#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using Int = long long; template ostream &operator<<(ostream &os, const pair &a) { return os << "(" << a.first << ", " << a.second << ")"; }; template void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; } template bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; } template bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; } //////////////////////////////////////////////////////////////////////////////// template struct ModInt { static constexpr unsigned M = M_; unsigned x; constexpr ModInt() : x(0U) {} constexpr ModInt(unsigned x_) : x(x_ % M) {} constexpr ModInt(unsigned long long x_) : x(x_ % M) {} constexpr ModInt(int x_) : x(((x_ %= static_cast(M)) < 0) ? (x_ + static_cast(M)) : x_) {} constexpr ModInt(long long x_) : x(((x_ %= static_cast(M)) < 0) ? (x_ + static_cast(M)) : x_) {} ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; } ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; } ModInt &operator*=(const ModInt &a) { x = (static_cast(x) * a.x) % M; return *this; } ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); } ModInt pow(long long e) const { if (e < 0) return inv().pow(-e); ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b; } ModInt inv() const { unsigned a = M, b = x; int y = 0, z = 1; for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast(q) * z; y = z; z = w; } assert(a == 1U); return ModInt(y); } ModInt operator+() const { return *this; } ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; } ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); } ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); } ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); } ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); } template friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); } template friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); } template friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); } template friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); } explicit operator bool() const { return x; } bool operator==(const ModInt &a) const { return (x == a.x); } bool operator!=(const ModInt &a) const { return (x != a.x); } friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; } }; //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // M: prime, G: primitive root, 2^K | M - 1 template struct Fft { static_assert(2U <= M_, "Fft: 2 <= M must hold."); static_assert(M_ < 1U << 30, "Fft: M < 2^30 must hold."); static_assert(1 <= K_, "Fft: 1 <= K must hold."); static_assert(K_ < 30, "Fft: K < 30 must hold."); static_assert(!((M_ - 1U) & ((1U << K_) - 1U)), "Fft: 2^K | M - 1 must hold."); static constexpr unsigned M = M_; static constexpr unsigned M2 = 2U * M_; static constexpr unsigned G = G_; static constexpr int K = K_; ModInt FFT_ROOTS[K + 1], INV_FFT_ROOTS[K + 1]; ModInt FFT_RATIOS[K], INV_FFT_RATIOS[K]; Fft() { const ModInt g(G); for (int k = 0; k <= K; ++k) { FFT_ROOTS[k] = g.pow((M - 1U) >> k); INV_FFT_ROOTS[k] = FFT_ROOTS[k].inv(); } for (int k = 0; k <= K - 2; ++k) { FFT_RATIOS[k] = -g.pow(3U * ((M - 1U) >> (k + 2))); INV_FFT_RATIOS[k] = FFT_RATIOS[k].inv(); } assert(FFT_ROOTS[1] == M - 1U); } // as[rev(i)] <- \sum_j \zeta^(ij) as[j] void fft(ModInt *as, int n) const { assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << K); int m = n; if (m >>= 1) { for (int i = 0; i < m; ++i) { const unsigned x = as[i + m].x; // < M as[i + m].x = as[i].x + M - x; // < 2 M as[i].x += x; // < 