#if !defined(MYLOCAL)//提出時用テンプレート
#pragma GCC optimize("Ofast")
#if defined(NDEBUG)
#undef NDEBUG
#endif
#include "bits/stdc++.h"
#if __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
#endif
using namespace std;
using ll=long long;
using dd=long double;
using pll=pair<ll,ll>;
using tll=tuple<ll,ll,ll>;
using qll=tuple<ll,ll,ll,ll>;
using ll2=array<ll,2>;
using ll3=array<ll,3>;
using ll4=array<ll,4>;
using namespace chrono;
constexpr ll INF = 1201001001001001001;
struct Fast{ Fast(){ cin.tie(0); ios::sync_with_stdio(false); cout<<fixed<<setprecision(numeric_limits<double>::max_digits10); } } fast;
#define EXPAND( x ) x//VS用おまじない
#define overload3(_1,_2,_3,name,...) name
#define overload4(_1,_2,_3,_4,name,...) name
#define overload5(_1,_2,_3,_4,_5,name,...) name
#define rep1(N) for (ll dmyi = 0; dmyi < (N); dmyi++)
#define rep2(i, N) for (ll i = 0; i < (N); i++)
#define rep3(i, S, E) for (ll i = (S); i <= (E); i++)
#define rep4(i, S, E, t) for (ll i = (S); i <= (E); i+=(t))
#define rep(...) EXPAND(overload4(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__))
#define dep3(i, E, S) for (ll i = (E); i >= (S); i--)
#define dep4(i, E, S, t) for (ll i = (E); i >= (S); i-=(t))
#define dep(...) EXPAND(overload4(__VA_ARGS__, dep4, dep3,_,_)(__VA_ARGS__))
#define ALL1(v) (v).begin(), (v).end()
#define ALL2(v,E) (v).begin(), (v).begin()+((E)+1)
#define ALL3(v,S,E) (v).begin()+(S), (v).begin()+((E)+1)
#define all(...) EXPAND(overload3(__VA_ARGS__, ALL3, ALL2, ALL1)(__VA_ARGS__))
#define RALL1(v) (v).rbegin(), (v).rend()
#define RALL2(v,E) (v).rbegin(), (v).rbegin()+((E)+1)
#define RALL3(v,S,E) (v).rbegin()+(S), (v).rbegin()+((E)+1)
#define rall(...) EXPAND(overload3(__VA_ARGS__, RALL3, RALL2, RALL1)(__VA_ARGS__))
template<class T> inline bool chmax(T &a, T b) { if (a < b) { a = b; return true; }return false; }
template<class T> inline bool chmin(T &a, T b) { if (a > b) { a = b; return true; }return false; }
template<class T> inline auto maxe(T &&v,ll S,ll E){ return *max_element(all(v,S,E)); }
template<class T> inline auto maxe(T &&v){ return *max_element(all(v)); }
template<class T> inline auto mine(T &&v,ll S,ll E){ return *min_element(all(v,S,E)); }
template<class T> inline auto mine(T &&v){ return *min_element(all(v)); }
template<class T,class U=typename remove_reference<T>::type::value_type>
inline U sum(T &&v,ll S,ll E) {return accumulate(all(v,S,E),U());}
template<class T> inline auto sum(T &&v) {return sum(v,0,v.end()-v.begin()-1);}
template<class T> inline ll sz(T &&v){ return (ll)v.size(); }
inline ll Ceil(ll a,ll b){ return (a<0) ? -(-a/b) : (a+b-1)/b; } //負もOK
inline ll Floor(ll a,ll b){ return -Ceil(-a,b); } //負もOK
inline ll Floormod(ll a,ll m){ return Floor(a,m)*m; } //負もOK
inline ll Ceilmod(ll a,ll m){ return Ceil(a,m)*m; } //負もOK
inline ll Mod(ll a,ll m){ ll r=a%m; if(r<0)r+=m; return r; } //負もOK
template<class T> inline T Pow(T a,ll n){ T r=1; for(; n>0; n>>=1,a*=a){ if(n&1)r*=a; } return r; }
inline ll Pow(int a,ll n){ return Pow((ll)a,n); }
inline ll limitmul(ll a,ll b,ll u){ return b==0||a<=u/b ? a*b : u; }//min(a*b,u) a,b,u≧0
//pair用テンプレート
template<class T,class S> inline pair<T,S>& operator+=(pair<T,S> &a,const pair<T,S> &b){ a.first+=b.first; a.second+=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator-=(pair<T,S> &a,const pair<T,S> &b){ a.first-=b.first; a.second-=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator*=(pair<T,S> &a,const pair<T,S> &b){ a.first*=b.first; a.second*=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator/=(pair<T,S> &a,const pair<T,S> &b){ a.first/=b.first; a.second/=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator%=(pair<T,S> &a,const pair<T,S> &b){ a.first%=b.first; a.second%=b.second; return a; }
template<class T,class S,class R> inline pair<T,S>& operator+=(pair<T,S> &a,R b){ a.first+=b; a.second+=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator-=(pair<T,S> &a,R b){ a.first-=b; a.second-=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator*=(pair<T,S> &a,R b){ a.first*=b; a.second*=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator/=(pair<T,S> &a,R b){ a.first/=b; a.second/=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator%=(pair<T,S> &a,R b){ a.first%=b; a.second%=b; return a; }
template<class T,class S,class R> inline pair<T,S> operator+(const pair<T,S> &a,R b){ pair<T,S> c=a; return c+=b; }
template<class T,class S,class R> inline pair<T,S> operator-(const pair<T,S> &a,R b){ pair<T,S> c=a; return c-=b; }
template<class T,class S,class R> inline pair<T,S> operator*(const pair<T,S> &a,R b){ pair<T,S> c=a; return c*=b; }
template<class T,class S,class R> inline pair<T,S> operator/(const pair<T,S> &a,R b){ pair<T,S> c=a; return c/=b; }
template<class T,class S,class R> inline pair<T,S> operator%(const pair<T,S> &a,R b){ pair<T,S> c=a; return c%=b; }
template<class T,class S,class R> inline pair<T,S> operator-(R b,const pair<T,S> &a){ pair<T,S> c=-a; return c+=b; }
template<class T,class S> inline pair<T,S> operator-(const pair<T,S> &a,const pair<T,S> &b){ pair<T,S> c=a; return c-=b; }
template<class T,class S> inline pair<T,S> operator-(const pair<T,S> &a){ pair<T,S> c=a; return c*=(-1); }
template<class T,class S> inline ostream &operator<<(ostream &os,const pair<T,S> &a){ return os << a.first << ' ' << a.second; }
//tuple用テンプレート 出力用のみ
template<class T,class S,class R> inline ostream &operator<<(ostream &os,const tuple<T,S,R> &a){ return os << get<0>(a) << ' ' << get<1>(a) << ' ' << get<2>(a); }
template<class T,class S,class R,class Q> inline ostream &operator<<(ostream &os,const tuple<T,S,R,Q> &a){ return os << get<0>(a) << ' ' << get<1>(a) << ' ' << get<2>(a) << ' ' << get<3>(a); }
//vector用テンプレート
template<class T> inline ostream &operator<<(ostream &os,const vector<T> &a){ for (ll i=0; i<(ll)a.size(); i++) os<<(i>0?" ":"")<<a[i]; return os; }
//array用テンプレート
template<class T,size_t S> inline array<T,S>& operator+=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]+=b[i]; return a; }
template<class T,size_t S> inline array<T,S>& operator-=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]-=b[i]; return a; }
template<class T,size_t S> inline array<T,S>& operator*=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]*=b[i]; return a; }
template<class T,size_t S> inline array<T,S>& operator/=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]/=b[i]; return a; }
