#include using namespace std; #define ALL(x) begin(x),end(x) #define rep(i,n) for(int i=0;i<(n);i++) #define debug(v) cout<<#v<<":";for(auto x:v){cout<bool chmax(T &a,const T &b){if(abool chmin(T &a,const T &b){if(b ostream &operator<<(ostream &os,const vector&v){ for(int i=0;i<(int)v.size();i++) os< istream &operator>>(istream &is,vector&v){ for(T &x:v)is>>x; return is; } namespace FastFourierTransform { using real = double; struct C { real x, y; C() : x(0), y(0) {} C(real x, real y) : x(x), y(y) {} inline C operator+(const C &c) const { return C(x + c.x, y + c.y); } inline C operator-(const C &c) const { return C(x - c.x, y - c.y); } inline C operator*(const C &c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); } inline C conj() const { return C(x, -y); } }; const real PI = acosl(-1); int base = 1; vector< C > rts = { {0, 0}, {1, 0} }; vector< int > rev = {0, 1}; void ensure_base(int nbase) { if(nbase <= base) return; rev.resize(1 << nbase); rts.resize(1 << nbase); for(int i = 0; i < (1 << nbase); i++) { rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1)); } while(base < nbase) { real angle = PI * 2.0 / (1 << (base + 1)); for(int i = 1 << (base - 1); i < (1 << base); i++) { rts[i << 1] = rts[i]; real angle_i = angle * (2 * i + 1 - (1 << base)); rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i)); } ++base; } } void fft(vector< C > &a, int n) { assert((n & (n - 1)) == 0); int zeros = __builtin_ctz(n); ensure_base(zeros); int shift = base - zeros; for(int i = 0; i < n; i++) { if(i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); } } for(int k = 1; k < n; k <<= 1) { for(int i = 0; i < n; i += 2 * k) { for(int j = 0; j < k; j++) { C z = a[i + j + k] * rts[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } } vector< int64_t > multiply(const vector< int > &a, const vector< int > &b) { int need = (int) a.size() + (int) b.size() - 1; int nbase = 1; while((1 << nbase) < need) nbase++; ensure_base(nbase); int sz = 1 << nbase; vector< C > fa(sz); for(int i = 0; i < sz; i++) { int x = (i < (int) a.size() ? a[i] : 0); int y = (i < (int) b.size() ? b[i] : 0); fa[i] = C(x, y); } fft(fa, sz); C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0); for(int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r; fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r; fa[i] = z; } for(int i = 0; i < (sz >> 1); i++) { C A0 = (fa[i] + fa[i + (sz >> 1)]) * t; C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i]; fa[i] = A0 + A1 * s; } fft(fa, sz >> 1); vector< int64_t > ret(need); for(int i = 0; i < need; i++) { ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x); } return ret; } }; template< typename T > struct ArbitraryModConvolution { using real = FastFourierTransform::real; using C = FastFourierTransform::C; ArbitraryModConvolution() = default; vector< T > multiply(const vector< T > &a, const vector< T > &b, int need = -1) { if(need == -1) need = a.size() + b.size() - 1; int nbase = 0; while((1 << nbase) < need) nbase++; FastFourierTransform::ensure_base(nbase); int sz = 1 << nbase; vector< C > fa(sz); for(int i = 0; i < a.size(); i++) { fa[i] = C(a[i].x & ((1 << 15) - 1), a[i].x >> 15); } fft(fa, sz); vector< C > fb(sz); if(a == b) { fb = fa; } else { for(int i = 0; i < b.size(); i++) { fb[i] = C(b[i].x & ((1 << 15) - 1), b[i].x >> 15); } fft(fb, sz); } real ratio = 0.25 / sz; C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1); for(int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); C a1 = (fa[i] + fa[j].conj()); C a2 = (fa[i] - fa[j].conj()) * r2; C b1 = (fb[i] + fb[j].conj()) * r3; C b2 = (fb[i] - fb[j].conj()) * r4; if(i != j) { C c1 = (fa[j] + fa[i].conj()); C c2 = (fa[j] - fa[i].conj()) * r2; C d1 = (fb[j] + fb[i].conj()) * r3; C d2 = (fb[j] - fb[i].conj()) * r4; fa[i] = c1 * d1 + c2 * d2 * r5; fb[i] = c1 * d2 + c2 * d1; } fa[j] = a1 * b1 + a2 * b2 * r5; fb[j] = a1 * b2 + a2 * b1; } fft(fa, sz); fft(fb, sz); vector< T > ret(need); for(int i = 0; i < need; i++) { int64_t aa = llround(fa[i].x); int64_t bb = llround(fb[i].x); int64_t cc = llround(fa[i].y); aa = T(aa).x, bb = T(bb).x, cc = T(cc).x; ret[i] = aa + (bb << 15) + (cc << 30); } return ret; } }; template< int mod > struct ModInt { int x; ModInt() : x(0) {} ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {} ModInt &operator+=(const ModInt &p) { if((x += p.x) >= mod) x -= mod; return *this; } ModInt &operator-=(const ModInt &p) { if((x += mod - p.x) >= mod) x -= mod; return *this; } ModInt &operator*=(const ModInt &p) { x = (int) (1LL * x * p.x % mod); return *this; } ModInt &operator/=(const ModInt &p) { *this *= p.inverse(); return *this; } ModInt operator-() const { return ModInt(-x); } ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; } ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; } ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; } ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; } bool operator==(const ModInt &p) const { return x == p.x; } bool operator!