#include #define rep(i,a,b) for(int i=a;i=b;i--) #define fore(i,a) for(auto &i:a) #define all(x) (x).begin(),(x).end() //#pragma GCC optimize ("-O3") using namespace std; void _main(); int main() { cin.tie(0); ios::sync_with_stdio(false); _main(); } typedef long long ll; const int inf = INT_MAX / 2; const ll infl = 1LL << 60; templatebool chmax(T& a, const T& b) { if (a < b) { a = b; return 1; } return 0; } templatebool chmin(T& a, const T& b) { if (b < a) { a = b; return 1; } return 0; } //--------------------------------------------------------------------------------------------------- template struct ModInt { static const int Mod = MOD; unsigned x; ModInt() : x(0) { } ModInt(signed sig) { x = sig < 0 ? sig % MOD + MOD : sig % MOD; } ModInt(signed long long sig) { x = sig < 0 ? sig % MOD + MOD : sig % MOD; } int get() const { return (int)x; } ModInt& operator+=(ModInt that) { if ((x += that.x) >= MOD) x -= MOD; return *this; } ModInt& operator-=(ModInt that) { if ((x += MOD - that.x) >= MOD) x -= MOD; return *this; } ModInt& operator*=(ModInt that) { x = (unsigned long long)x * that.x % MOD; return *this; } ModInt& operator/=(ModInt that) { return *this *= that.inverse(); } ModInt operator+(ModInt that) const { return ModInt(*this) += that; } ModInt operator-(ModInt that) const { return ModInt(*this) -= that; } ModInt operator*(ModInt that) const { return ModInt(*this) *= that; } ModInt operator/(ModInt that) const { return ModInt(*this) /= that; } ModInt inverse() const { long long a = x, b = MOD, u = 1, v = 0; while (b) { long long t = a / b; a -= t * b; std::swap(a, b); u -= t * v; std::swap(u, v); } return ModInt(u); } bool operator==(ModInt that) const { return x == that.x; } bool operator!=(ModInt that) const { return x != that.x; } ModInt operator-() const { ModInt t; t.x = x == 0 ? 0 : Mod - x; return t; } }; template ostream& operator<<(ostream& st, const ModInt a) { st << a.get(); return st; }; template ModInt operator^(ModInt a, unsigned long long k) { ModInt r = 1; while (k) { if (k & 1) r *= a; a *= a; k >>= 1; } return r; } typedef ModInt<1000000007> mint; template struct FormalPowerSeries { using Poly = vector; using Conv = function; Conv conv; FormalPowerSeries(Conv conv) :conv(conv) {} Poly pre(const Poly& as, int deg) { return Poly(as.begin(), as.begin() + min((int)as.size(), deg)); } Poly add(Poly as, Poly bs) { int sz = max(as.size(), bs.size()); Poly cs(sz, T(0)); for (int i = 0; i < (int)as.size(); i++) cs[i] += as[i]; for (int i = 0; i < (int)bs.size(); i++) cs[i] += bs[i]; return cs; } Poly sub(Poly as, Poly bs) { int sz = max(as.size(), bs.size()); Poly cs(sz, T(0)); for (int i = 0; i < (int)as.size(); i++) cs[i] += as[i]; for (int i = 0; i < (int)bs.size(); i++) cs[i] -= bs[i]; return cs; } Poly mul(Poly as, Poly bs) { return conv(as, bs); } Poly mul(Poly as, T k) { for (auto& a : as) a *= k; return as; } // F(0) must not be 0 Poly inv(Poly as, int deg) { assert(as[0] != T(0)); Poly rs({ T(1) / as[0] }); for (int i = 1; i < deg; i <<= 1) rs = pre(sub(add(rs, rs), mul(mul(rs, rs), pre(as, i << 1))), i << 1); return