template<class T,size_t S> inline array<T,S>& operator%=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]%=b[i]; return a; }
template<class T,size_t S,class R> inline array<T,S>& operator+=(array<T,S> &a,R b){ for (T &e: a) e+=b; return a; }
template<class T,size_t S,class R> inline array<T,S>& operator-=(array<T,S> &a,R b){ for (T &e: a) e-=b; return a; }
template<class T,size_t S,class R> inline array<T,S>& operator*=(array<T,S> &a,R b){ for (T &e: a) e*=b; return a; }
template<class T,size_t S,class R> inline array<T,S>& operator/=(array<T,S> &a,R b){ for (T &e: a) e/=b; return a; }
template<class T,size_t S,class R> inline array<T,S>& operator%=(array<T,S> &a,R b){ for (T &e: a) e%=b; return a; }
template<class T,size_t S,class R> inline array<T,S> operator+(const array<T,S> &a,R b){ array<T,S> c=a; return c+=b; }
template<class T,size_t S,class R> inline array<T,S> operator-(const array<T,S> &a,R b){ array<T,S> c=a; return c-=b; }
template<class T,size_t S,class R> inline array<T,S> operator*(const array<T,S> &a,R b){ array<T,S> c=a; return c*=b; }
template<class T,size_t S,class R> inline array<T,S> operator/(const array<T,S> &a,R b){ array<T,S> c=a; return c/=b; }
template<class T,size_t S,class R> inline array<T,S> operator%(const array<T,S> &a,R b){ array<T,S> c=a; return c%=b; }
template<class T,size_t S,class R> inline array<T,S> operator-(R b,const array<T,S> &a){ array<T,S> c=-a; return c+=b; }
template<class T,size_t S> inline array<T,S> operator-(const array<T,S> &a,const array<T,S> &b){ array<T,S> c=a; return c-=b; }
template<class T,size_t S> inline array<T,S> operator-(const array<T,S> &a){ array<T,S> c=a; return c*=(-1); }
template<class T,size_t S> inline ostream &operator<<(ostream &os,const array<T,S> &a){ for (ll i=0; i<(ll)S; i++) os<<(i>0?" ":"")<<a[i]; return os; }
inline struct{
system_clock::time_point st = system_clock::now();
ll operator()()const{return duration_cast<microseconds>(system_clock::now()-st).count()/1000;}
} timeget;
struct cinutil{
template<class T> static void cin1core(T &a){ cin>>a; }
template<class T,class S> static void cin1core(pair<T,S> &a){
cin1core(a.first), cin1core(a.second);
}
template<class... Args> static void cin1core(tuple<Args...> &a){
cinTplRec<tuple<Args...>,sizeof...(Args)-1>()(a);
}
template<class T,size_t N>
static void cin1core(array<T,N> &a){for(int i=0;i<(int)N;++i) cin>>a[i];}
private:
template<class Tpl,int i> struct cinTplRec{
void operator()(Tpl &a){ cinTplRec<Tpl,i-1>()(a); cin1core(get<i>(a)); }
};
template<class Tpl> struct cinTplRec<Tpl,0>{
void operator()(Tpl &a){ cin1core(get<0>(a)); }
};
};
template<class T> T cin1(){ T a; cinutil::cin1core(a); return a; }
template<class... Args> tuple<Args...> cins(){ return cin1<tuple<Args...>>(); }
template<long long MOD> struct mll_{
using Int = long long;
using ll = long long;
ll val_=0;
/*---- utility ----*/
mll_ &norm(){ return normR().normS(); }//正規化
mll_ &normR(){ val_%=MOD; return *this; }//剰余正規化のみ
mll_ &normS(){ if (val_<0) val_+=MOD; return *this; }//正負正規化のみ
mll_ &normP(){ if (val_>=MOD) val_-=MOD; return *this; }//加算時正規化
mll_ &invsg(){ val_=-val_; return normS(); }//正負反転
ll modinv(int a){//a^-1 mod MOD
int ypre=0,y=1,apre=MOD;
while (a>1){
int t=apre/a;
apre-=a*t,swap(a,apre);
ypre-=y*t,swap(y,ypre);
}
return y<0 ? y+MOD: y;
}
/*---- I/F ----*/
constexpr mll_(){}
mll_(ll v): val_(v){ norm(); }
constexpr mll_(ll v,bool b): val_(v){} //正規化無のコンストラクタ
Int val()const{ return (Int)val_; }
bool isnone() const { return val_==-1; } //true:値なし
mll_ &none() { val_=-1; return *this; } //値なしにする
mll_ &inv(){ val_=modinv((int)val_); return *this; }
mll_ &operator+=(mll_ b){ val_+=b.val_; return normP(); }
mll_ &operator-=(mll_ b){ val_-=b.val_; return normS(); }
mll_ &operator*=(mll_ b){ val_*=b.val_; return normR(); }
mll_ &operator/=(mll_ b){ return *this*=b.inv(); }
mll_ &operator+=(ll b){ return *this+=mll_(b); }
mll_ &operator-=(ll b){ return *this-=mll_(b); }
mll_ &operator*=(ll b){ return *this*=mll_(b); }
mll_ &operator/=(ll b){ return *this/=mll_(b); }
mll_ &operator--(int){ return *this-=1; }
mll_ &operator++(int){ return *this+=1; }
mll_ operator-()const{ return mll_(*this).invsg(); }
mll_ operator+(mll_ b)const{ return mll_(*this)+=b; }
mll_ operator-(mll_ b)const{ return mll_(*this)-=b; }
mll_ operator*(mll_ b)const{ return mll_(*this)*=b; }
mll_ operator/(mll_ b)const{ return mll_(*this)/=b; }
mll_ operator+(ll b)const{ return mll_(*this)+=b; }
mll_ operator-(ll b)const{ return mll_(*this)-=b; }
mll_ operator*(ll b)const{ return mll_(*this)*=b; }
mll_ operator/(ll b)const{ return mll_(*this)/=b; }
friend mll_ operator+(ll a,mll_ b){ return b+a; }
friend mll_ operator-(ll a,mll_ b){ return -b+a; }
friend mll_ operator*(ll a,mll_ b){ return b*a; }
friend mll_ operator/(ll a,mll_ b){ return mll_(a)/b; }
bool operator==(mll_ b)const{ return val_==b.val_; }
bool operator!=(mll_ b)const{ return val_!=b.val_; }
bool operator==(ll b)const{ return *this==mll_(b); }
bool operator!=(ll b)const{ return *this!=mll_(b); }
friend bool operator==(ll a,mll_ b){ return mll_(a)==b; }
friend bool operator!=(ll a,mll_ b){ return mll_(a)!=b; }
friend ostream &operator<<(ostream &os,mll_ a){ return os << a.val_; }
friend istream &operator>>(istream &is,mll_ &a){ return is >> a.val_; }
mll_ pow(ll k)const{
mll_ ret(1,false),a(*this);
for (; k>0; k>>=1,a*=a) if (k&1)ret*=a;
return ret;
}
static constexpr int mod() { return MOD; }
//enum{ modll=MOD };
};
struct bll{
ll s=0;
bll(ll s_=0): s(s_){}
bll(int s_): s(s_){}
bll(const string &bitstr): s(str2val(bitstr)){}
bll(const char *bitstr): s(str2val(bitstr)){}
struct ref {
bll &b; const ll msk;
ref(bll &b_,ll pos):b(b_),msk(1LL<<pos){}
operator ll() const { return (b.s&msk)!=0; }
ref &operator=(bool x){ if(x) b.s|=msk; else b.s&=~msk; return *this; }
};
ref operator[](ll pos){ return ref(*this,pos); }
ll operator[](ll pos) const { return (s>>pos)&1; }
bll &operator=(int b){ s=b; return *this; }
bll &operator=(ll b){ s=b; return *this; }
bll &operator=(const string &bitstr){ s=str2val(bitstr); return *this; }
bll &operator=(const char *bitstr){ s=str2val(bitstr); return *this; }
bll operator++(int){ bll b(*this); s++; return b; }
bll operator--(int){ bll b(*this); s--; return b; }
operator ll() const noexcept { return s; }
bll &operator&=(ll b){ s&=b; return *this; }
bll &operator|=(ll b){ s|=b; return *this; }
