#include using namespace std; // -------------------------------------------------------------------------------------------------------------- /*#pragma GCC target("avx") #pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")//*/ void wa_haya_exe() { cin.tie(nullptr); ios::sync_with_stdio(false); } mt19937 Random() { random_device seed_gen; mt19937 engine {seed_gen()}; return engine; } // -------------------------------------------------------------------------------------------------------------- // alias template using V = vector; template using P = pair; template using pq = priority_queue; template using pqr = priority_queue,greater>; using ll = long long; using ld = long double; using uint = unsigned; using ul = unsigned long long; using pi = P; using pl = P; using pd = P; using pc = P; using ps = P; // Input template void scan(T& a) { cin >> a; } template void scan(P& p) { scan(p.first); scan(p.second); } template void scan(V& a) { for(auto &i: a) scan(i); } void in(){} template void in(Head& head, Tail&... tail) { scan(head); in(tail...); } #define INT(...) int __VA_ARGS__;in(__VA_ARGS__) #define LL(...) ll __VA_ARGS__;in(__VA_ARGS__) #define UL(...) ul __VA_ARGS__;in(__VA_ARGS__) #define CHR(...) char __VA_ARGS__;in(__VA_ARGS__) #define STR(...) string __VA_ARGS__;in(__VA_ARGS__) #define DBL(...) double __VA_ARGS__;in(__VA_ARGS__) #define LD(...) ld __VA_ARGS__;in(__VA_ARGS__) // Output template void print(const T& a) { cout << a; } template void print(const P& p) { print(p.first); cout<<" "; print(p.second); } template void print(const V& a) { if(!a.empty()){ print(a[0]); for(auto i=a.begin(); ++i!=a.end();){ cout<<" "; print(*i); } } } void out(){ cout< void out(const T& t){ print(t); cout< void out(const Head& head, const Tail&... tail) { print(head); cout<<" "; out(tail...); } #define fix(x) cout<::infinity(); const double PI=acos(-1); // for #define overload4(_1,_2,_3,_4,name,...) name #define overload3(_1,_2,_3,name,...) name #define rep1(n) for(ll i=0; i=0; i--) #define rvp2(i,n) for(ll i=n-1; i>=0; i--) #define rvp3(i,a,b) for(ll i=b-1; i>=a; i--) #define rvp4(i,a,b,c) for(ll i=b-1; i>=a; i-=c) #define rvp(...) overload4(__VA_ARGS__,rvp4,rvp3,rvp2,rvp1)(__VA_ARGS__) #define all1(v) v.begin(),v.end() #define all2(v,a) v.begin(),v.begin()+a #define all3(v,a,b) v.begin()+a,v.begin()+b #define all(...) overload3(__VA_ARGS__,all3,all2,all1)(__VA_ARGS__) #define each1(elem,v) for(auto &elem: v) #define each2(x,y,v) for(auto &[x,y]: v) #define each3(x,y,z,v) for(auto &[x,y,z]: v) #define each(...) overload4(__VA_ARGS__,each3,each2,each1)(__VA_ARGS__) #define sqrp1(n) for(ll i=0; i*i name(__VA_ARGS__) #define VEC(type,name,size) vector name(size); in(name) #define wec(type,name,h,...) vector> name(h,vector(__VA_ARGS__)) #define Wec(type,name,h) vector> name(h) #define WEC(type,name,h,w) vector> name(h,vector(w)); in(name) #define rall(v) v.rbegin(), v.rend() #define rev(v) reverse(all(v)) #define Sort(v) sort(all(v)) #define Sortt(v) sort(rall(v)) #define Min(v) *min_element(all(v)) #define Max(v) *max_element(all(v)) #define LB(v,x) lower_bound(all(v),x) #define UB(v,x) upper_bound(all(v),x) #define rtt(v,idx) rotate(all(v,idx),v.end()) #define adf(v,res) adjacent_difference(all(v),res.begin()) #define psum(v,s) partial_sum(all(v),s.begin()); rtt(s,v.size()) #define