#include using namespace std; namespace Rec{ template class y_combinator_result { Fun fun_; public: template explicit y_combinator_result(T &&fun): fun_(std::forward(fun)) {} template decltype(auto) operator()(Args &&...args) { return fun_(std::ref(*this), std::forward(args)...); } }; template decltype(auto) y_combinator(Fun &&fun) { return y_combinator_result>(std::forward(fun)); } } //fast IO by yosupo struct Scanner { FILE* fp = nullptr; char line[(1 << 15) + 1]; size_t st = 0, ed = 0; void reread() { memmove(line, line + st, ed - st); ed -= st; st = 0; ed += fread(line + ed, 1, (1 << 15) - ed, fp); line[ed] = '\0'; } bool succ() { while (true) { if (st == ed) { reread(); if (st == ed) return false; } while (st != ed && isspace(line[st])) st++; if (st != ed) break; } if (ed - st <= 50) reread(); return true; } template ::value, int> = 0> bool read_single(T& ref) { if (!succ()) return false; while (true) { size_t sz = 0; while (st + sz < ed && !isspace(line[st + sz])) sz++; ref.append(line + st, sz); st += sz; if (!sz || st != ed) break; reread(); } return true; } template ::value, int> = 0> bool read_single(T& ref) { if (!succ()) return false; bool neg = false; if (line[st] == '-') { neg = true; st++; } ref = T(0); while (isdigit(line[st])) { ref = 10 * ref + (line[st++] - '0'); } if (neg) ref = -ref; return true; } template bool read_single(vector& ref) { for (auto& d : ref) { if (!read_single(d)) return false; } return true; } void read() {} template void read(H& h, T&... t) { bool f = read_single(h); assert(f); read(t...); } Scanner(FILE* _fp) : fp(_fp) {} }; struct Printer { public: template void write() {} template void write(const H& h, const T&... t) { if (F) write_single(' '); write_single(h); write(t...); } template void writeln(const T&... t) { write(t...); write_single('\n'); } Printer(FILE* _fp) : fp(_fp) {} ~Printer() { flush(); } private: static constexpr size_t SIZE = 1 << 15; FILE* fp; char line[SIZE], small[50]; size_t pos = 0; void flush() { fwrite(line, 1, pos, fp); pos = 0; } void write_single(const char& val) { if (pos == SIZE) flush(); line[pos++] = val; } template ::value, int> = 0> void write_single(T val) { if (pos > (1 << 15) - 50) flush(); if (val == 0) { write_single('0'); return; } if (val < 0) { write_single('-'); val = -val; // todo min } size_t len = 0; while (val) { small[len++] = char('0' + (val % 10)); val /= 10; } for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; } pos += len; } void write_single(const string& s) { for (char c : s) write_single(c); } void write_single(const char* s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) write_single(s[i]); } template void write_single(const vector& val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write_single(' '); write_single(val[i]); } } }; using ll=long long; //#define int ll #define rng(i,a,b) for(int i=int(a);i<=int(b);i++) #define rep(i,b) rng(i,0,b-1) #define gnr(i,b,a) for(int i=int(b);i>=int(a);i--) #define per(i,b) gnr(i,b-1,0) #define pb push_back #define eb emplace_back #define fi first #define se second #define bg begin() #define ed end() #define all(x) x.bg,x.ed #define si(x) int(x.size()) #ifdef LOCAL #define dmp(x) cerr<<__LINE__<<" "<<#x<<" "< void chmax(t&a,u b){if(a void chmin(t&a,u b){if(b using vc=vector; template using vvc=vc>; using pii=pair; using vi=vc; template ostream& operator<<(ostream& os,const pair& p){ return os<<"{"< ostream& operator<<(ostream& os,const vc& v){ os<<"{"; for(auto e:v)os< void dmpr(ostream&os,const T&t,const Args&... args){ os<; using pli=pair; using pll=pair; template ostream& operator<<(ostream&os,const array&a){ return os<(all(a)); } template void print_tuple(ostream&,const T&){ } template void print_tuple(ostream&os,const T&t){ if(i)os<<","; os<(t); print_tuple(os,t); } template ostream& operator<<(ostream&os,const tuple&t){ os<<"{"; print_tuple<0,tuple,Args...>(os,t); return os<<"}"; } template void print(t x,int suc=1){ cout<>i; return i; } vi readvi(int n,int off=0){ vi v(n); rep(i,n)v[i]=read()+off; return v; } template void print(const vector&v,int suc=1){ rep(i,v.size()) print(v[i],i==int(v.size())-1?suc:2); } string readString(){ string s; cin>>s; return s; } template T sq(const T& t){ return t*t; } //#define CAPITAL void yes(bool ex=true){ #ifdef CAPITAL cout<<"YES"<<"\n"; #else cout<<"Yes"<<"\n"; #endif if(ex)exit(0); } void no(bool ex=true){ #ifdef CAPITAL cout<<"NO"<<"\n"; #else cout<<"No"<<"\n"; #endif if(ex)exit(0); } void possible(bool ex=true){ #ifdef CAPITAL cout<<"POSSIBLE"<<"\n"; #else cout<<"Possible"<<"\n"; #endif if(ex)exit(0); } void impossible(bool ex=true){ #ifdef CAPITAL cout<<"IMPOSSIBLE"<<"\n"; #else cout<<"Impossible"<<"\n"; #endif if(ex)exit(0); } constexpr ll ten(int n){ return n==0?1:ten(n-1)*10; } const ll infLL=LLONG_MAX/3; #ifdef int const int inf=infLL; #else const int inf=INT_MAX/2-100; #endif int topbit(signed t){ return t==0?-1:31-__builtin_clz(t); } int topbit(ll t){ return t==0?-1:63-__builtin_clzll(t); } int botbit(signed a){ return a==0?32:__builtin_ctz(a); } int botbit(ll a){ return a==0?64:__builtin_ctzll(a); } int popcount(signed t){ return __builtin_popcount(t); } int popcount(ll t){ return __builtin_popcountll(t); } bool ispow2(int i){ return i&&(i&-i)==i; } ll mask(int i){ return (ll(1)< bool inc(t a,t b,t c){ return a<=b&&b<=c; } template void mkuni(vc&v){ sort(all(v)); v.erase(unique(all(v)),v.ed); } ll rand_int(ll l, ll r) { //[l, r] #ifdef LOCAL static mt19937_64 gen; #else static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count()); #endif return uniform_int_distribution(l, r)(gen); } template int lwb(const vc&v,const t&a){ return lower_bound(all(v),a)-v.bg; } struct modinfo{uint mod,root;}; template struct modular{ static constexpr uint const &mod=ref.mod; static modular root(){return modular(ref.root);} uint v; //modular(initializer_listls):v(*ls.bg){} modular(ll vv=0){s(vv%mod+mod);} modular& s(uint vv){ v=vv>=1; } return res; } modular inv()const{return pow(mod-2);} /*modular inv()const{ int x,y; int g=extgcd(v,mod,x,y); assert(g==1); if(x<0)x+=mod; return modular(x); }*/ friend modular operator+(int x,const modular&y){ return modular(x)+y; } friend modular operator-(int x,const modular&y){ return modular(x)-y; } friend modular operator*(int x,const modular&y){ return modular(x)*y; } friend modular operator/(int x,const modular&y){ return modular(x)/y; } friend ostream& operator<<(ostream&os,const modular&m){ return os<>(istream&is,modular&m){ ll x;is>>x; m=modular(x); return is; } bool operator<(const modular&r)const{return v; int main(){ cin.tie(0); ios::sync_with_stdio(0); cout<