bll &operator^=(ll b){ s^=b; return *this; }
bll &operator+=(ll b){ s+=b; return *this; }
bll &operator-=(ll b){ s-=b; return *this; }
bll &operator<<=(ll i){ s<<=i; return *this; }
bll &operator>>=(ll i){ s>>=i; return *this; }
bll operator&(ll b)const{ return s&b; }
bll operator|(ll b)const{ return s|b; }
bll operator^(ll b)const{ return s^b; }
bll operator+(ll b)const{ return s+b; }
bll operator-(ll b)const{ return s-b; }
bll operator<<(ll i)const{ return s<<i; }
bll operator>>(ll i)const{ return s>>i; }
bll operator&(int b)const{ return s&b; }
bll operator|(int b)const{ return s|b; }
bll operator^(int b)const{ return s^b; }
bll operator+(int b)const{ return s+b; }
bll operator-(int b)const{ return s-b; }
bll operator<<(int i)const{ return s<<i; }
bll operator>>(int i)const{ return s>>i; }
bll operator~()const{ return ~s; }
bll &oneq (bll msk){ s|= msk.s; return *this; }
bll &offeq (bll msk){ s&=~msk.s; return *this; }
bll &flipeq(bll msk){ s^= msk.s; return *this; }
bll on (bll msk)const{ return bll(s).oneq (msk); }
bll off (bll msk)const{ return bll(s).offeq (msk); }
bll flip (bll msk)const{ return bll(s).flipeq(msk); }
bool any0(bll msk)const{ return ~s&msk.s; }
bool any1(bll msk)const{ return s&msk.s; }
bool all0(bll msk)const{ return !any1(msk); }
bool all1(bll msk)const{ return !any0(msk); }
bll &oneq (ll l,ll r){ return oneq (rngmsk(l,r)); }
bll &offeq (ll l,ll r){ return offeq (rngmsk(l,r)); }
bll &flipeq(ll l,ll r){ return flipeq(rngmsk(l,r)); }
bll on (ll l,ll r)const{ return on (rngmsk(l,r)); }
bll off (ll l,ll r)const{ return off (rngmsk(l,r)); }
bll flip (ll l,ll r)const{ return flip(rngmsk(l,r)); }
bool any0(ll l,ll r)const{ return any0(rngmsk(l,r)); }
bool any1(ll l,ll r)const{ return any1(rngmsk(l,r)); }
bool all0(ll l,ll r)const{ return all0(rngmsk(l,r)); }
bool all1(ll l,ll r)const{ return all1(rngmsk(l,r)); }
bll &maskeq(ll l,ll r){ s&=rngmsk(l,r); return *this; }
bll mask(ll l,ll r)const{ return bll(s).maskeq(l,r); }
bll &oneq (ll i){ s|= (1LL<<i); return *this; }
bll &offeq (ll i){ s&=~(1LL<<i); return *this; }
bll &flipeq(ll i){ s^= (1LL<<i); return *this; }
bll on (ll i)const{ return s| (1LL<<i); }
bll off (ll i)const{ return s&~(1LL<<i); }
bll flip(ll i)const{ return s^ (1LL<<i); }
bool contains(ll b)const{ return (s&b)==b; }
bll substr(ll l,ll r)const{ return (s&rngmsk(l,r))>>r; }
static bll rngmsk(ll l,ll r){ return (1LL<<(l+1))-(1LL<<r); }
ll msbit()const{
for(ll x=63,o=-1;;){
ll m=(x+o)/2;
if((1LL<<m)<=s) o=m; else x=m;
if(x-o==1) return o;
}
}
ll lsbit()const{ return bll(lsb()).msbit(); }
ll msb()const{ ll pos=msbit(); return (pos<0) ? 0LL : 1LL<<pos; }
ll lsb()const{ return s&-s; }
ll count()const{ return bitset<64>(s).count(); }
ll count(bll msk)const{ return (msk&s).count(); }
ll count(ll l,ll r)const{ return mask(l,r).count(); }
vector<ll> idxes()const{
vector<ll> v;
for(ll i=0,t=s; t; t>>=1,i++) if(t&1)v.push_back(i);
return v;
}
string to_string(ll wd=-1)const{
wd=max({wd,msbit()+1,1LL});
string ret;
for(ll i=wd-1;i>=0;--i) ret += '0'+char((s>>i)&1);
return ret;
}
private:
ll str2val(const string &bitstr){
ll val=0, len=(ll)bitstr.size();
for(ll i=0;i<len;++i) val|=ll(bitstr[i]-'0')<<(len-1-i);
return val;
}
};
template<class T> struct SET: set<T>{
using P=set<T>;
typename P::iterator it=P::end();
template<class...Args> SET(Args...args): P(args...){}
SET(initializer_list<T> a): P(a.begin(),a.end()){}
ll size() const { return (ll)P::size(); }
bool insert(const T &x){ bool r; tie(it,r)=P::insert(x); return r; }
template <class It> void insert(It st,It en){ P::insert(st,en); }
void insert(initializer_list<T> a){ P::insert(a.begin(),a.end()); }
template<class...A> bool emplace(A&&...a){ bool r; tie(it,r)=P::emplace(a...); return r; }
void eraseit(){ it=P::erase(it); }
void find(const T &x){ it=P::find(x); }
bool contains(const T &x){ return P::count(x)==1; }
void lower_bound(const T &x){ it=P::lower_bound(x); }
void upper_bound(const T &x){ it=P::upper_bound(x); }
bool isend() { return it==P::end(); }
T getit() { return *it; }
T next() { return *(++it); }
T prev() { return *(--it); }
bool nextok() { return !isend() && it!=--P::end(); }
bool prevok() { return it!=P::begin(); }
T front() { return *(it=P::begin()); }
T back() { return *(it=--P::end()); }
void pop_front(){ front(); eraseit(); }
void pop_back(){ back(); eraseit(); }
void push_front(const T &x){ it=P::insert(P::begin(),x); }
void push_back (const T &x){ it=P::insert(P::end(),x); }
void push_out(SET &b){ b.push_front(back()); pop_back(); }
void pull_in(SET &b){ push_back(b.front()); b.pop_front(); }
};
template<class T> struct cumulativesum{
using Int = long long;
using ll = long long;
ll n=0; vector<T> c;
cumulativesum():c(1){}
template<class S> cumulativesum(S &&v): n((ll)v.size()),c(n+1) { Ini(v); }
template<class S> void init(S &&v){ n=(ll)v.size(); c.resize(n+1); Ini(v); }
void add(T x) { n++; c.push_back(c.back()+x); }
T operator()(Int l,Int r){ return c[max(min(n,r+1),0LL)]-c[min(max(0LL,l),n)]; }
pair<Int,T> group(T i){
ll g=upper_bound(c.begin(),c.end(),i)-c.begin()-1;
T r = g>=0 ? i-c[g] : i;
return {g,r};
}
T mx(){//区間和max
T mn=T(),samx=0;
for(ll i=1;i<=n;++i){
chmax(samx,c[i]-mn);
chmin(mn,c[i]);
}
return samx;
}
template<class S> void Ini(S &&v) { for(ll i=0;i<n;++i) c[i+1]=c[i]+v[i]; }
};
template<class S> cumulativesum(S) -> cumulativesum<typename remove_reference<S>::type::value_type>;
template<class T> vector<T> powers(T m,ll n){
vector<T> ret(n+1,1);
for(ll i=1;i<=n;++i) ret[i]=ret[i-1]*m;
return ret;
}
template <class T> auto runlength(T &&v){
vector<pair<typename remove_reference<T>::type::value_type,ll>> ret;
for(auto&&e:v){
if(ret.empty() or ret.back().first!=e) ret.emplace_back(e,1);
else ret.back().second++;
}
return ret;
}
inline vector<ll> str2num(string &s,char base,const string &etc){
vector<ll> v(s.size());
for(ll i=0;i<(ll)s.size();++i){
size_t pos=etc.find(s[i]);
if(pos==etc.npos) v[i]=s[i]-(ll)base;
else v[i]=-((ll)pos+1);
}
return v;
}
template<class T> struct combination{
vector<T> f,g; ll mxN=0;
combination(){}
combination(ll maxN): f(maxN+1,1),g(maxN+1),mxN(maxN) {
for (ll i=1;i<=mxN;++i) { f[i]=f[i-1]*i; }
g[mxN]=1/f[mxN];
for (ll i=mxN;i>=1;--i) { g[i-1]=g[i]*i; }
}
T P(ll n,ll r){ return (n<0 || r<0 || n<r) ? T(0) : f[n]*g[n-r]; } //nPr
T H(ll n,ll r){ return operator()(n+r-1,n-1); }//nHr
T inv(ll n) { return f[n-1] * g[n]; } //1/n
T fact(ll n) { return f[n]; } //n!
