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#ifndef Wh_XcjmHeader_H
#define Wh_XcjmHeader_H

//#define Memory_Unlimited_Mode_On true
//#define Init_Factorial_List_On true
//#define Init_Prime_List_On true
//#define WindowsUI_Mode_On true
//#define Urgent_Mode_On true

#include<stdio.h>
#include<algorithm>
#include<string.h>
#include<math.h>

typedef long long ll;
typedef unsigned long long ull;

#ifdef Memory_Unlimited_Mode_On
//Caution: You should use "signed main(){}" with this mode on
typedef long long int;
#endif

#ifndef mo
#define mo 1000000007
#endif

#if defined(Urgent_Mode_On)&&Urgent_Mode_On
	#define qr fastIO::read()
	#define gc fastIO::nc()
	#define pc(a) fastIO::Ostream.out(a)
	#define ps pc(' ')
	#define pe pc('
')
	#define pt(a) fastIO::print(a)
	#define pd(a) fastIO::print(a)
#else
	#define qr read()
	#define gc getchar()
	#define pc(a) putchar(a)
	#define ps putchar(' ')
	#define pe putchar('
')
	#define pt(a) print(a)
	#define pd(a) printf("%lf",a)
#endif

#ifdef _STL_PAIR_H
typedef std::pair<int,int> pii;
typedef std::pair<ll,ll> pll;
#define mk(a,b) std::make_pair(a,b)
#endif

#if defined(WindowsUI_Mode_On)&&WindowsUI_Mode_On
#include<windows.h>
#define sys(opt) system(opt)
void gotoxy(int x,int y){
    COORD pos = {x,y};
    HANDLE hOut = GetStdHandle(STD_OUTPUT_HANDLE);// 获取标准输出设备句柄
    SetConsoleCursorPosition(hOut, pos);//两个参数分别是指定哪个窗体，具体位置
}
void SetConsoleCursor(bool opt=false){
	HANDLE handle = GetStdHandle(STD_OUTPUT_HANDLE);
	CONSOLE_CURSOR_INFO CursorInfo;
	GetConsoleCursorInfo(handle, &CursorInfo);//获取控制台光标信息
	CursorInfo.bVisible = opt; //隐藏控制台光标
	SetConsoleCursorInfo(handle, &CursorInfo);//设置控制台光标状态
}
#endif

#ifndef C
#define C(b,a) mymath::Combination(mymath::min((a),(b)-(a)),(b))//choose a elements from b equivalent elements
#endif
#ifndef P
#define P(b,a) mymath::Permutation((a),(b))//choose a elements from b inequivalent elements
#endif

#if defined(Init_Factorial_List_On)&&Init_Factorial_List_On
#ifndef Factorial_List_Limit
#define Factorial_List_Limit 1000005
#endif
ll Factorial_List[Factorial_List_Limit];
ll Inv_Factorial_List[Factorial_List_Limit];
bool Has_Calculated_the_Factorial_List=false;
void Init_Factorial_List();
#endif