vsum1(v) reduce(all(v),0LL) #define vsum2(v,a) reduce(all(v,a),0LL) #define vsum3(v,a,b) reduce(all(v,a,b),0LL) #define vsum(...) overload3(__VA_ARGS__,vsum3,vsum2,vsum1)(__VA_ARGS__) #define vdsum1(v) reduce(all(v),0.0L) #define vdsum2(v,a) reduce(all(v,a),0.0LL) #define vdsum3(v,a,b) reduce(all(v,a,b),0.0L) #define vdsum(...) overload3(__VA_ARGS__,vdsum3,vdsum2,vdsum1)(__VA_ARGS__) #define vmul1(v) reduce(all(v),1LL,[](ll acc,ll i){ return acc*i; }) #define vmul2(v,a) reduce(all(v,a),1LL,[](ll acc,ll i){ return acc*i; }) #define vmul3(v,a,b) reduce(all(v,a,b),1LL,[](ll acc,ll i){ return acc*i; }) #define vmul(...) overload3(__VA_ARGS__,vmul3,vmul2,vmul1)(__VA_ARGS__) #define iot(a,init) iota(all(a),init) #define ers(v,x) v.erase(remove(all(v),x),v.end()) #define unq(v) Sort(v);v.erase(unique(all(v)),v.end()) #define cp(v,w) copy(all(v),back_inserter(w)) #define smp(v,w,size,eng) sample(all(v),back_inserter(w),size,eng); #define rpl(v,f,r) replace_if(all(v),f,r) #define All(v,f) all_of(all(v),f) #define Exist(v,f) any_of(all(v),f) #define Find(v,x) find(all(v),x) #define cntif(v,f) count_if(all(v),f) template inline T Count(V& v, ll x) { Sort(v); return UB(v,x)-LB(v,x); } template inline void vout(V& v) { FE(v,[](T x){out(x);}); } // renew max(old < new) or min(old > new) template bool chmax(T& a, const T& b) { if (a bool chmax(T& a, const U& b) { if (a bool chmin(T& a, const T& b) { if (a>b) { a=b; return 1; } return 0; } template bool chmin(T& a, const U& b) { if (a>b) { a=b; return 1; } return 0; } // overflow template bool overflow_if_add(T a, T b) { return (numeric_limits::max() - a) < b; } template bool overflow_if_mul(T a, T b) { return (numeric_limits::max() / a) < b; } // tips #define elif else if #define scp(a,x,b) a<=x&&x<=b #define Mod(x,m) (x+m)%m #define each_idx(el,v) &el-&v[0] #define Ceil(x) ceil((ld)x) #define PW(a,b) (ll)pow(a,b) #define IP(v,u,init) inner_product(all(v),u.begin(),init) #define Mid(a,b) midpoint(a,b) #define dst(v,itr) distance(v.begin(),itr) #define mp make_pair #define mt make_tuple #define bit_check(bit,tar) (tar>>bit)&1 #define popcnt(x) __builtin_popcountll(x) #define str_rpl(s,a,b) regex_replace(s,regex(a),b) #define ast(bool) assert(bool) // min, max #define key_min(map) map.begin()->first #define key_max(map) map.rbegin()->first #define set_min(set) *set.begin() #define set_max(set) *set.rbegin() template constexpr auto mymin(T... a) { return min(initializer_list>{a...}); } template constexpr auto mymax(T... a) { return max(initializer_list>{a...}); } /* Permutation Sample template inline void nPr(V& v) { Sort(v); do {out(v);} while(next_permutation(all(v))); }//*/ /* Divisor Sample template V divisor(T n) { V div; sqrp(i,1,n+1) { if(n%i==0) { div.emplace_back(i); if(i*i!=n) div.emplace_back(n/i); } } Sort(div); return div; }//*/ /* Prime Sample bool is_prime(ul n) { sqrp(i,2,n+1) if(n%i==0) return 0; return 1; }//*/ // Siege of Eratosthenes V SoE(ll n) { V prime(n+1,1); if(n>=0) prime[0]=0; if(n>=1) prime[1]=0; sqrp(i,2,n+1) { if(!prime[i]) continue; rep(j,i*i,n+1,i) prime[j]=0; } return prime; } /* Prime Factor Sample template V> prmfct(T n) { V> res; rep(i,2,n+1) { if(n%i!=0) continue; T tmp=0; while(n%i==0) { tmp++; n/=i; } res.emplace_back(i,tmp); } if(n!