#include #include using namespace std; namespace my{ using ml=atcoder::modint1000000007; auto&operator>>(istream&i,ml&x){int t;i>>t;x=t;return i;} auto&operator<<(ostream&o,const ml&x){return o<<(int)x.val();} #define eb emplace_back #define LL(...) ll __VA_ARGS__;lin(__VA_ARGS__) #define VL(n,...) vec__VA_ARGS__;setsize({n},__VA_ARGS__);lin(__VA_ARGS__) #define FO(n) for(ll ij=n;ij-->0;) #define FOR(i,...) for(auto[i,i##stop,i##step]=range(0,__VA_ARGS__);isync_with_stdio(0);cout<r{0,0,1};ll I=0;((r[I++]=a),...);if(!s&&I==1)swap(r[0],r[1]);r[0]-=s;return r;} constexpr char newline=10; constexpr char space=32; auto max(const auto&...a){return max(initializer_list>{a...});} auto min(const auto&...a){return min(initializer_list>{a...});} templatestruct pair{ A a;B b; pair()=default; pair(A a,B b):a(a),b(b){} pair(const std::pair&p):a(p.first),b(p.second){} auto operator<=>(const pair&)const=default; pair operator+(const pair&p)const{return{a+p.a,b+p.b};} friend ostream&operator<<(ostream&o,const pair&p){return o<ostream&operator<<(ostream&o,const std::pair&p){return o<concept vectorial=is_base_of_v,V>; templatestruct vec_attr{using core_type=T;static constexpr int d=0;}; templatestruct vec_attr{using core_type=typename vec_attr::core_type;static constexpr int d=vec_attr::d+1;}; templateusing core_t=vec_attr::core_type; templateistream&operator>>(istream&i,vector&v){fe(v,e)i>>e;return i;} templateostream&operator<<(ostream&o,const vector&v){fe(v,e)o<?newline:space);return o;} templatestruct vec:vector{ using vector::vector; vec(const vector&v){vector::operator=(v);} vec&operator^=(const vec&u){this->insert(this->end(),u.begin(),u.end());return*this;} vec operator^(const vec&u)const{return vec{*this}^=u;} vec&operator+=(const vec&u){vec&v=*this;fo(i,v.size())v[i]+=u[i];return v;} vec&operator-=(const vec&u){vec&v=*this;fo(i,v.size())v[i]-=u[i];return v;} vec operator+(const vec&u)const{return vec{*this}+=u;} vec operator-(const vec&u)const{return vec{*this}-=u;} vec&operator++(){fe(*this,e)++e;return*this;} vec&operator--(){fe(*this,e)--e;return*this;} vec operator-()const{vec v=*this;fe(v,e)e=-e;return v;} auto scan(const auto&f)const{pair,bool>r{};fe(*this,e)if constexpr(!vectorial)r.b?f(r.a,e),r:r={e,1};else if(auto s=e.scan(f);s.b)r.b?f(r.a,s.a),r:r=s;return r;} auto max()const{return scan([](auto&a,const auto&b){ab?a=b:0;;}).a;} }; templateauto make_vec(const ll(&s)[n],T x={}){if constexpr(n==i+1)return vec(s[i],x);else{auto X=make_vec(s,x);return vec(s[i],X);}} templatevoid setsize(const ll(&l)[n],A&...a){((a=make_vec(l,core_t())),...);} void lin(auto&...a){(cin>>...>>a);} templatevoid pp(const auto&...a){ll n=sizeof...(a);((cout<0,c)),...);cout<rts={{0,0},{1,0}}; vectorfft_rev={0,1}; void ensure_base(int nbase){ if(nbase<=base)return; fft_rev.resize(1<>1]>>1)+((i&1)<<(nbase-1)); while(base&a,int n){ assert((n&(n-1))==0); int zeros=__builtin_ctz(n); ensure_base(zeros); int shift=base-zeros; fo(i,n)if(i<(fft_rev[i]>>shift))swap(a[i],a[fft_rev[i]>>shift]); for(int k=1;kstruct arbitrary_mod_convolution{ using real=fft::real; using complex=fft::complex; arbitrary_mod_convolution(){} std::vectormultiply(const std::vector&a,const std::vector&b,int need=-1){ if(need==-1)need=a.size()+b.size()-1; int nbase=0; while((1<fa(sz); fo(i,a.size())fa[i]=complex(a[i].val()&((1<<15)-1),a[i].val()>>15); fft::fast_fourier_transform(fa,sz); std::vectorfb(sz); if(a==b){ fb=fa; }else{ fo(i,b.size())fb[i]=complex(b[i].val()&((1<<15)-1),b[i].val()>>15); fft::fast_fourier_transform(fb,sz); } real ratio=0.25/sz; complex 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); complex a1=(fa[i]+fa[j].conj()); complex a2=(fa[i]-fa[j].conj())*r2; complex b1=(fb[i]+fb[j].conj())*r3; complex b2=(fb[i]-fb[j].conj())*r4; if(i!