=(const ModInt &p) const { return x != p.x; } ModInt inverse() const { int a = x, b = mod, u = 1, v = 0, t; while(b > 0) { t = a / b; swap(a -= t * b, b); swap(u -= t * v, v); } return ModInt(u); } ModInt pow(int64_t n) const { ModInt ret(1), mul(x); while(n > 0) { if(n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } friend ostream &operator<<(ostream &os, const ModInt &p) { return os << p.x; } friend istream &operator>>(istream &is, ModInt &a) { int64_t t; is >> t; a = ModInt< mod >(t); return (is); } static int get_mod() { return mod; } }; using mint = ModInt< 1000000007 >; template< typename T > struct FormalPowerSeries : vector< T > { using vector< T >::vector; using P = FormalPowerSeries; using MULT = function< P(P, P) >; using FFT = function< void(P &) >; using SQRT = function< T(T) >; static MULT &get_mult() { static MULT mult = nullptr; return mult; } static void set_mult(MULT f) { get_mult() = f; } static FFT &get_fft() { static FFT fft = nullptr; return fft; } static FFT &get_ifft() { static FFT ifft = nullptr; return ifft; } static void set_fft(FFT f, FFT g) { get_fft() = f; get_ifft() = g; } static SQRT &get_sqrt() { static SQRT sqr = nullptr; return sqr; } static void set_sqrt(SQRT sqr) { get_sqrt() = sqr; } void shrink() { while(this->size() && this->back() == T(0)) this->pop_back(); } P operator+(const P &r) const { return P(*this) += r; } P operator+(const T &v) const { return P(*this) += v; } P operator-(const P &r) const { return P(*this) -= r; } P operator-(const T &v) const { return P(*this) -= v; } P operator*(const P &r) const { return P(*this) *= r; } P operator*(const T &v) const { return P(*this) *= v; } P operator/(const P &r) const { return P(*this) /= r; } P operator%(const P &r) const { return P(*this) %= r; } P &operator+=(const P &r) { if(r.size() > this->size()) this->resize(r.size()); for(int i = 0; i < r.size(); i++) (*this)[i] += r[i]; return *this; } P &operator+=(const T &r) { if(this->empty()) this->resize(1); (*this)[0] += r; return *this; } P &operator-=(const P &r) { if(r.size() > this->size()) this->resize(r.size()); for(int i = 0; i < r.size(); i++) (*this)[i] -= r[i]; shrink(); return *this; } P &operator-=(const T &r) { if(this->empty()) this->resize(1); (*this)[0] -= r; shrink(); return *this; } P &operator*=(const T &v) { const int n = (int) this->size(); for(int k = 0; k < n; k++) (*this)[k] *= v; return *this; } P &operator*=(const P &r) { if(this->empty() || r.empty()) { this->clear(); return *this; } assert(get_mult() != nullptr); return *this = get_mult()(*this, r); } P &operator%=(const P &r) { return *this -= *this / r * r; } P operator-() const { P ret(this->size()); for(int i = 0; i < this->size(); i++) ret[i] = -(*this)[i]; return ret; } P &operator/=(const P &r) { if(this->size() < r.size()) { this->clear(); return *this; } int n = this->size() - r.size() + 1; return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n); } P dot(P r) const { P ret(min(this->size(), r.size())); for(int i = 0; i < ret.size(); i++) ret[i] = (*this)[i] * r[i]; return ret; } P pre(int sz) const { return P(begin(*this), begin(*this) + min((int) this->size(), sz)); } P operator>>(int sz) const { if(this->size() <= sz) return {}; P ret(*this); ret.erase(ret.begin(), ret.begin() + sz); return ret; } P operator<<(int sz) const { P ret(*this); ret.insert(ret.begin(), sz, T(0)); return ret; } P rev(int deg = -1) const { P ret(*this); if(deg != -1) ret.resize(deg, T(0)); reverse(begin(ret), end(ret)); return ret; } P diff() const { const int n = (int) this->size(); P ret(max(0, n - 1)); for(int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i); return ret; } P integral() const { const int n = (int) this->size(); P ret(n + 1); ret[0] = T(0); for(int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1); return ret; } // F(0) must not be 0 P inv(int deg = -1) const { assert(((*this)[0]) != T(0)); const