=1) res.emplace_back(n,1); return res; }//*/ /* BFS Sample template inline void bfs(V>& g, T start, V& visited) { queue que; que.emplace(start); visited[start]=1; while(!que.empty()) { T v=que.front(); que.pop(); each(nv,g[v]) { if(!visited[nv]) { visited[nv]=1; que.emplace(nv); } } } }//*/ /* DFS Sample template inline void dfs(V>& g, T v, V& visited) { visited[v]=1; each(nv,g[v]) if(!visited[nv]) dfs(g,nv,visited); }//*/ // UnionFind-Tree struct UnionFind { V par,sz; UnionFind(ll n) { par.resize(n,0); sz.resize(n,0); rep(n) makeTree(i); } void makeTree(ll x) { par[x]=x; sz[x]=1; } bool same(ll x, ll y) { return root(x)==root(y); } bool unite(ll x, ll y) { x=root(x); y=root(y); if(x==y) return 0; if(sz[x]>sz[y]) { par[y]=x; sz[x]+=sz[y]; } else { par[x]=y; sz[y]+=sz[x]; } return 1; } ll root(ll x) { if(x!=par[x]) par[x]=root(par[x]); return par[x]; } ll size(ll x) { return sz[root(x)]; } }; // mint constexpr const int mod=MOD; struct mint{ uint num = 0; constexpr mint() noexcept {} constexpr mint(const mint &x) noexcept : num(x.num){} constexpr operator ll() const noexcept { return num; } constexpr mint& operator+=(mint x) noexcept { num += x.num; if(num >= mod) num -= mod; return *this; } constexpr mint& operator++() noexcept { if(num == mod - 1) num = 0; else num++; return *this; } constexpr mint operator++(int) noexcept { mint ans(*this); operator++(); return ans; } constexpr mint operator-() const noexcept { return mint(0) -= *this; } constexpr mint& operator-=(mint x) noexcept { if(num < x.num) num += mod; num -= x.num; return *this; } constexpr mint& operator--() noexcept { if(num == 0) num = mod - 1; else num--; return *this; } constexpr mint operator--(int) noexcept { mint ans(*this); operator--(); return ans; } constexpr mint& operator*=(mint x) noexcept { num = ul(num) * x.num % mod; return *this; } constexpr mint& operator/=(mint x) noexcept { return operator*=(x.inv()); } template constexpr mint(T x) noexcept { using U = typename conditional= 4, T, int>::type; U y = x; y %= U(mod); if(y < 0) y += mod; num = uint(y); } template constexpr mint operator+(T x) const noexcept { return mint(*this) += x; } template constexpr mint& operator+=(T x) noexcept { return operator+=(mint(x)); } template constexpr mint operator-(T x) const noexcept { return mint(*this) -= x; } template constexpr mint& operator-=(T x) noexcept { return operator-=(mint(x)); } template constexpr mint operator*(T x) const noexcept { return mint(*this) *= x; } template constexpr mint& operator*=(T x) noexcept { return operator*=(mint(x)); } template constexpr mint operator/(T x) const noexcept { return mint(*this) /= x; } template constexpr mint& operator/=(T x) noexcept { return operator/=(mint(x)); } constexpr mint inv() const noexcept { ll x = 0, y = 0; extgcd(num, mod, x, y); return x; } static constexpr ll extgcd(ll a, ll b, ll &x, ll &y) noexcept { ll g = a; x = 1; y = 0; if(b){ g = extgcd(b, a % b, y, x); y -= a / b * x; } return g; } constexpr mint pow(ul x) const noexcept { mint ans = 1, cnt = *this; while(x){ if(x & 1) ans *= cnt; cnt *= cnt; x /= 2; } return ans; } }; ostream& operator<<(ostream& os, const mint& m){ os << m.num; return os; } /*V fac(1,1),inv(1,1); void reserve(ll a){ if(fac.size()>=a) return; if(a=mod) a=mod; while(fac.size()>24){ mint ans=1; rep(r) ans*=n--; return ans; } reserve(n+1); return fac[n]*inv[n-r]; } mint comb(ll n,ll r){ if(r<0 || n