=j){ complex c1=(fa[j]+fa[i].conj()); complex c2=(fa[j]-fa[i].conj())*r2; complex d1=(fb[j]+fb[i].conj())*r3; complex 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::fast_fourier_transform(fa,sz); fft::fast_fourier_transform(fb,sz); std::vectorret(need); fo(i,need){ int64_t aa=llround(fa[i].x); int64_t bb=llround(fb[i].x); int64_t cc=llround(fa[i].y); aa=T(aa).val(),bb=T(bb).val(),cc=T(cc).val(); ret[i]=aa+(bb<<15)+(cc<<30); } return ret; } }; templatestruct formal_power_series:vec{ using vec::vec; using fps=formal_power_series; static constexpr ll SPARSE_THRESHOLD=20; static inline arbitrary_mod_convolutionfft; static fps mul(const fps&a,const fps&b){ if constexpr(T::mod()==998244353)return convolution(a,b); else return fft.multiply(a,b); } auto operator<=>(const fps&f)const{return this->size()<=>f.size();} fps pre(ll deg)const{fps r(this->begin(),this->begin()+min((ll)this->size(),deg));r.resize(deg);return r;} fps&operator+=(const fps&g){if(g.size()>this->size())this->resize(g.size());fo(i,g.size())(*this)[i]+=g[i];return*this;} fps&operator-=(const fps&g){if(g.size()>this->size())this->resize(g.size());fo(i,g.size())(*this)[i]-=g[i];return*this;} fps&operator*=(const fps&g){return*this=(this->size()&&g.size()?mul(*this,g):fps{});} fps&operator>>=(ll sz){if((ll)this->size()<=sz)return*this=fps{};this->erase(this->begin(),this->begin()+sz);return*this;} fps&operator<<=(ll sz){this->insert(this->begin(),sz,T{});return*this;} fps&operator/=(const fps&g){ ll I1=0,I2=0; while(I1size()&&(*this)[I1]==0)++I1; while(I2=I2); ll L=max(this->size(),g.size()); return*this=((*this>>I2)*(g>>I2).inv(L)).pre(L); } fps operator+(const fps&g)const{return fps{*this}+=g;} fps operator-(const fps&g)const{return fps{*this}-=g;} fps operator*(const fps&g)const{return fps{*this}*=g;} fps operator/(const fps&g)const{return fps{*this}/=g;} fps operator-()const{auto r=*this;fe(r,x)x=-x;return r;} fps operator>>(ll sz)const{return fps{*this}>>=sz;} fps operator<<(ll sz)const{return fps{*this}<<=sz;} fps&operator+=(const T&c){if(!this->size())this->resize(1);(*this)[0]+=c;return*this;} fps&operator-=(const T&c){if(!this->size())this->resize(1);(*this)[0]-=c;return*this;} fps&operator*=(const T&c){fo(i,this->size())(*this)[i]*=c;return*this;} fps&operator/=(const T&c){T c_inv=T{1}/c;fo(i,this->size())(*this)[i]*=c_inv;return*this;} fps operator+(const T&c)const{return fps{*this}+=c;} fps operator-(const T&c)const{return fps{*this}-=c;} fps operator*(const T&c)const{return fps{*this}*=c;} fps operator/(const T&c)const{return fps{*this}/=c;} T operator()(T x)const{T r=0,xi=1;fe(*this,ai)r+=ai*xi,xi*=x;return r;} fps inv_sparse(ll deg=-1)const{ assert((*this)[0]!=T{}); ll n=this->size(); if(deg==-1)deg=n; vec>p; fo(i,1,n)if((*this)[i]!=T{})p.eb(i,(*this)[i]); fps r(deg); r[0]=T{1}/(*this)[0]; fo(i,1,deg){ T t{}; fe(p,k,fk){ if(i-k<0)break; t-=fk*r[i-k]; } r[i]=r[0]*t; } return r; } ll nonzero_terms_count()const{ll r=0;fe(*this,e)r+=(e!=T{});return r;} fps inv(ll deg=-1)const{ assert((*this)[0]!=T{}); if(deg==-1)deg=this->size(); if(nonzero_terms_count()pre(i<<1)*(r*r)).pre(i<<1); return r.pre(deg); } }; templateusing fps=formal_power_series; single_testcase void solve(){ LL(K,N); VL(N,a); fpsf(K+1); fo(i,N)if(a[i]<=K)f[a[i]]++; pp((-f+1).inv()[K]); }}