int n = (int) this->size(); if(deg == -1) deg = n; if(get_fft() != nullptr) { P ret(*this); ret.resize(deg, T(0)); return ret.inv_fast(); } P ret({T(1) / (*this)[0]}); for(int i = 1; i < deg; i <<= 1) { ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1); } return ret.pre(deg); } // F(0) must be 1 P log(int deg = -1) const { assert((*this)[0] == 1); const int n = (int) this->size(); if(deg == -1) deg = n; return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } P sqrt(int deg = -1) const { const int n = (int) this->size(); if(deg == -1) deg = n; if((*this)[0] == T(0)) { for(int i = 1; i < n; i++) { if((*this)[i] != T(0)) { if(i & 1) return {}; if(deg - i / 2 <= 0) break; auto ret = (*this >> i).sqrt(deg - i / 2); if(ret.empty()) return {}; ret = ret << (i / 2); if(ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return P(deg, 0); } P ret; if(get_sqrt() == nullptr) { assert((*this)[0] == T(1)); ret = {T(1)}; } else { auto sqr = get_sqrt()((*this)[0]); if(sqr * sqr != (*this)[0]) return {}; ret = {T(sqr)}; } T inv2 = T(1) / T(2); for(int i = 1; i < deg; i <<= 1) { ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2; } return ret.pre(deg); } // F(0) must be 0 P exp(int deg = -1) const { assert((*this)[0] == T(0)); const int n = (int) this->size(); if(deg == -1) deg = n; if(get_fft() != nullptr) { P ret(*this); ret.resize(deg, T(0)); return ret.exp_rec(); } P ret({T(1)}); for(int i = 1; i < deg; i <<= 1) { ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1); } return ret.pre(deg); } P online_convolution_exp(const P &conv_coeff) const { const int n = (int) conv_coeff.size(); assert((n & (n - 1)) == 0); vector< P > conv_ntt_coeff; auto& fft = get_fft(); auto& ifft = get_ifft(); for(int i = n; i >= 1; i >>= 1) { P g(conv_coeff.pre(i)); fft(g); conv_ntt_coeff.emplace_back(g); } P conv_arg(n), conv_ret(n); auto rec = [&](auto rec, int l, int r, int d) -> void { if(r - l <= 16) { for(int i = l; i < r; i++) { T sum = 0; for(int j = l; j < i; j++) sum += conv_arg[j] * conv_coeff[i - j]; conv_ret[i] += sum; conv_arg[i] = i == 0 ? T(1) : conv_ret[i] / i; } } else { int m = (l + r) / 2; rec(rec, l, m, d + 1); P pre(r - l); for(int i = 0; i < m - l; i++) pre[i] = conv_arg[l + i]; fft(pre); for(int i = 0; i < r - l; i++) pre[i] *= conv_ntt_coeff[d][i]; ifft(pre); for(int i = 0; i < r - m; i++) conv_ret[m + i] += pre[m + i - l]; rec(rec, m, r, d + 1); } }; rec(rec, 0, n, 0); return conv_arg; } P exp_rec() const { assert((*this)[0] == T(0)); const int n = (int) this->size(); int m = 1; while(m < n) m *= 2; P conv_coeff(m); for(int i = 1; i < n; i++) conv_coeff[i] = (*this)[i] * i; return online_convolution_exp(conv_coeff).pre(n); } P inv_fast() const { assert(((*this)[0]) != T(0)); const int n = (int) this->size(); P res{T(1) / (*this)[0]}; for(int d = 1; d < n; d <<= 1) { P f(2 * d), g(2 * d); for(int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j]; for(int j = 0; j < d; j++) g[j] = res[j]; get_fft()(f); get_fft()(g); for(int j = 0; j < 2 * d; j++) f[j] *= g[j]; get_ifft()(f); for(int j = 0; j < d; j++) { f[j] = 0; f[j + d] = -f[j + d]; } get_fft()(f); for(int j = 0; j < 2 * d; j++) f[j] *= g[j]; get_ifft()(f); for(int j = 0; j < d; j++) f[j] = res[j]; res = f; } return res.pre(n); } P pow(int64_t k, int deg = -1) const { const int n = (int) this->size(); if(deg == -1) deg = n; for(int i = 0; i < n; i++) { if((*this)[i] != T(0)) { T rev = T(1) / (*this)[i]; P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k)); if(i * k > deg) return P(deg, T(0)); ret = (ret << (i * k)).pre(deg); if(ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return *this; } T eval(T x) const { T r = 0, w = 1; for(auto &v : *this) { r += w * v; w *= x; } return r; } P pow_mod(int64_t n, P mod) const { P modinv = mod.rev().inv(); auto get_div = [&](P base) { if(base.size() < mod.size()) { base.clear(); return base; } int n = base.size() - mod.size() + 1; return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n); }; P x(*this), ret{1}; while(n > 0) { if(n & 1) { ret *= x; ret -= get_div(ret) * mod; } x *= x; x -= get_div(x) * mod; n >>= 1; } return ret; } }; signed main(){ int k,n;cin>>k>>n; vector v(n); cin>>v; using FPS=FormalPowerSeries; FPS f(k+1); f[0]=1; rep(i,n) f[v[i]]=-1; ArbitraryModConvolution ntt; auto mult=[&](const FPS::P &a,const FPS::P &b){ auto ret=ntt.multiply(a,b); return FPS::P(begin(ret),end(ret)); }; FPS::set_mult(mult); cout<