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// ModInt {{{ template<int MD> struct ModInt { using ll = long long; int x; constexpr ModInt() : x(0) {} constexpr ModInt(ll v) { _set(v % MD + MD); } constexpr static int mod() { return MD; } constexpr explicit operator bool() const { return x != 0; } constexpr ModInt operator + (const ModInt& a) const { return ModInt()._set((ll) x + a.x); } constexpr ModInt operator - (const ModInt& a) const { return ModInt()._set((ll) x - a.x + MD); } constexpr ModInt operator * (const ModInt& a) const { return ModInt()._set((ll) x * a.x % MD); } constexpr ModInt operator / (const ModInt& a) const { return ModInt()._set((ll) x * a.inv().x % MD); } constexpr ModInt operator - () const { return ModInt()._set(MD - x); } constexpr ModInt& operator += (const ModInt& a) { return *this = *this + a; } constexpr ModInt& operator -= (const ModInt& a) { return *this = *this - a; } constexpr ModInt& operator *= (const ModInt& a) { return *this = *this * a; } constexpr ModInt& operator /= (const ModInt& a) { return *this = *this / a; } friend constexpr ModInt operator + (ll a, const ModInt& b) { return ModInt()._set(a % MD + b.x); } friend constexpr ModInt operator - (ll a, const ModInt& b) { return ModInt()._set(a % MD - b.x + MD); } friend constexpr ModInt operator * (ll a, const ModInt& b) { return ModInt()._set(a % MD * b.x % MD); } friend constexpr ModInt operator / (ll a, const ModInt& b) { return ModInt()._set(a % MD * b.inv().x % MD); } constexpr bool operator == (const ModInt& a) const { return x == a.x; } constexpr bool operator != (const ModInt& a) const { return x != a.x; } friend std::istream& operator >> (std::istream& is, ModInt& other) { ll val; is >> val; other = ModInt(val); return is; } constexpr friend std::ostream& operator << (std::ostream& os, const ModInt& other) { return os << other.x; } constexpr ModInt pow(ll k) const { ModInt ans = 1, tmp = x; while (k) { if (k & 1) ans *= tmp; tmp *= tmp; k >>= 1; } return ans; } constexpr ModInt inv() const { if (x < 1000111) { _precalc(1000111); return invs[x]; } int a = x, b = MD, ax = 1, bx = 0; while (b) { int q = a/b, t = a%b; a = b; b = t; t = ax - bx*q; ax = bx; bx = t; } assert(a == 1); if (ax < 0) ax += MD; return ax; } static std::vector<ModInt> factorials, inv_factorials, invs; constexpr static void _precalc(int n) { if (factorials.empty()) { factorials = {1}; inv_factorials = {1}; invs = {0}; } if (n > MD) n = MD; int old_sz = factorials.size(); if (n <= old_sz) return; factorials.resize(n); inv_factorials.resize(n); invs.resize(n); for (int i = old_sz; i < n; ++i) factorials[i] = factorials[i-1] * i; inv_factorials[n-1] = factorials.back().pow(MD - 2); for (int i = n - 2; i >= old_sz; --i) inv_factorials[i] = inv_factorials[i+1] * (i+1); for (int i = n-1; i >= old_sz; --i) invs[i] = inv_factorials[i] * factorials[i-1]; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> fac; int v = MD - 1; for (ll i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < MD; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((MD - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } static ModInt C(int n, int k) { _precalc(n + 1); return factorials[n] * inv_factorials[k] * inv_factorials[n-k]; } private: // Internal, DO NOT USE. // val must be in [0, 2*MD) constexpr inline __attribute__((always_inline)) ModInt& _set(ll v) { x = v >= MD ? v - MD : v; return *this; } }; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::factorials = {1}; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::inv_factorials = {1}; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::invs = {0}; // }}}
#line 1 "Math/modint.h" // ModInt {{{ template<int MD> struct ModInt { using ll = long long; int x; constexpr ModInt() : x(0) {} constexpr ModInt(ll v) { _set(v % MD + MD); } constexpr static int mod() { return MD; } constexpr explicit operator bool() const { return x != 0; } constexpr ModInt operator + (const ModInt& a) const { return ModInt()._set((ll) x + a.x); } constexpr ModInt operator - (const ModInt& a) const { return ModInt()._set((ll) x - a.x + MD); } constexpr ModInt operator * (const ModInt& a) const { return ModInt()._set((ll) x * a.x % MD); } constexpr ModInt operator / (const ModInt& a) const { return ModInt()._set((ll) x * a.inv().x % MD); } constexpr ModInt operator - () const { return ModInt()._set(MD - x); } constexpr ModInt& operator += (const ModInt& a) { return *this = *this + a; } constexpr ModInt& operator -= (const ModInt& a) { return *this = *this - a; } constexpr ModInt& operator *= (const ModInt& a) { return *this = *this * a; } constexpr ModInt& operator /= (const ModInt& a) { return *this = *this / a; } friend constexpr ModInt operator + (ll a, const ModInt& b) { return ModInt()._set(a % MD + b.x); } friend constexpr ModInt operator - (ll a, const ModInt& b) { return ModInt()._set(a % MD - b.x + MD); } friend constexpr ModInt operator * (ll a, const ModInt& b) { return ModInt()._set(a % MD * b.x % MD); } friend constexpr ModInt operator / (ll a, const ModInt& b) { return ModInt()._set(a % MD * b.inv().x % MD); } constexpr bool operator == (const ModInt& a) const { return x == a.x; } constexpr bool operator != (const ModInt& a) const { return x != a.x; } friend std::istream& operator >> (std::istream& is, ModInt& other) { ll val; is >> val; other = ModInt(val); return is; } constexpr friend std::ostream& operator << (std::ostream& os, const ModInt& other) { return os << other.x; } constexpr ModInt pow(ll k) const { ModInt ans = 1, tmp = x; while (k) { if (k & 1) ans *= tmp; tmp *= tmp; k >>= 1; } return ans; } constexpr ModInt inv() const { if (x < 1000111) { _precalc(1000111); return invs[x]; } int a = x, b = MD, ax = 1, bx = 0; while (b) { int q = a/b, t = a%b; a = b; b = t; t = ax - bx*q; ax = bx; bx = t; } assert(a == 1); if (ax < 0) ax += MD; return ax; } static std::vector<ModInt> factorials, inv_factorials, invs; constexpr static void _precalc(int n) { if (factorials.empty()) { factorials = {1}; inv_factorials = {1}; invs = {0}; } if (n > MD) n = MD; int old_sz = factorials.size(); if (n <= old_sz) return; factorials.resize(n); inv_factorials.resize(n); invs.resize(n); for (int i = old_sz; i < n; ++i) factorials[i] = factorials[i-1] * i; inv_factorials[n-1] = factorials.back().pow(MD - 2); for (int i = n - 2; i >= old_sz; --i) inv_factorials[i] = inv_factorials[i+1] * (i+1); for (int i = n-1; i >= old_sz; --i) invs[i] = inv_factorials[i] * factorials[i-1]; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> fac; int v = MD - 1; for (ll i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < MD; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((MD - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } static ModInt C(int n, int k) { _precalc(n + 1); return factorials[n] * inv_factorials[k] * inv_factorials[n-k]; } private: // Internal, DO NOT USE. // val must be in [0, 2*MD) constexpr inline __attribute__((always_inline)) ModInt& _set(ll v) { x = v >= MD ? v - MD : v; return *this; } }; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::factorials = {1}; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::inv_factorials = {1}; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::invs = {0}; // }}}