rs; } // not zero Poly div(Poly as, Poly bs) { while (as.back() == T(0)) as.pop_back(); while (bs.back() == T(0)) bs.pop_back(); if (bs.size() > as.size()) return Poly(); reverse(as.begin(), as.end()); reverse(bs.begin(), bs.end()); int need = as.size() - bs.size() + 1; Poly ds = pre(mul(as, inv(bs, need)), need); reverse(ds.begin(), ds.end()); return ds; } // F(0) must be 1 Poly sqrt(Poly as, int deg) { assert(as[0] == T(1)); T inv2 = T(1) / T(2); Poly ss({ T(1) }); for (int i = 1; i < deg; i <<= 1) { ss = pre(add(ss, mul(pre(as, i << 1), inv(ss, i << 1))), i << 1); for (T& x : ss) x *= inv2; } return ss; } Poly diff(Poly as) { int n = as.size(); Poly res(n - 1); for (int i = 1; i < n; i++) res[i - 1] = as[i] * T(i); return res; } Poly integral(Poly as) { int n = as.size(); Poly res(n + 1); res[0] = T(0); for (int i = 0; i < n; i++) res[i + 1] = as[i] / T(i + 1); return res; } // F(0) must be 1 Poly log(Poly as, int deg) { return pre(integral(mul(diff(as), inv(as, deg))), deg); } // F(0) must be 0 Poly exp(Poly as, int deg) { Poly f({ T(1) }); as[0] += T(1); for (int i = 1; i < deg; i <<= 1) f = pre(mul(f, sub(pre(as, i << 1), log(f, i << 1))), i << 1); return f; } Poly partition(int n) { Poly rs(n + 1); rs[0] = T(1); for (int k = 1; k <= n; k++) { if (1LL * k * (3 * k + 1) / 2 <= n) rs[k * (3 * k + 1) / 2] += T(k % 2 ? -1LL : 1LL); if (1LL * k * (3 * k - 1) / 2 <= n) rs[k * (3 * k - 1) / 2] += T(k % 2 ? -1LL : 1LL); } return inv(rs, n + 1); } }; #define FOR(i,n) for(int i = 0; i < (n); i++) #define sz(c) ((int)(c).size()) #define ten(x) ((int)1e##x) template T extgcd(T a, T b, T& x, T& y) { for (T u = y = 1, v = x = 0; a;) { T q = b / a; swap(x -= q * u, u); swap(y -= q * v, v); swap(b -= q * a, a); } return b; } template T mod_inv(T a, T m) { T x, y; extgcd(a, m, x, y); return (m + x % m) % m; } ll mod_pow(ll a, ll n, ll mod) { ll ret = 1; ll p = a % mod; while (n) { if (n & 1) ret = ret * p % mod; p = p * p % mod; n >>= 1; } return ret; } struct MathsNTTModAny { template class NTT { public: int get_mod() const { return mod; } void _ntt(vector& a, int sign) { const int n = sz(a); assert((n ^ (n & -n)) == 0); //n = 2^k const int g = 3; //g is primitive root of mod int h = (int)mod_pow(g, (mod - 1) / n, mod); // h^n = 1 if (sign == -1) h = (int)mod_inv(h, mod); //h = h^-1 % mod //bit reverse int i = 0; for (int j = 1; j < n - 1; ++j) { for (int k = n >> 1; k > (i ^= k); k >>= 1); if (j < i) swap(a[i], a[j]); } for (int m = 1; m < n; m *= 2) { const int m2 = 2 * m; const ll base = mod_pow(h, n / m2, mod); ll w = 1; FOR(x, m) { for (int s = x; s < n; s += m2) { ll u = a[s]; ll d = a[s + m] * w % mod; a[s] = u + d; if (a[s] >= mod) a[s] -= mod; a[s + m] = u - d; if (a[s + m] < 0) a[s + m] += mod; } w = w * base % mod; } } for (auto& x : a) if (x < 0) x += mod; } void ntt(vector& input) { _ntt(input, 1); } void intt(vector& input) { _ntt(input, -1); const int n_inv = mod_inv(sz(input), mod); for (auto& x : input) x = x * n_inv % mod; } vector convolution(const vector& a, const vector& b) { int ntt_size = 1; while (ntt_size < sz(a) + sz(b)) ntt_size *= 2; vector _a = a, _b = b; _a.resize(ntt_size); _b.resize(ntt_size); ntt(_a); ntt(_b); FOR(i, ntt_size) { (_a[i] *= _b[i]) %= mod; } intt(_a); return _a; } }; ll garner(vector> mr, int mod) { mr.emplace_back(mod, 0); vector coffs(sz(mr), 1); vector constants(sz(mr), 0); FOR(i, sz(mr) - 1) { // coffs[i] * v + constants[i] == mr[i].second (mod mr[i].first) ll v = (mr[i].second - constants[i]) * mod_inv(coffs[i], mr[i].first) % mr[i].first; if (v < 0) v += mr[i].first; for (int j = i + 1; j < sz(mr); j++) { (constants[j] += coffs[j] * v) %= mr[j].first; (coffs[j] *= mr[i].first) %= mr[j].first; } } return constants[sz(mr) - 1]; } typedef NTT<167772161, 3> NTT_1; typedef NTT<469762049, 3> NTT_2; typedef NTT<1224736769, 3> NTT_3; vector solve(vector a, vector b, int mod = 1000000007) { for (auto& x : a) x %= mod; for (auto& x : b) x %= mod; NTT_1 ntt1; NTT_2 ntt2; NTT_3 ntt3; assert(ntt1.get_mod() < ntt2.get_mod() && ntt2.get_mod() < ntt3.get_mod()); auto x = ntt1.convolution(a, b); auto y = ntt2.convolution(a, b); auto z = ntt3.convolution(a, b); const ll m1 = ntt1.get_mod(), m2 = ntt2.get_mod(), m3 = ntt3.get_mod(); const ll m1_inv_m2 = mod_inv(m1, m2); const ll m12_inv_m3 = mod_inv(m1 * m2, m3); const ll m12_mod = m1 * m2 % mod; vector ret(sz(x)); FOR(i, sz(x)) { ll v1 = (y[i] - x[i]) * m1_inv_m2 % m2; if (v1 < 0) v1 += m2; ll v2 = (z[i] - (x[i] + m1 * v1) % m3) * m12_inv_m3 % m3; if (v2 < 0) v2 += m3; ll constants3 = (x[i] + m1 * v1 + m12_mod * v2) % mod; if (constants3 < 0) constants3 += mod; ret[i] = constants3; } return ret; } vector solve(vector a, vector b, int mod = 1000000007) { vector x(all(a)); vector y(all(b)); auto z = solve(x, y, mod); vector res; fore(aa, z) res.push_back(aa % mod); return res; } vector solve(vector a, vector b, int mod = 1000000007) { int n = a.size(); vector x(n); rep(i, 0, n) x[i] = a[i].get(); n = b.size(); vector y(n); rep(i, 0, n) y[i] = b[i].get(); auto z = solve(x, y, mod); vector res; fore(aa, z) res.push_back(aa % mod); vector res2; fore(x, res) res2.push_back(x); return res2; } }; /* using T = mint; FormalPowerSeries FPS([&](auto a, auto b) { MathsNTTModAny ntt; return ntt.solve(a, b); }); */ /*---------------------------------------------------------------------------------------------------             ∧_∧       ∧_∧  (´<_` )  Welcome to My Coding Space!      ( ´_ゝ`) /  ⌒i @hamayanhamayan     /   \    | |     /   / ̄ ̄ ̄ ̄/  |   __(__ニつ/  _/ .| .|____      \/____/ (u ⊃ ---------------------------------------------------------------------------------------------------*/ int K, N, x[101010]; //--------------------------------------------------------------------------------------------------- void _main() { using T = mint; FormalPowerSeries FPS([&](auto a, auto b) { MathsNTTModAny ntt; return ntt.solve(a, b); }); cin >> K >> N; rep(i, 0, N) cin >> x[i]; /*vector fx(K + 1); fx[0] = 1; vector fy(K + 1); rep(i, 0, N) fy[x[i]] = 1; mint ans = 0; rep(k, 0, K) { fx = FPS.mul(fx, fy); fx.resize(K + 1); ans += fx[K]; }*/ vector fy(K + 1); fy[0] = 1; rep(i, 0, N) fy[x[i]] = -1; fy = FPS.inv(fy, K + 1); cout << fy[K] << endl; }