/*** author: yuji9511 ***/ #include // #include // using namespace atcoder; using namespace std; using ll = long long; using lpair = pair; using vll = vector; const ll MOD = 1e9+7; const ll INF = 1e18; #define rep(i,m,n) for(ll i=(m);i<(n);i++) #define rrep(i,m,n) for(ll i=(m);i>=(n);i--) ostream& operator<<(ostream& os, lpair& h){ os << "(" << h.first << ", " << h.second << ")"; return os;} #define printa(x,n) for(ll i=0;i void print(H&& h, T&&... t){cout<(t)...);} templatebool chmax(T &a, const T &b) { if (abool chmin(T &a, const T &b) { if (b using is_modint = std::is_base_of; template using is_modint_t = std::enable_if_t::value>; 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; 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; constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } 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; } 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); constexpr std::pair inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; 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; auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } if (m0 < 0) m0 += b / s; return {s, m0}; } } // 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()); } static_modint(bool 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; }; using mint = static_modint<1000000007>; 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 sz(c) ((int)(c).size()) 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; rep(x,0,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); rep(i,0,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); rep(i,0,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)); rep(i,0,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(a.begin() ,a.end()); vector y(b.begin(), b.end()); auto z = solve(x, y, mod); vector res; for(auto &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].val(); n = b.size(); vector y(n); rep(i, 0, n) y[i] = b[i].val(); auto z = solve(x, y, mod); vector res; for(auto &aa: z) res.push_back(aa % mod); vector res2; for(auto &x: res) res2.push_back(x); return res2; } }; void solve(){ ll K,N; cin >> K >> N; vll x(N); rep(i,0,N) cin >> x[i]; FormalPowerSeries FPS([&](auto a, auto b) { MathsNTTModAny ntt; return ntt.solve(a, b); }); vector f(K+1, 0); f[0] = 1; rep(i,0,N) f[x[i]] = -1; f = FPS.inv(f, K+1); print(f[K].val()); } int main(){ cin.tie(0); ios::sync_with_stdio(false); solve(); }