#include #include typedef long long int ll; typedef long double ld; using namespace std; using namespace atcoder; template struct LazyMontgomeryModInt { using mint = LazyMontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 ret = mod; for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret; return ret; } static constexpr u32 r = get_r(); static constexpr u32 n2 = -u64(mod) % mod; static_assert(r * mod == 1, "invalid, r * mod != 1"); static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30"); static_assert((mod & 1) == 1, "invalid, mod % 2 == 0"); u32 a; constexpr LazyMontgomeryModInt() : a(0) {} constexpr LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)){}; static constexpr u32 reduce(const u64 &b) { return (b + u64(u32(b) * u32(-r)) * mod) >> 32; } constexpr mint &operator+=(const mint &b) { if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } constexpr mint &operator-=(const mint &b) { if (i32(a -= b.a) < 0) a += 2 * mod; return *this; } constexpr mint &operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } constexpr mint &operator/=(const mint &b) { *this *= b.inverse(); return *this; } constexpr mint operator+(const mint &b) const { return mint(*this) += b; } constexpr mint operator-(const mint &b) const { return mint(*this) -= b; } constexpr mint operator*(const mint &b) const { return mint(*this) *= b; } constexpr mint operator/(const mint &b) const { return mint(*this) /= b; } constexpr bool operator==(const mint &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } constexpr bool operator!=(const mint &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } constexpr mint operator-() const { return mint() - mint(*this); } constexpr mint pow(u64 n) const { mint ret(1), mul(*this); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } constexpr mint inverse() const { return pow(mod - 2); } friend ostream &operator<<(ostream &os, const mint &b) { return os << b.get(); } friend istream &operator>>(istream &is, mint &b) { int64_t t; is >> t; b = LazyMontgomeryModInt(t); return (is); } constexpr u32 get() const { u32 ret = reduce(a); return ret >= mod ? ret - mod : ret; } static constexpr u32 get_mod() { return mod; } }; template struct NTT { static constexpr uint32_t get_pr() { uint32_t _mod = mint::get_mod(); using u64 = uint64_t; u64 ds[32] = {}; int idx = 0; u64 m = _mod - 1; for (u64 i = 2; i * i <= m; ++i) { if (m % i == 0) { ds[idx++] = i; while (m % i == 0) m /= i; } } if (m != 1) ds[idx++] = m; uint32_t _pr = 2; while (1) { int flg = 1; for (int i = 0; i < idx; ++i) { u64 a = _pr, b = (_mod - 1) / ds[i], r = 1; while (b) { if (b & 1) r = r * a % _mod; a = a * a % _mod; b >>= 1; } if (r == 1) { flg = 0; break; } } if (flg == 1) break; ++_pr; } return _pr; }; static constexpr uint32_t mod = mint::get_mod(); static constexpr uint32_t pr = get_pr(); static constexpr int level = __builtin_ctzll(mod - 1); mint dw[level], dy[level]; void setwy(int k) { mint w[level], y[level]; w[k - 1] = mint(pr).pow((mod - 1) / (1 << k)); y[k - 1] = w[k - 1].inverse(); for (int i = k - 2; i > 0; --i) w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1]; dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2]; for (int i = 3; i < k; ++i) { dw[i] = dw[i - 1] * y[i - 2] * w[i]; dy[i] = dy[i - 1] * w[i - 2] * y[i]; } } NTT() { setwy(level); } void fft4(vector &a, int k) { if ((int)a.size() <= 1) return; if (k == 1) { mint a1 = a[1]; a[1] = a[0] - a[1]; a[0] = a[0] + a1; return; } if (k & 1) { int v = 1 << (k - 1); for (int j = 0; j < v; ++j) { mint ajv = a[j + v]; a[j + v] = a[j] - ajv; a[j] += ajv; } } int u = 1 << (2 + (k & 1)); int v = 1 << (k - 2 - (k & 1)); mint one = mint(1); mint imag = dw[1]; while (v) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = j1 + v; int j3 = j2 + v; for (; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3; } } // jh >= 1 mint ww = one, xx = one * dw[2], wx = one; for (int jh = 4; jh < u;) { ww = xx * xx, wx = ww * xx; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for (; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3; } xx *= dw[__builtin_ctzll((jh += 4))]; } u <<= 2; v >>= 2; } } void ifft4(vector &a, int k) { if ((int)a.size() <= 1) return; if (k == 1) { mint a1 = a[1]; a[1] = a[0] - a[1]; a[0] = a[0] + a1; return; } int u = 1 << (k - 2); int v = 1; mint one = mint(1); mint imag = dy[1]; while (u) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = v + v; int j3 = j2 + v; for (; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag; a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3; a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3; } } // jh >= 1 mint ww = one, xx = one * dy[2], yy = one; u <<= 2; for (int jh = 4; jh < u;) { ww = xx * xx, yy = xx * imag; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for (; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy; a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww; a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww; } xx *= dy[__builtin_ctzll(jh += 4)]; } u >>= 4; v <<= 2; } if (k & 1) { u = 1 << (k - 1); for (int j = 0; j < u; ++j) { mint ajv = a[j] - a[j + u]; a[j] += a[j + u]; a[j + u] = ajv; } } } void ntt(vector &a) { if ((int)a.size() <= 1) return; fft4(a, __builtin_ctz(a.size())); } void intt(vector &a) { if ((int)a.size() <= 1) return; ifft4(a, __builtin_ctz(a.size())); mint iv = mint(a.size()).inverse(); for (auto &x : a) x *= iv; } vector multiply(const vector &a, const vector &b) { int l = a.size() + b.size() - 1; if (min(a.size(), b.size()) <= 40) { vector s(l); for (int i = 0; i < (int)a.size(); ++i) for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j]; return s; } int k = 2, M = 4; while (M < l) M <<= 1, ++k; setwy(k); vector s(M), t(M); for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i]; for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i]; fft4(s, k); fft4(t, k); for (int i = 0; i < M; ++i) s[i] *= t[i]; ifft4(s, k); s.resize(l); mint invm = mint(M).inverse(); for (int i = 0; i < l; ++i) s[i] *= invm; return s; } void ntt_doubling(vector &a) { int M = (int)a.size(); auto b = a; intt(b); mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1)); for (int i = 0; i < M; i++) b[i] *= r, r *= zeta; ntt(b); copy(begin(b), end(b), back_inserter(a)); } }; namespace ArbitraryNTT { using i64 = int64_t; using u128 = __uint128_t; constexpr int32_t m0 = 167772161; constexpr int32_t m1 = 469762049; constexpr int32_t m2 = 754974721; using mint0 = LazyMontgomeryModInt; using mint1 = LazyMontgomeryModInt; using mint2 = LazyMontgomeryModInt; constexpr int r01 = mint1(m0).inverse().get(); constexpr int r02 = mint2(m0).inverse().get(); constexpr int r12 = mint2(m1).inverse().get(); constexpr int r02r12 = i64(r02) * r12 % m2; constexpr i64 w1 = m0; constexpr i64 w2 = i64(m0) * m1; template vector mul(const vector &a, const vector &b) { static NTT ntt; vector s(a.size()), t(b.size()); for (int i = 0; i < (int)a.size(); ++i) s[i] = i64(a[i] % submint::get_mod()); for (int i = 0; i < (int)b.size(); ++i) t[i] = i64(b[i] % submint::get_mod()); return ntt.multiply(s, t); } template vector multiply(const vector &s, const vector &t, int mod) { auto d0 = mul(s, t); auto d1 = mul(s, t); auto d2 = mul(s, t); int n = d0.size(); vector ret(n); const int W1 = w1 % mod; const int W2 = w2 % mod; for (int i = 0; i < n; i++) { int n1 = d1[i].get(), n2 = d2[i].get(), a = d0[i].get(); int b = i64(n1 + m1 - a) * r01 % m1; int c = (i64(n2 + m2 - a) * r02r12 + i64(m2 - b) * r12) % m2; ret[i] = (i64(a) + i64(b) * W1 + i64(c) * W2) % mod; } return ret; } template vector multiply(const vector &a, const vector &b) { if (a.size() == 0 && b.size() == 0) return {}; if (min(a.size(), b.size()) < 128) { vector ret(a.size() + b.size() - 1); for (int i = 0; i < (int)a.size(); ++i) for (int j = 0; j < (int)b.size(); ++j) ret[i + j] += a[i] * b[j]; return ret; } vector s(a.size()), t(b.size()); for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i].get(); for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i].get(); vector u = multiply(s, t, mint::get_mod()); vector ret(u.size()); for (int i = 0; i < (int)u.size(); ++i) ret[i] = mint(u[i]); return ret; } template vector multiply_u128(const vector &s, const vector &t) { if (s.size() == 0 && t.size() == 0) return {}; if (min(s.size(), t.size()) < 128) { vector ret(s.size() + t.size() - 1); for (int i = 0; i < (int)s.size(); ++i) for (int j = 0; j < (int)t.size(); ++j) ret[i + j] += i64(s[i]) * t[j]; return ret; } auto d0 = mul(s, t); auto d1 = mul(s, t); auto d2 = mul(s, t); int n = d0.size(); vector ret(n); for (int i = 0; i < n; i++) { i64 n1 = d1[i].get(), n2 = d2[i].get(); i64 a = d0[i].get(); i64 b = (n1 + m1 - a) * r01 % m1; i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2; ret[i] = a + b * w1 + u128(c) * w2; } return ret; } } // namespace ArbitraryNTT template struct FormalPowerSeries : vector { using vector::vector; using FPS = FormalPowerSeries; 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]; return *this; } FPS &operator+=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] += r; 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]; return *this; } FPS &operator-=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] -= r; return *this; } FPS &operator*=(const mint &v) { for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v; return *this; } FPS &operator/=(const FPS &r) { if (this->size() < r.size()) { this->clear(); return *this; } int n = this->size() - r.size() + 1; if ((int)r.size() <= 64) { FPS f(*this), g(r); g.shrink(); mint coeff = g.back().inverse(); for (auto &x : g) x *= coeff; int deg = (int)f.size() - (int)g.size() + 1; int gs = g.size(); FPS quo(deg); for (int i = deg - 1; i >= 0; i--) { quo[i] = f[i + gs - 1]; for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j]; } *this = quo * coeff; this->resize(n, mint(0)); return *this; } return *this = ((*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 mint &v) const { return FPS(*this) += v; } FPS operator-(const FPS &r) const { return FPS(*this) -= r; } FPS operator-(const mint &v) const { return FPS(*this) -= v; } FPS operator*(const FPS &r) const { return FPS(*this) *= r; } FPS operator*(const mint &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->size()); for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i]; return ret; } void shrink() { while (this->size() && this->back() == mint(0)) this->pop_back(); } FPS rev() const { FPS ret(*this); reverse(begin(ret), end(ret)); return ret; } FPS dot(FPS r) const { FPS ret(min(this->size(), r.size())); for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i]; return ret; } FPS pre(int sz) const { return FPS(begin(*this), begin(*this) + min((int)this->size(), sz)); } FPS operator>>(int sz) const { if ((int)this->size() <= sz) return {}; FPS ret(*this); ret.erase(ret.begin(), ret.begin() + sz); return ret; } FPS operator<<(int sz) const { FPS ret(*this); ret.insert(ret.begin(), sz, mint(0)); return ret; } FPS diff() const { const int n = (int)this->size(); FPS ret(max(0, n - 1)); mint one(1), coeff(1); for (int i = 1; i < n; i++) { ret[i - 1] = (*this)[i] * coeff; coeff += one; } return ret; } FPS integral() const { const int n = (int)this->size(); FPS ret(n + 1); ret[0] = mint(0); if (n > 0) ret[1] = mint(1); auto mod = mint::get_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; } mint eval(mint x) const { mint r = 0, w = 1; for (auto &v : *this) r += w * v, w *= x; return r; } FPS log(int deg = -1) const { assert((*this)[0] == mint(1)); if (deg == -1) deg = (int)this->size(); return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } FPS pow(int64_t k, int deg = -1) const { const int n = (int)this->size(); if (deg == -1) deg = n; if (k == 0) { FPS ret(deg); if (deg) ret[0] = 1; return ret; } for (int i = 0; i < n; i++) { if ((*this)[i] != mint(0)) { mint rev = mint(1) / (*this)[i]; FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg); ret *= (*this)[i].pow(k); ret = (ret << (i * k)).pre(deg); if ((int)ret.size() < deg) ret.resize(deg, mint(0)); return ret; } if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0)); } return FPS(deg, mint(0)); } static void *ntt_ptr; static void set_fft(); FPS &operator*=(const FPS &r); void ntt(); void intt(); void ntt_doubling(); static int ntt_pr(); FPS inv(int deg = -1) const; FPS exp(int deg = -1) const; }; template void *FormalPowerSeries::ntt_ptr = nullptr; template void FormalPowerSeries::set_fft() { ntt_ptr = nullptr; } template void FormalPowerSeries::ntt() { exit(1); } template void FormalPowerSeries::intt() { exit(1); } template void FormalPowerSeries::ntt_doubling() { exit(1); } template int FormalPowerSeries::ntt_pr() { exit(1); } template FormalPowerSeries& FormalPowerSeries::operator*=( const FormalPowerSeries& r) { if (this->empty() || r.empty()) { this->clear(); return *this; } auto ret = ArbitraryNTT::multiply(*this, r); return *this = FormalPowerSeries(ret.begin(), ret.end()); } template FormalPowerSeries FormalPowerSeries::inv(int deg) const { assert((*this)[0] != mint(0)); if (deg == -1) deg = (*this).size(); FormalPowerSeries ret({mint(1) / (*this)[0]}); for (int i = 1; i < deg; i <<= 1) ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1); return ret.pre(deg); } template FormalPowerSeries FormalPowerSeries::exp(int deg) const { assert((*this).size() == 0 || (*this)[0] == mint(0)); if (deg == -1) deg = (int)this->size(); FormalPowerSeries ret({mint(1)}); for (int i = 1; i < deg; i <<= 1) { ret = (ret * (pre(i << 1) + mint(1) - ret.log(i << 1))).pre(i << 1); } return ret.pre(deg); } // f *= (1 + c x^n) template void sparse_mul(FormalPowerSeries& f, int n, mint c, int expand = false) { if (expand) f.resize(f.size() + n); for (int i = (int)f.size() - 1; i >= 0; --i) { if (i - n >= 0) f[i] += f[i - n] * c; } } // f /= (1 + c x^n) template void sparse_div(FormalPowerSeries& f, int n, mint c) { for (int i = 0; i < (int)f.size(); ++i) { if (i + n < (int)f.size()) f[i + n] -= f[i] * c; } } // FPS(形式的べき級数)のライブラリ Nyaanさん // mod1000000007で使える // resize(m+1) m乗の項までセット // rev() reverse // dot(r) dot積(項ごとに積) // pre(sz) szまでを残して後を切り捨て // diff() 微分 // integral() 積分 // eval(x) xを代入した時の合計を求める // log(deg) 定数項が1である必要がある、log(f)を返す // f = f.pow(k,deg) k乗する // inv(deg) // exp(deg) // (degの項まで計算する,-1で元のサイズ) // sparse_mul(FPS f,n,(mint)c,expand) *= (1+cx^n) (expand=1であればresize) // sparse_div(FPS f,n,(mint)c) /= (1+cx^n) // retやmodのマクロがバグるので消しておくこと! // using mint = LazyMontgomeryModInt<1000000007>; // FormalPowerSeries f; #define inf 1010000000 #define llinf 1001000000000000000ll #define pi 3.141592653589793238 #define rep(i, n) for(ll i = 0; i < (n); i++) #define rep1(i, n) for(ll i = 1; i <= (n); i++) #define rep2(i,l,r) for(ll i = (l); i < (r); i++) #define per(i, n) for(ll i = (n)-1; i >= 0; i--) #define each(x, v) for (auto&& x : v) #define rng(a) a.begin(),a.end() #define fi first #define se second #define pb push_back #define eb emplace_back #define pob pop_back #define st string #define pcnt __builtin_popcountll #define bit(n) (1LL<<(n)) template inline T in(){ T x; cin >> x; return (x);} #define vcin(x,n) {for(ll loop=0; loop<(n); loop++) cin>>x[loop];} #define dame { puts("-1"); return 0;} #define yes { puts("Yes"); return 0;} #define no { puts("No"); return 0;} #define ret(x) { cout<<(x)<