T finv(ll n) { return g[n]; } //1/n!
T operator()(ll n,ll r){
if (r<0) return 0;
if (n<0) return operator()(-n+r-1,r) * ((r&1)?-1:1); //-nＣr = (-1)^r * n+r-1Ｃr
if (n<r) return 0;
if (n<=mxN) return f[n]*g[n-r]*g[r]; //通常
//n巨大、rかn-r小
if (n-r<r) r=n-r;
T bunsi=1,bunbo=1;
for (ll i=0;i<r;++i) bunsi*=n-i;
for (ll i=0;i<r;++i) bunbo*=i+1;
return bunsi/bunbo;
}
template<class SP>
vector<T> CnLnR(long long nL,long long nR,long long r,SP sp){
if (nR-nL+1<=0) return vector<T>();
if (r<0) return vector<T>(nR-nL+1,0);
vector<T> v=sp(nL-r+1,nR-r+1,r);
for (T& e: v) e*=finv(r);
return v;
}
template<class SP>
vector<T> HrLrR(long long n,long long rL,long long rR,SP sp){//r<0不可
return CnLnR(n-1+rL,n-1+rR,n-1,sp);
}
};
template<class T> struct wrapVector1d{
using S=typename T::value_type;
using Int = long long;
const T *v;
S Ini;
wrapVector1d(const T &v_,S ini_=S()):v(&v_),Ini(ini_){}
S operator[](Int i)const{ return (i<0 || (Int)v->size()<=i) ? Ini : (*v)[i]; }
};
template<class T> struct wrapVector2d{
using S=typename T::value_type;
using Int = long long;
const vector<T> *v;
S Ini;
T dmy;
wrapVector2d(const vector<T> &v_,S ini_=S()):v(&v_),Ini(ini_){}
wrapVector1d<T> operator[](ll i)const{
return (i<0 || (Int)v->size()<=i) ?
wrapVector1d(dmy,Ini) : wrapVector1d((*v)[i],Ini);
}
};
namespace dumpstring{//dummy
inline string stringf(const char *format,...){
char bf[1000];
va_list ap;
va_start(ap,format);
vsprintf(bf,format,ap);
va_end(ap);
return string(bf);
}
template <class T> string stringfx(T x,int wd=1){ return ""; }
struct args{
using Int = long long;
args(){}
args &wd(Int wd__){ (void)wd__; return *this; }
args &sx(Int s){ (void)s; return *this; }
template<size_t DIM> args &rngs(array<array<Int,DIM>,2> rngs){ return *this; }
args &tr(vector<Int> tr__){ (void)tr__; return *this; }
args &tr(){ return *this; }
args &labels(vector<string> labels__){ (void)labels__; return *this; }
args &xrev(){ return *this; }
args &yrev(){ return *this; }
args &zrev(){ return *this; }
args &wrev(){ return *this; }
};
template<class NdT>
void dumpNd(const string &h,const NdT &fd,const args &p=args(),ostream &os=cerr){}
};
using dumpstring::stringf; using dumpstring::stringfx;
using dumpstring::args; using dumpstring::dumpNd;
#endif//テンプレートend
template<class T> struct Vector: vector<T>{
using Int = long long;
using vT=vector<T>;
using cvT=const vector<T>;
using cT=const T;
using vT::vT; //親クラスのコンストラクタの隠蔽を回避
using vT::begin,vT::end,vT::insert,vT::erase;
auto it(Int i){ return begin()+i; }
auto it(Int i)const{ return begin()+i; }
Vector(cvT& b):vT(b){}
Vector(vT&& b):vT(move(b)){}
Vector(int n,cT& x):vT(n,x){}// ┬ 型推論のためラッパー
Vector(long long n,cT& x):vT(n,x){}
template<class S> Vector(const Vector<S>& b):vT(b.begin(),b.end()){}
template<class S> Vector(const vector<S>& b):vT(b.begin(),b.end()){}
Vector(Int n,T s,T d){ iota(n,s,d); }
Vector(Int n,function<T(Int)> g):vT(n){ for(Int i=0;i<n;++i) (*this)[i]=g(i); }
Vector &operator+=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]+=b[i]; return *this; }
Vector &operator-=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]-=b[i]; return *this; }
Vector &operator*=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]*=b[i]; return *this; }
Vector &operator/=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]/=b[i]; return *this; }
Vector &operator%=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]%=b[i]; return *this; }
Vector &operator+=(const Vector<T> &b){ return *this+=(cvT&)b; }
Vector &operator-=(const Vector<T> &b){ return *this-=(cvT&)b; }
Vector &operator*=(const Vector<T> &b){ return *this*=(cvT&)b; }
Vector &operator/=(const Vector<T> &b){ return *this/=(cvT&)b; }
Vector &operator%=(const Vector<T> &b){ return *this%=(cvT&)b; }
Vector operator+(cvT &b){ return Vector(*this)+=b; }
Vector operator-(cvT &b){ return Vector(*this)-=b; }
Vector operator*(cvT &b){ return Vector(*this)*=b; }
Vector operator/(cvT &b){ return Vector(*this)/=b; }
Vector operator%(cvT &b){ return Vector(*this)%=b; }
Vector operator+(const Vector<T> &b){ return Vector(*this)+=b; }
Vector operator-(const Vector<T> &b){ return Vector(*this)-=b; }
Vector operator*(const Vector<T> &b){ return Vector(*this)*=b; }
Vector operator/(const Vector<T> &b){ return Vector(*this)/=b; }
Vector operator%(const Vector<T> &b){ return Vector(*this)%=b; }
template<class S> Vector &operator+=(S x){ for(T &e: *this) e+=x; return *this; }
template<class S> Vector &operator-=(S x){ for(T &e: *this) e-=x; return *this; }
template<class S> Vector &operator*=(S x){ for(T &e: *this) e*=x; return *this; }
template<class S> Vector &operator/=(S x){ for(T &e: *this) e/=x; return *this; }
template<class S> Vector &operator%=(S x){ for(T &e: *this) e%=x; return *this; }
template<class S> Vector operator+(S x)const{ return Vector(*this)+=x; }
template<class S> Vector operator-(S x)const{ return Vector(*this)-=x; }
template<class S> Vector operator*(S x)const{ return Vector(*this)*=x; }
template<class S> Vector operator/(S x)const{ return Vector(*this)/=x; }
template<class S> Vector operator%(S x)const{ return Vector(*this)%=x; }
Vector &operator--(int){ return *this-=1; }
Vector &operator++(int){ return *this+=1; }
Vector operator-()const{ return Vector(*this)*=-1; }
template<class S> friend Vector operator-(S x,const Vector &a){ return -a+=x; }
Vector slice(Int l,Int r,Int d=1)const{
Vector ret;
for(Int i=l;(d>0&&i<=r)||(d<0&&r<=i);i+=d) ret.push_back((*this)[i]);
return ret;
}
