] = f[0].second.pow(k); T coef = f[0].second.inv(); for (int i = 1; i < n; i++) { for (int j = 1; j < d; j++) { if (i - f[j].first < 0) break; res[i] += f[j].second * res[i - f[j].first] * (T(k) * f[j].first - (i - f[j].first)); } res[i] *= invs[i] * coef; } return res; } FPS taylor_shift(T c) const { FPS f(*this); const int n = f.size(); std::vector fac(n), finv(n); fac[0] = 1; for (int i = 1; i < n; i++) { fac[i] = fac[i - 1] * i; f[i] *= fac[i]; } finv[n - 1] = fac[n - 1].inv(); for (int i = n - 1; i > 0; i--) finv[i - 1] = finv[i] * i; std::reverse(f.begin(), f.end()); FPS g(n); g[0] = T(1); for (int i = 1; i < n; i++) g[i] = g[i - 1] * c * finv[i] * fac[i - 1]; f = (f * g).pre(n); std::reverse(f.begin(), f.end()); for (int i = 0; i < n; i++) f[i] *= finv[i]; return f; } }; using namespace std; typedef long long ll; #define all(x) begin(x), end(x) constexpr int INF = (1 << 30) - 1; constexpr long long IINF = (1LL << 60) - 1; constexpr int dx[4] = {1, 0, -1, 0}, dy[4] = {0, 1, 0, -1}; template istream& operator>>(istream& is, vector& v) { for (auto& x : v) is >> x; return is; } template ostream& operator<<(ostream& os, const vector& v) { auto sep = ""; for (const auto& x : v) os << exchange(sep, " ") << x; return os; } template bool chmin(T& x, U&& y) { return y < x and (x = forward(y), true); } template bool chmax(T& x, U&& y) { return x < y and (x = forward(y), true); } template void mkuni(vector& v) { sort(begin(v), end(v)); v.erase(unique(begin(v), end(v)), end(v)); } template int lwb(const vector& v, const T& x) { return lower_bound(begin(v), end(v), x) - begin(v); } using mint = atcoder::modint998244353; using FPS = FormalPowerSeries; int main() { ios::sync_with_stdio(false); cin.tie(nullptr); int N; cin >> N; vector A(N); for (int& x : A) cin >> x; int k = 1; while (k < N) k <<= 1; vector prod(k << 1, {1}); for (int i = 0; i < N; i++) prod[k + i] = {-A[i], 1}; for (int i = k - 1; i > 0; i--) prod[i] = prod[i << 1] * prod[i << 1 | 1]; vector f(k << 1); f[0] = {1}; for (int i = 1; i < k + N; i++) { f[i] = f[i >> 1]; if (~i & 1) f[i] *= prod[i | 1]; f[i] %= prod[i]; } mint ans = 1; for (int i = 0; i < N; i++) ans *= f[k + i][0]; cout << ans.val() << '\n'; return 0; }