2 M } } if (m >>= 1) { ModInt prod = 1U; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned x = (prod * as[i + m]).x; // < M as[i + m].x = as[i].x + M - x; // < 3 M as[i].x += x; // < 3 M } prod *= FFT_RATIOS[__builtin_ctz(++h)]; } } for (; m; ) { if (m >>= 1) { ModInt prod = 1U; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned x = (prod * as[i + m]).x; // < M as[i + m].x = as[i].x + M - x; // < 4 M as[i].x += x; // < 4 M } prod *= FFT_RATIOS[__builtin_ctz(++h)]; } } if (m >>= 1) { ModInt prod = 1U; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned x = (prod * as[i + m]).x; // < M as[i].x = (as[i].x >= M2) ? (as[i].x - M2) : as[i].x; // < 2 M as[i + m].x = as[i].x + M - x; // < 3 M as[i].x += x; // < 3 M } prod *= FFT_RATIOS[__builtin_ctz(++h)]; } } } for (int i = 0; i < n; ++i) { as[i].x = (as[i].x >= M2) ? (as[i].x - M2) : as[i].x; // < 2 M as[i].x = (as[i].x >= M) ? (as[i].x - M) : as[i].x; // < M } } // as[i] <- (1/n) \sum_j \zeta^(-ij) as[rev(j)] void invFft(ModInt *as, int n) const { assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << K); int m = 1; if (m < n >> 1) { ModInt prod = 1U; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned long long y = as[i].x + M - as[i + m].x; // < 2 M as[i].x += as[i + m].x; // < 2 M as[i + m].x = (prod.x * y) % M; // < M } prod *= INV_FFT_RATIOS[__builtin_ctz(++h)]; } m <<= 1; } for (; m < n >> 1; m <<= 1) { ModInt prod = 1U; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + (m >> 1); ++i) { const unsigned long long y = as[i].x + M2 - as[i + m].x; // < 4 M as[i].x += as[i + m].x; // < 4 M as[i].x = (as[i].x >= M2) ? (as[i].x - M2) : as[i].x; // < 2 M as[i + m].x = (prod.x * y) % M; // < M } for (int i = i0 + (m >> 1); i < i0 + m; ++i) { const unsigned long long y = as[i].x + M - as[i + m].x; // < 2 M as[i].x += as[i + m].x; // < 2 M as[i + m].x = (prod.x * y) % M; // < M } prod *= INV_FFT_RATIOS[__builtin_ctz(++h)]; } } if (m < n) { for (int i = 0; i < m; ++i) { const unsigned y = as[i].x + M2 - as[i + m].x; // < 4 M as[i].x += as[i + m].x; // < 4 M as[i + m].x = y; // < 4 M } } const ModInt invN = ModInt(n).inv(); for (int i = 0; i < n; ++i) { as[i] *= invN; } } void fft(vector> &as) const { fft(as.data(), as.size()); } void invFft(vector> &as) const { invFft(as.data(), as.size()); } vector> convolve(vector> as, vector> bs) const { if (as.empty() || bs.empty()) return {}; const int len = as.size() + bs.size() - 1; int n = 1; for (; n < len; n <<= 1) {} as.resize(n); fft(as); bs.resize(n); fft(bs); for (int i = 0; i < n; ++i) as[i] *= bs[i]; invFft(as); as.resize(len); return as; } vector> square(vector> as) const { if (as.empty()) return {}; const int len = as.size() + as.size() - 1; int n = 1; for (; n < len; n <<= 1) {} as.resize(n); fft(as); for (int i = 0; i < n; ++i) as[i] *= as[i]; invFft(as); as.resize(len); return as; } }; // M0 M1 M2 = 789204840662082423367925761 (> 7.892 * 10^26, > 2^89) // M0 M3 M4 M5 M6 = 797766583174034668024539679147517452591562753 (> 7.977 * 10^44, > 2^149) const Fft<998244353U, 3U, 23> FFT0; const Fft<897581057U, 3U, 23> FFT1; const Fft<880803841U, 26U, 23> FFT2; const Fft<985661441U, 3U, 22> FFT3; const Fft<943718401U, 7U, 22> FFT4; const Fft<935329793U, 3U, 22> FFT5; const Fft<918552577U, 5U, 22> FFT6; // T = unsigned, unsigned long long, ModInt template T garner(ModInt a0, ModInt a1, ModInt a2) { static const ModInt INV_M0_M1 = ModInt(M0).inv(); static const ModInt INV_M0M1_M2 = (ModInt(M0) * M1).inv(); const ModInt b1 = INV_M0_M1 * (a1 - a0.x); const ModInt b2 = INV_M0M1_M2 * (a2 - (ModInt(b1.x) * M0 + a0.x)); return (T(b2.x) * M1 + b1.x) * M0 + a0.x; } template T garner(ModInt a0, ModInt a1, ModInt a2, ModInt a3, ModInt a4) { static const ModInt INV_M0_M1 = ModInt(M0).inv(); static const ModInt INV_M0M1_M2 = (ModInt(M0) * M1).inv(); static const ModInt INV_M0M1M2_M3 = (ModInt(M0) * M1 * M2).inv(); static const ModInt INV_M0M1M2M3_M4 = (ModInt(M0) * M1 * M2 * M3).inv(); const ModInt b1 = INV_M0_M1 * (a1 - a0.x); const ModInt b2 = INV_M0M1_M2 * (a2 - (ModInt(b1.x) * M0 + a0.x)); const ModInt b3 = INV_M0M1M2_M3 * (a3 - ((ModInt(b2.x) * M1 + b1.x) * M0 + a0.x)); const ModInt b4 = INV_M0M1M2M3_M4 * (a4 - (((ModInt(b3.x) * M2 + b2.x) * M1 + b1.x) * M0 + a0.x)); return (((T(b4.x) * M3 + b3.x) * M2 + b2.x) * M1 + b1.x) * M0 + a0.x; } template vector> convolve(const vector> &as, const vector> &bs) { static constexpr unsigned M0 = decltype(FFT0)::M; static constexpr unsigned M1 = decltype(FFT1)::M; static constexpr unsigned M2 = decltype(FFT2)::M; if (as.empty() || bs.empty()) return {}; const int asLen = as.size(), bsLen = bs.size(); vector> as0(asLen), bs0(bsLen); for (int i = 0; i < asLen; ++i) as0[i] = as[i].x; for (int i = 0; i < bsLen; ++i) bs0[i] = bs[i].x; const vector> cs0 = FFT0.convolve(as0, bs0); vector> as1(asLen), bs1(bsLen); for (int i = 0; i < asLen; ++i) as1[i] = as[i].x; for (int i = 0; i < bsLen; ++i) bs1[i] = bs[i].x; const vector> cs1 = FFT1.convolve(as1, bs1); vector> as2(asLen), bs2(bsLen); for (int i = 0; i < asLen; ++i) as2[i] = as[i].x; for (int i = 0; i < bsLen; ++i) bs2[i] = bs[i].x; const vector> cs2 = FFT2.convolve(as2, bs2); vector> cs(asLen + bsLen - 1); for (int i = 0; i < asLen + bsLen - 1; ++i) { cs[i] = garner>(cs0[i], cs1[i], cs2[i]); } return cs; } template vector> square(const vector> &as) { static constexpr unsigned M0 = decltype(FFT0)::M; static constexpr unsigned M1 = decltype(FFT1)::M; static constexpr unsigned M2 = decltype(FFT2)::M; if (as.empty()) return {}; const int asLen = as.size(); vector> as0(asLen); for (int i = 0; i < asLen; ++i) as0[i] = as[i].x; const vector> cs0 = FFT0.square(as0); vector> as1(asLen); for (int i = 0; i < asLen; ++i) as1[i] = as[i].x; const vector> cs1 = FFT1.square(as1); vector> as2(asLen); for (int i = 0; i < asLen; ++i) as2[i] = as[i].x; const vector> cs2 = FFT2.square(as2); vector> cs(asLen + asLen - 1); for (int i = 0; i < asLen + asLen - 1; ++i) { cs[i] = garner>(cs0[i], cs1[i], cs2[i]); } return cs; } constexpr unsigned MO = 90001; using Mint = ModInt; constexpr int L = 60; const Mint G = Mint(13).pow((MO - 1) / L); char S[200'010]; int M; int K[10]; int A[10][10]; int main() { for (; ~scanf("%s%d", S, &M); ) { for (int i = 0; i < M; ++i) { scanf("%d", &K[i]); for (int k = 0; k < K[i]; ++k) { scanf("%d", &A[i][k]); } } vector> fss(M); for (int i = 0; i < M; ++i) { fss[i].assign(MO, 0); for (int p = 1; p < 1 << K[i]; ++p) { int l = 1; for (int k = 0; k < K[i]; ++k) if (p >> k & 1) { l = l / __gcd(l, A[i][k]) * A[i][k]; } const Mint coef = -(__builtin_parity(p) ? -1 : +1) * Mint(l).inv(); // cerr< prod(MO, 0); prod[1] = 1; for (int i = 0; i < M; ++i) { prod = convolve(prod, fss[i]); for (int j = MO; j < (int)prod.size(); ++j) { prod[j - MO] += prod[j]; } prod.resize(MO); } Int n = 0; for (int h = 0; S[h]; ++h) { n = n * 10 + (S[h] - '0'); n = 1 + (n - 1) % (MO - 1); } Mint ans = 0; for (int x = 0; x < (int)MO; ++x) { ans += prod[x] * Mint(x).pow(n); } printf("%u\n", ans.x); } return 0; } // tsurai