Int size()const{ return (Int)vT::size(); }
Vector &push_back(cT& x,Int n=1){ for(Int i=0;i<n;++i){ vT::push_back(x); } return *this; }
Vector &pop_back(Int n=1){ for(Int i=0;i<n;++i){ vT::pop_back(); } return *this; }
Vector &push_front(cT& x,Int n=1){ this->insert(0,x,n); return *this; }
Vector &pop_front(Int n=1){ erase(0,n-1); return *this; }
T pull_back(){ T x=move(vT::back()); vT::pop_back(); return x; }
T pull_front(){ T x=move(vT::front()); erase(0); return x; }
Vector &insert(Int i,cT& x,Int n=1){ insert(it(i),n,x); return *this; }
Vector &insert(Int i,cvT& b){ insert(it(i),b.begin(),b.end()); return *this; }
Vector &erase(Int i){ erase(it(i)); return *this; }
Vector &erase(Int l,Int r){ erase(it(l),it(r+1)); return *this; }
Vector &concat(cvT &b,Int n=1){
cvT B = (&b==this) ? *this : vT{};
for(int i=0;i<n;++i) this->insert(size(),(&b==this)?B:b);
return *this;
}
Vector repeat(Int n){ return Vector{}.concat(*this,n); }
Vector &reverse(Int l=0,Int r=-1){ r+=r<0?size():0; std::reverse(it(l),it(r+1)); return *this; }
Vector &rotate(Int m){ return rotate(0,size()-1,m); }
Vector &rotate(Int l,Int r,Int m){ std::rotate(it(l),it(m),it(r+1)); return *this; }
Vector &sort(Int l=0,Int r=-1){ r+=r<0?size():0; std::sort(it(l),it(r+1)); return *this; }
Vector &rsort(Int l=0,Int r=-1){ return sort(l,r).reverse(l,r); }
template<class Pr> Vector &sort(Pr pr){ return sort(0,size()-1,pr); }
template<class Pr> Vector &sort(Int l,Int r,Pr pr){ std::sort(it(l),it(r+1),pr); return *this; }
template<int key> Vector &sortbykey(Int l=0,Int r=-1){
r+=r<0?size():0;
sort(l,r,[](cT &x,cT &y){return get<key>(x)<get<key>(y);});
return *this;
}
Vector &uniq(){ erase(unique(begin(),end()),end()); return *this; }
Vector &sortq(){ return sort().uniq(); }
Vector &fill(cT& x){ return fill(0,size()-1,x); }
Vector &fill(Int l,Int r,cT& x){ std::fill(it(l),it(r+1),x); return *this; }
template<class S=Int> Vector &iota(Int n,T s=0,S d=1){
vT::resize(n);
if(n==0) return *this;
(*this)[0]=s;
for(int i=1;i<n;++i) (*this)[i]=(*this)[i-1]+d;
return *this;
}
Int count(cT& x)const{ return count(0,size()-1,x); }
Int count(Int l,Int r,cT& x)const{ return Int(std::count(it(l),it(r+1),x)); }
template<class Pr> Int countif(Pr pr)const{ return countif(0,size()-1,pr); }
template<class Pr> Int countif(Int l,Int r,Pr pr)const{ return Int(count_if(it(l),it(r+1),pr)); }
Int find(cT& x)const{ return find(0,size()-1,x); }
Int find(Int l,Int r,cT& x)const{ return Int(std::find(it(l),it(r+1),x)-begin()); }
template<class Pr> Int findif(Pr pr)const{ return findif(0,size()-1,pr); }
template<class Pr> Int findif(Int l,Int r,Pr pr)const{ return Int(find_if(it(l),it(r+1),pr)-begin()); }
Vector<Int> findall(cT& x)const{ return findall(0,size()-1,x); }
Vector<Int> findall(Int l,Int r,cT& x)const{ return findallif(l,r,[&](cT& y){return y==x;}); }
template<class Pr> Vector<Int> findallif(Pr pr)const{ return findallif(0,size()-1,pr); }
template<class Pr> Vector<Int> findallif(Int l,Int r,Pr pr)const{
Vector<Int> ret;
for(Int i=l;i<=r;++i) if(pr((*this)[i])) ret.push_back(i);
return ret;
}
Int flooridx(cT& x)const{ return Int(upper_bound(begin(),end(),x)-begin()-1); }
Int ceilidx(cT& x)const{ return Int(lower_bound(begin(),end(),x)-begin()); }
Int leftnmof(cT& x)const{ return flooridx(x)+1; }
Int rightnmof(cT& x)const{ return size()-ceilidx(x); }
bool contains(cT& x)const{ Int i=flooridx(x); return i>=0 && (*this)[i]==x; }
template<class Pr> Int flooridx(cT& x,Pr pr)const{ return Int(upper_bound(begin(),end(),x,pr)-begin()-1); }
template<class Pr> Int ceilidx(cT& x,Pr pr)const{ return Int(lower_bound(begin(),end(),x,pr)-begin()); }
template<class Pr> Int leftnmof(cT& x,Pr pr)const{ return flooridx(x,pr)+1; }
template<class Pr> Int rightnmof(cT& x,Pr pr)const{ return size()-ceilidx(x,pr); }
template<class Pr> bool contains(cT& x,Pr pr)const{ Int i=flooridx(x,pr); return i>=0 && (*this)[i]==x; }
template<class S> using VV = Vector<Vector<S>>; template<class S> using sVV = vector<vector<S>>;
template<class S> using VVV = Vector<VV<S>>; template<class S> using sVVV = vector<sVV<S>>;
template<class S> using VVVV = Vector<VVV<S>>; template<class S> using sVVVV = vector<sVVV<S>>;
template<class S> using VVVVV = Vector<VVVV<S>>; template<class S> using sVVVVV = vector<sVVVV<S>>;
auto tostd()const{ return tov(*this); }
template <class S> static vector<S> tov(const Vector<S>&v){ return v; }
template <class S> static sVV<S> tov(const VV<S> &v){ sVV<S> ret; for(auto&& e:v) ret.push_back(e); return ret; }
template <class S> static sVVV<S> tov(const VVV<S> &v){ sVVV<S> ret; for(auto&& e:v) ret.push_back(e.tostd()); return ret; }
template <class S> static sVVVV<S> tov(const VVVV<S> &v){ sVVVV<S> ret; for(auto&& e:v) ret.push_back(e.tostd()); return ret; }
template <class S> static sVVVVV<S> tov(const VVVVV<S> &v){ sVVVVV<S> ret; for(auto&& e:v) ret.push_back(e.tostd()); return ret; }
};
/*
vll a={9,8,7},b={1,2,3};
vpll p={{5,3},{7,8},{0,2},};
- -------- 操作系 --------
a+=x a-=x a*=x a/=x a%=x a+x a-x a*x a/x a%x -a x-a a++ a-- //∀i a[i]にxを演算
a+=b a-=b a*=b a/=b a%=b a+b a-b a*b a/b a%b //要素毎演算(同サイズ時)
a.push_front(x,n); //n個先頭追加 省略時1
a.push_back(x,n); //n個末尾追加 省略時1
a.pop_front(n); //n個先頭削除 省略時1
a.pop_back(n); //n個末尾削除 省略時1
ll x=a.pull_front(); //pop_front()と同時に値取得
ll x=a.pull_back(); //pop_back()と同時に値取得
a.insert(i,x,n); //a[i]にn個x挿入 n省略時1
a.insert(i,b); //a[i]にvll b挿入
a.erase(i); //a[i]削除
a.erase(l,r); //区間[l,r]削除
a.concat(b); //aにbを結合 b=a可
a.concat(b,n); //aにbをn回結合 b=a可
a.reverse(l,r); //[l,r]を反転 l,r省略可
a.rotate(m); //a[m]を先頭にするrotate
a.rotate(l,r,m); //a[m]を先頭にするrotate 範囲[l,r]
a.sort(l,r); //[l,r]をソート l,r省略可
a.rsort(l,r); //[l,r]を逆順ソート l,r省略可
p.sort(l,r,[&](pll x,pll y){return x.second<y.second;});//比較関数指定sort l,r省略可
a.uniq(); //連続同値を1つにする
a.sortq(); //ソートしてユニーク
a.fill(l,r,x); //[l,r]にx代入 l,r省略可
a.iota(n,s,d); //aを等差数列にする 長さn,初項s,公差d
vll a(n,s,d); //コンストラクタ版iota
vll b=a.slice(st,en,d); //a[st:en:d] d省略時1
vll b=a.repeat(n); //aをn回繰り返す
- -------- 検索系 --------
auto pr=[&](auto &x){ return x>0; }; //検索条件
ll m=a.count(x); //xの個数
ll m=a.count(l,r,x); //xの個数in[l,r]
ll m=a.countif(pr); //条件満たす個数
ll m=a.countif(l,r,pr); //条件満たす個数in[l,r]
ll i=a.find(x); //xの最左位置i ない時N(配列長)
ll i=a.find(l,r,x); //xの最左位置i in[l,r] ない時r+1
ll i=a.findif(pr); //条件満たす最左位置i ない時N(配列長)
ll i=a.findif(l,r,pr); //条件満たす最左位置i in[l,r] ない時r+1
vll is=a.findall(x); //xの位置i列挙
vll is=a.findall(l,r,x); //xの位置i列挙in[l,r]
vll is=a.findallif(pr); //条件満たす位置i列挙
vll is=a.findallif(l,r,pr); //条件満たす位置i列挙in[l,r]
- -------- 昇順sort済み配列用 --------
ll i=a.flooridx(x); //x以下の最近傍位置i ない時-1
ll i=a.ceilidx(x); //x以上の最近傍位置i ない時N(配列長)
ll m=a.leftnmof(x); //x以下の個数
ll m=a.rightnmof(x); //x以上の個数
bool b=a.contains(x); //xを含む
- -------- 比較関数prでsort済みの配列用 --------
auto pr=[&](auto &x,auto &y){ return x>y; }; //降順ソート時
ll i=a.flooridx(x,pr); //x以左の最近傍位置i ない時-1
ll i=a.ceilidx(x,pr); //x以右の最近傍位置i ない時N(配列長)
ll m=a.leftnmof(x,pr); //x以左の個数
ll m=a.rightnmof(x,pr); //x以右の個数
bool b=a.contains(x,pr); //xを含む
a.concat(b,n).pop_back().rsort().uniq(); //連続適用できる
auto aa=a.tostd(); //N次元VectorをN次元vectorに変換(N≦5)
*/
template<class T> struct wrapv: Vector<T>{
using Int = long long;
T def=T();
T defIF=T();
wrapv(const Vector<T> &b):Vector<T>(b){}
wrapv(Vector<T> &&b):Vector<T>(move(b)){}
wrapv(const std::vector<T> &b):Vector<T>(b){}
wrapv(std::vector<T> &&b):Vector<T>(move(b)){}
T &operator[](Int i){
return (i<0 || this->size()<=i) ? (defIF=def) : Vector<T>::operator[](i);
}
void setdef(const T& x){ def=x; }
};
/*
wrapv v=vll(N,0,1); //vllなどでコンストラクトしてから代入する
v.setdef(INF); //範囲外での値セット
*/
#if 0
#define MODLL (1000000007LL)
#else
#define MODLL (998244353LL)
#endif
using mll = mll_<MODLL>;
//using mll = fraction;
// １
//０┼２
// ３ 左 上 右 下
vector<pll> dxys={{0,-1},{-1,0},{0,1},{1,0},};
namespace SolvingSpace{
template<class T> using vector = Vector<T>;
using vll=vector< ll>; using vmll=vector< mll>; using vdd=vector< dd>;
using vvll=vector< vll>; using vvmll=vector< vmll>; using vvdd=vector< vdd>;
using vvvll=vector< vvll>; using vvvmll=vector< vvmll>; using vvvdd=vector< vvdd>;
using vvvvll=vector<vvvll>; using vvvvmll=vector<vvvmll>; using vvvvdd=vector<vvvdd>;
using vpll=vector< pll>; using vtll=vector< tll>; using vqll=vector< qll>;
using vvpll=vector< vpll>; using vvtll=vector< vtll>; using vvqll=vector< vqll>;
using vll2=vector< ll2>; using vll3=vector< ll3>; using vll4=vector< ll4>;
using vvll2=vector< vll2>; using vvll3=vector< vll3>; using vvll4=vector< vll4>;
using vvvll2=vector<vvll2>; using vvvll3=vector< vvll3>; using vvvll4=vector<vvll4>;
using vss=vector<string>;
template<class T> vector<T> cinv(ll nm){ return vector<T>(nm,[](ll i){ (void)i; return cin1<T>(); }); }
template<class T> vector<vector<T>> cinvv(ll H,ll W){ return vector<vector<T>>(H,[&](ll i){ (void)i; return cinv<T>(W); }); }
template<int DIM> struct rangeNd{
using Int = long long;
using NInt = array<Int,DIM>;
NInt sv,ev,lv,dv; //各変数の初期値,終了値,項数,公差(通常±1)
Int size_=1;
NInt cyclemask;
rangeNd(const NInt &sv_,const NInt &ev_,Int cycle=-1): sv(sv_),ev(ev_){
for(ll i=0;i<DIM;++i) lv[i]=abs(sv[i]-ev[i])+1;
for(ll i=0;i<DIM;++i) dv[i]=sign(ev[i]-sv[i]);
for(ll i=0;i<DIM;++i) size_*=lv[i];
for(ll i=0;i<DIM;++i) cyclemask[i]=~0;
if(cycle!=-1) cyclemask[0]=cycle-1;
}
Int size()const{ return size_; }
Int csize()const{ return size_/lv[0]; }
bool isCycle()const{ return cyclemask[0]!=~0; }
Int cycle()const{ return isCycle() ? cyclemask[0]+1 : -1; }
tuple<Int,Int,Int,Int> SELDx()const{ return SELDof(0); }
tuple<Int,Int,Int,Int> SELDy()const{ return SELDof(1); }
tuple<Int,Int,Int,Int> SELDz()const{ return SELDof(2); }
tuple<Int,Int,Int,Int> SELDw()const{ return SELDof(3); }
array<NInt,2> rngs()const{ return {sv,dv}; }
NInt ls()const{ return lv; }
NInt v2i(const NInt &vx)const{
NInt ret;
for(Int i=0;i<DIM;++i) ret[i]=(vx[i]-sv[i])*dv[i];
return ret;
}
auto toID(){
return [this](const NInt &vx)->Int{
Int id=0;
for(Int i=0;i<DIM;++i){
Int x=vx[i],s=sv[i],l=lv[i],d=dv[i],xx=(x-s)*d;
if(xx<0 || l<=xx) return -1;
id=id*l+(xx&cyclemask[i]);
}
return id;
};
}
auto fromID(){
return [this](Int id)->NInt{
NInt ret;
for(Int i=DIM-1;i>=0;--i){
Int s=sv[i],l=lv[i],d=dv[i],nid=id/l,xx=id-nid*l;
ret[i]=s+xx*d;
id=nid;
}
return ret;
};
}
bool out(const NInt &vx){
for(Int i=0;i<DIM;++i){
Int x=vx[i],s=sv[i],l=lv[i],d=dv[i];
Int xx=(x-s)*d;
if(xx<0 || l<=xx) return true;
}
return false;
}
NInt start(Int xx=0)const{
NInt ret=sv;
ret[0]=sv[0]+dv[0]*xx;
return ret;
}
bool next(NInt &vx,Int dMn=0)const{ //次へ進める 対象:第dMn以降 @return true:次有
for(int d=DIM-1;d>=dMn;--d){
if(vx[d]==ev[d]) vx[d]=sv[d];
else{ vx[d]+=dv[d]; return true; }
}
return false;
}
private:
tuple<Int,Int,Int,Int> SELDof(Int i)const{
if(i>=DIM) return {-1,-1,-1,-1};
return {sv[i],ev[i],lv[i],dv[i]};
}
static Int sign(Int x){ return x>=0 ? 1 : -1; }
};
template<class STATE,class VAL,class Ope,class Tran,class ToID,class FromID>
struct memoDP{
using Int = long long;
using ll = long long;
Ope ope; //貰うときの演算
Tran tran;
ToID toID;
FromID fromID;
VAL yet,ini,dmy; //yet:未計算印の値 ini:ope演算の初期値
vector<VAL> dp_;
bool autoResize = true;
bool nextdp = false; //true:nextDP
//-------- コンストラクタ --------
memoDP(STATE,VAL,Ope ope_,Tran tran_,ToID toID_,FromID fromID_,VAL yet_,VAL ini_)
: ope(ope_),tran(tran_),toID(toID_),fromID(fromID_),yet(yet_),ini(ini_),dmy(ini_){}
//事前メモリ確保用コンストラクタ
template<class RNG> memoDP(STATE,VAL,Ope ope_,Tran tran_,ToID toID_,
FromID fromID_,VAL yet_,VAL ini_,const RNG &rng)
: ope(ope_),tran(tran_),toID(toID_),fromID(fromID_),yet(yet_),ini(ini_),dmy(ini_),
dp_(rng.isCycle() ? rng.csize()*rng.cycle() : rng.size(),yet),
autoResize(false),nextdp(rng.isCycle()){}
//-------- アクセス・計算実行 --------
template<class... Args> VAL &operator()(Args&&... args){ return Dp({args...}); }
template<class... Args> VAL calcIfAny(Args&&... args){ return calcIfAnySt({args...}); }
template<class... Args> void calc(Args&&... args){ calcSt({args...},true); }
template<class RNG> void calcall(const RNG &rng){
if(nextdp) calcallNextDp(rng);
else calcallNormal(rng);
}
using vVAL = vector< VAL>; using vvVAL = vector< vVAL>;
using vvvVAL = vector<vvVAL>; using vvvvVAL = vector<vvvVAL>;
vVAL tov(array<Int,1> ls){ return tov(ls[0]); }
vvVAL tov(array<Int,2> ls){ return tov(ls[0],ls[1]); }
vvvVAL tov(array<Int,3> ls){ return tov(ls[0],ls[1],ls[2]); }
vvvvVAL tov(array<Int,4> ls){ return tov(ls[0],ls[1],ls[2],ls[3]); }
vVAL tov(Int xl){
dp_.resize(xl,yet); return vVAL(dp_);
}
vvVAL tov(Int xl,Int yl){
dp_.resize(xl*yl,yet);
vvVAL ret(xl,vVAL(yl));
for(ll i=0;i<xl;++i) for(ll j=0;j<yl;++j) ret[i][j]=dp_[i*yl+j];
return ret;
}
vvvVAL tov(Int xl,Int yl,Int zl){
dp_.resize(xl*yl*zl,yet);
vvvVAL ret(xl,vvVAL(yl,vVAL(zl)));
for(ll i=0;i<xl;++i) for(ll j=0;j<yl;++j) for(ll k=0;k<zl;++k){
ret[i][j][k]=dp_[i*yl*zl+j*zl+k];
}
return ret;
}
vvvvVAL tov(Int xl,Int yl,Int zl,Int wl){
dp_.resize(xl*yl*zl*wl,yet);
vvvvVAL ret(xl,vvvVAL(yl,vvVAL(zl,vVAL(wl))));
for(ll i=0;i<xl;++i) for(ll j=0;j<yl;++j) for(ll k=0;k<zl;++k){
for(ll l=0;l<wl;++l) ret[i][j][k][l]=dp_[i*yl*zl*wl+j*zl*wl+k*wl+l];
}
return ret;
}
private:
VAL &Dp(const STATE &vx){ //dp_がreallocされると参照が無効になることに注意
ll id=toID(vx);
if(autoResize && (ll)dp_.size()<=id) dp_.resize(id+1,yet);
return id<0 ? (dmy=ini) : dp_[id];
}
VAL calcIfAnySt(const STATE &vx){
VAL &Dpvx=Dp(vx);
if(Dpvx!=yet) return Dpvx;
Dpvx=ini;
auto src_calc2srcVa=[&](const STATE &src,auto calc2srcVa){
VAL srcVa=calcIfAnySt(src);
VAL &DpvxL = autoResize ? Dp(vx) : Dpvx; //resizeすると参照が無効になるため
VAL nVa=ope(DpvxL,calc2srcVa(srcVa));
if(DpvxL!=nVa) DpvxL=nVa,preRenew(vx,src);
};
auto dpWrap=[this](auto &&... args)->VAL{ return calcIfAny(args...); };
auto setVal=[&](VAL va){
(autoResize ? Dp(vx) : Dpvx) = va; //resizeすると参照が無効になるため
};
tran(vx,src_calc2srcVa,dpWrap,setVal);
return autoResize ? Dp(vx) : Dpvx; //resizeすると参照が無効になるため
}
void calcSt(const STATE &vx,bool force=false){ //force=true:計算済でも再計算
VAL &Dpvx=Dp(vx);
if(!force && Dpvx!=yet) return;
Dpvx=ini;
auto src_calc2srcVa=[&](const STATE &src,auto calc2srcVa){
VAL srcVa=Dp(src);
VAL nVa=ope(Dpvx,calc2srcVa(srcVa));
if(Dpvx!=nVa) Dpvx=nVa,preRenew(vx,src);
};
auto dpWrap=[this](auto &&... args)->VAL{ return (*this)(args...); };
auto setVal=[&](VAL va){ Dpvx=va; };
tran(vx,src_calc2srcVa,dpWrap,setVal);
}
template<class RNG> void calcallNormal(const RNG &rng){
STATE vx=rng.start();
do calcSt(vx); while(rng.next(vx));
}
template<class RNG> void calcallNextDp(const RNG &rng){
assert(rng.isCycle());
for(ll xx=0;xx<rng.ls()[0];++xx){
STATE vx=rng.start(xx);
bool force = rng.cycle()<=xx;
do calcSt(vx,force); while(rng.next(vx,1));
}
}
//-------- ダンプ --------
public:
#if 1
template<int N>
void dump(rangeNd<N> &rng,vector<string> labels={},const vector<ll> axes={0,1,2,3}){
auto ls=rng.ls();
if(nextdp) ls[0]=rng.cycle();
dumpstring::dumpNd("",tov(ls).tostd(),args().rngs(rng.rngs()).tr(axes).labels(labels));
}
#endif
//-------- 最適パス --------
#if 0
private:
vector<ll> pre;
ll preini=-1,predmy=-1;
ll &Pre(const STATE &vx){
ll id=toID(vx);
if((ll)pre.size()<=id) pre.resize(id+1,preini);
return id<0 ? (predmy=preini) : pre[id];
}
void preRenew(const STATE &vx,const STATE &src){ Pre(vx)=toID(src); }
public:
template<class... Args> vector<STATE> pathget(Args&&... endargs){
vector<ll> ids{toID({endargs...})};
while(ids.back()<(ll)pre.size() && pre[ids.back()]!=preini) ids.push_back(pre[ids.back()]);
reverse(ids.begin(),ids.end());
vector<STATE> path;
for(auto&& id: ids) path.push_back(fromID(id));
return path;
}
#else
void preRenew(const STATE &vx,const STATE &src){ (void)vx,(void)src; }
#endif
};
struct unionfind{
vll par,rank,size;
ll ccNum;
unionfind(ll n): par(n),rank(n),size(n,1),ccNum(n){
for(ll i=0;i<n;++i) par[i]=i;
}
bool issame(ll u,ll v) { return root(u) == root(v); }
ll root(ll v) { return (par[v] == v) ? v : par[v]=root(par[v]); }
ll ccsize(ll v){ return size[root(v)]; }//頂点vを含む連結成分のサイズ
ll ccnum() { return ccNum; } //連結成分数
void merge(ll u,ll v){
u = root(u);
v = root(v);
if(u == v) return;
if(rank[u] < rank[v]) par[u] = v;
else par[v] = u;
if(rank[u] == rank[v]) rank[u]++;
size[u] = size[v] = size[u] + size[v];
ccNum--;
}
};
/*
unionfind uft(n);
uft.merge(u,v);
bool b = uft.issame(u,v);
ll rt = uft.root(v);
ll ccsz= uft.ccsize(v); //頂点vを含む連結成分のサイズ
ll ccnm= uft.ccnum(); //連結成分数
*/
namespace baseNspace{
using Int=long long;
using ll=long long;
vector<Int> to_vector(ll x,bool rev=false,Int d=10){
vector<Int> dgs;
for(; x>0; x/=d) dgs.push_back(Int(x%d));
if(!rev) reverse(dgs.begin(),dgs.end());
return dgs;
}
ll vtoll(const vector<Int> &v,Int d=10){
ll x=0;
for(Int dg:v) x=x*d+dg;
return x;
};
string v_to_string(const vector<Int> &v){
if(v.empty()) return "0";
string s;
for(auto&& e: v) s += e<10 ? char(e+'0') : char(e-10+'A');
return s;
}
vector<Int> s_to_vector(const string &s){
vector<Int> dgs;
if(s=="0") return dgs;
for(auto&& c: s) dgs.push_back(c<='9' ? Int(c-'0') : Int(c-'A')+10);
return dgs;
}
string to_string(ll x,bool rev,Int d){ //overload
return v_to_string(to_vector(x,rev,d));
}
template<ll d> struct basen{
ll x=0;
basen(){}
basen(ll x): x(x){}
basen(int x): x(x){}
basen(const char *s){ fromString(s); }
basen(const string &s){ fromString(s); }
Int operator[](Int i)const{
ll y = valid(i+1) ? x%bases[i+1] : x;
return valid(i) ? Int(y/bases[i]) : Int(0);
}
operator Int()const{ return Int(x); }
basen operator++(int){ basen b(*this); x++; return b; }
basen operator--(int){ basen b(*this); x--; return b; }
basen &operator+=(Int b){ x+=b; return *this; }
basen &operator-=(Int b){ x-=b; return *this; }
basen operator+(ll b)const{ return x+b; }
basen operator-(ll b)const{ return x-b; }
basen operator+(int b)const{ return x+b; }
basen operator-(int b)const{ return x-b; }
basen operator+(const basen &b)const{ return x+b.x; }
basen operator-(const basen &b)const{ return x-b.x; }
basen &seteq(Int i,Int dg){ assert(valid(i)); return addeq(i,dg-(*this)[i]); }
basen &addeq(Int i,Int dg){ assert(valid(i)); x+=dg*bases[i]; return *this; }
[[nodiscard]] basen set(Int i,Int dg)const{ return basen(*this).seteq(i,dg); }
[[nodiscard]] basen add(Int i,Int dg)const{ return basen(*this).addeq(i,dg); }
static void setbase(const vector<Int> &v){
for(auto&& x: v) bases.push_back(bases.back()*x);
}
Int ndigits()const{//桁数
for(Int ng=-1,ok=(Int)bases.size();;){
Int m=(ng+ok)/2;
if(bases[m]>x) ok=m; else ng=m;
if(ok-ng==1) return ok;
}
}
string to_string(Int wd=-1)const{
wd=max({wd,ndigits(),1LL});
string ret;
for(ll i=wd-1;i>=0;--i) ret += '0'+char((*this)[i]);
return ret;
}
static vector<ll> setbaseD(){
vector<ll> ret{1};
if(d>=2) while(ret.back()<=LLONG_MAX/d) ret.push_back(ret.back()*d);
return ret;
}
bool valid(Int i)const{ return i<(Int)bases.size(); }
void fromString(const string &s){
ll N=(ll)s.size();
for(ll i=0;i<N;++i) addeq(i,s[N-1-i]-'0');
}
static inline vector<ll> bases = setbaseD();
};
}
using baseNspace::to_vector; using baseNspace::v_to_string; using baseNspace::vtoll;
using baseNspace::to_string; using baseNspace::s_to_vector;
using std::to_string;
using baseNspace::basen;
/*
- ---------------- ll・vll・string d進法変換ライブラリ ----------------
ll:常に10進、string,vll:d進法 として変換
ll 3041 ⇔ vll {3,0,4,1} ⇔ string "3041" (10進法)
ll 0 ⇔ vll {} ⇔ string "0" (10進法)
ll 12 ⇔ vll {1,1,0,0} ⇔ string "1100" (2進法)
ll 46 ⇔ vll {2,14} ⇔ string "2E" (16進法)
- -------- ll→vll to_vector --------
vll dgs=to_vector(x);
vll dgs=to_vector(x,true);
. reverse有↑
vll dgs=to_vector(x,false,d);
. ↑d進法 省略時10
- -------- vll→ll vtoll --------
ll x=vtoll(dgs);
ll x=vtoll(dgs,d);
. ↑d進法 省略時10
- -------- string→vll s_to_vector --------
vll dgs=s_to_vector(s);
- -------- vll→string v_to_string --------
string s=v_to_string(dgs);
- -------- ll→string -------- std::to_stringのoverload
string s=to_string(x);
. ↑引数1つ時はstd::to_stringが呼ばれる
string s=to_string(x,false,d);
. ↑d進法
string s=to_string(x,true,d);
. reverse有↑
- -------- string→ll -------- std::stoll使う
ll x=stoll(s);
ll x=stoll(s,nullptr,d);
. ↑d進法
- ---------------- llのままd進法各桁を操作するライブラリ ----------------
- -------- 定義 --------
using base=basen<d>; //d進法のとき
using base=basen<-1>; //┬各桁別進法のとき
base::setbase(v); //┘ 第i桁の範囲が[0,v[i])になる
base b=x; //整数から
base b="101"; //文字列から
- -------- 操作 --------
ll dg=base(x)[i]; //xの第i桁
base b=x;
ll n=b.ndigits(); //桁数 201(3)なら3
b.seteq(i,dg); //第i桁dgに変更
b.addeq(i,dg); //第i桁にdg加算
base bb=b.set(i,dg); //第i桁dgに変更したもの返す
base bb=b.add(i,dg); //第i桁にdg加算したもの返す
ll y=base(x).set(i,dg); //llのまま扱うことも可能(キャストで実現)
b+=x b-=x b++ b--
b+x b-x //base型を返す
(b==x b!=x) //llへのcastを経由し可能
string s=b.to_string(); //文字列出力
string s=b.to_string(6); //文字列出力 桁数指定(0埋め)
*/
void cin2solve()
{
auto [N,M]=cins<ll,ll>();
auto uvc=cinv<tuple<ll,ll,ll>>(M);
for(auto&&[u,v,c]:uvc) u--,v--,c--;
uvc.sortbykey<2>();
ll L=6;
vll cnt(L);
unionfind uft(N);
for(auto&&[u,v,c]:uvc){
if(!uft.issame(u,v)){
cnt[c]++;
uft.merge(u,v);
}
}
using STATE=array<ll,1>;
using VAL=dd; VAL yet=(dd)-INF*7;
auto operation=[&](VAL a,VAL b){return a+b;}; VAL ini=0;
using base=basen<-1>; //┬各桁別進法のとき
vll baseini=cnt+1;
base::setbase(baseini); //┘ 第i桁の範囲が[0,v[i])になる
auto transition=[&](const STATE &vx,auto src_calc2srcVa,auto dp,auto setVal){
auto[s]=vx;
//dp(現在位置)をdp(…)から計算
VAL va=0;
if (s==0){ //初期化時
va=0.;
}
else{ //通常時
ll lz=0;
rep(i,0,L-1){
if(base(s)[i]==0) lz++;
else break;
}
assert(lz<L);
va+=dd(L)/(L-lz);
ll curde=lz;
rep(de,lz,L-1){
if(base(s)[de]>0)curde=de;
va+=dd(1)/(L-lz)*dp(base(s).add(curde,-1));
}
}
//結果セット
setVal(va);
};
auto [s1,e1]=pll{0,base::bases.back()-1}; //1つ目の変数の範囲
rangeNd<1> rng({s1,},{e1,});
memoDP dp(STATE(),VAL(),operation,transition,rng.toID(),rng.fromID(),yet,ini,rng);
//VAL ans=dp.calcIfAny(e1);
dp.calcall(rng);
VAL ans=dp(e1);
cout << ans << '\n';
return;
}
}//SorvingSpace
//////////////////////////////////////////
int main(){
#if 1
//SolvingSpace::labo();
SolvingSpace::cin2solve();
//SolvingSpace::generand();
#else
ll t; cin >> t;
rep(i,0,t-1){
SolvingSpace::cin2solve();
//SolvingSpace::generand();
}
#endif
cerr << timeget() <<"ms"<< '\n';
return 0;
}