namespace mymath{//一些方便的数学运算
	inline void Add(ll &a,ll b){
		a=a+b;a%=mo;
	}
	inline ll add(ll a,ll b){
		a=a+b;a%=mo;
        return a;
	}
	inline void Add(ll &a,ll b,ll p){
		a=a+b;a%=p;
	}
	inline ll add(ll a,ll b,ll p){
		a=a+b;a%=p;
        return a;
	}
	template<typename _Tp>_Tp abs(_Tp x){
		return x<0?-x:x;
	}
	inline void Abs(double &x){x<0?x=-x:0;}
	inline void Abs(ll &x){x<0?x=-x:0;}
	inline void Mul(ll &a,ll b,ll p){
		if(p<=3e9)
			a=a*b,a%=p;
		else{
			#if defined(Urgent_Mode_On)&&Urgent_Mode_On
			a=(a*b-(ll)((long double)a/p*b+1.0e-8)*p+p)%p;  
			#else
			ll ret=0,k=a;
			while(b){
				if(b&1)Add(ret,k,p);
				b>>=1;
				Add(k,k,p);
			}
			a=ret;
			#endif
		}
	}
	inline void Mul(ll &a,ll b){
		Mul(a,b,mo);
	}
	inline ll mul(ll a,ll b,ll p){
		if(p<=3e9)
			a=a*b,a%=p;
		else{
			#if defined(Urgent_Mode_On)&&Urgent_Mode_On
			a=(a*b-(ll)((long double)a/p*b+1.0e-8)*p+p)%p;  
			#else
			ll ret=0,k=a;
			while(b){
				if(b&1)Add(ret,k,p);
				b>>=1;
				Add(k,k,p);
			}
			a=ret;
			#endif
		}
        return a;
	}
	inline ll mul(ll a,ll b){
		return mul(a,b,mo);
	}
	inline void Min(ll &a,ll b){
		a=a<b?a:b;
	}
	inline void Max(ll &a,ll b){
		a=a>b?a:b;
	}
    template<typename _Tp> inline _Tp min(_Tp a1,_Tp a2){
        return a1<a2?a1:a2;
    }
	inline ll min(ll a1,ll a2){return a1<a2?a1:a2;}
	inline double min(double a1,double a2){return a1<a2?a1:a2;}
	inline ll min(ll a1,ll a2,ll a3){
		return min(a1,min(a2,a3));
	}
	inline ll min(ll a1,ll a2,ll a3,ll a4){
		return min(a1,min(a2,a3,a4));
	}
	inline ll min(ll a1,ll a2,ll a3,ll a4,ll a5){
		return min(a1,min(a2,a3,a4,a5));
	}
	inline ll min(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6){
		return min(a1,min(a2,a3,a4,a5,a6));
	}
	inline ll min(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7){
		return min(a1,min(a2,a3,a4,a5,a6,a7));
	}
	inline ll min(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7,ll a8){
		return min(a1,min(a2,a3,a4,a5,a6,a7,a8));
	}
	inline ll min(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7,ll a8,ll a9){
		return min(a1,min(a2,a3,a4,a5,a6,a7,a8,a9));
	}
	inline ll min(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7,ll a8,ll a9,ll a10){
		return min(a1,min(a2,a3,a4,a5,a6,a7,a8,a9,a10));
	}
	inline ll max(ll a1,ll a2){return a1>a2?a1:a2;}
	inline double max(double a1,double a2){return a1>a2?a1:a2;}
	inline ll max(ll a1,ll a2,ll a3){
		return max(a1,max(a2,a3));
	}
	inline ll max(ll a1,ll a2,ll a3,ll a4){
		return max(a1,max(a2,a3,a4));
	}
	inline ll max(ll a1,ll a2,ll a3,ll a4,ll a5){
		return max(a1,max(a2,a3,a4,a5));
	}
	inline ll max(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6){
		return max(a1,max(a2,a3,a4,a5,a6));
	}
	inline ll max(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7){
		return max(a1,max(a2,a3,a4,a5,a6,a7));
	}
	inline ll max(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7,ll a8){
		return max(a1,max(a2,a3,a4,a5,a6,a7,a8));
	}
	inline ll max(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7,ll a8,ll a9){
		return max(a1,max(a2,a3,a4,a5,a6,a7,a8,a9));
	}
	inline ll max(ll a1,ll a2,ll a3,ll a4,ll a5,ll a6,ll a7,ll a8,ll a9,ll a10){
		return max(a1,max(a2,a3,a4,a5,a6,a7,a8,a9,a10));
	}
	ll ksm(ll a,ll k){
		ll ret=1;
		while(k){
			if(k&1)Mul(ret,a);
			Mul(a,a);
			k>>=1;
		}
		return ret;
	}
	ll ksm(ll a,ll k,ll p){
		ll ret=1;
		while(k){
			if(k&1)Mul(ret,a,p);
			Mul(a,a,p);
			k>>=1;
		}
		return ret;
	}
	ll CalcInv(ll x){return ksm(x,mo-2);}
	ll Combination(ll a,ll b){//组合 b选a
		if(a>b||a<0||b<0)return -1;
		ll ret=1;
        #if defined(Init_Factorial_List_On)&&Init_Factorial_List_On
            if(!Has_Calculated_the_Factorial_List)Init_Factorial_List();
            Mul(ret,Factorial_List[b]);
            Mul(ret,Inv_Factorial_List[b-a]);
            Mul(ret,Inv_Factorial_List[a]);
            return ret;
        #endif
		for(ll k=b;b-k<a;k--){
			mymath::Mul(ret,k);
		}
		for(int i=2;i<=a;i++)
			mymath::Mul(ret,CalcInv(i));
		return ret;
	}
	ll Permutation(ll a,ll b){//排列 b选a
		if(a>b||a<0||b<0)return -1;
		ll ret=1;
        #if defined(Init_Factorial_List_On)&&Init_Factorial_List_On
            if(~Has_Calculated_the_Factorial_List)Init_Factorial_List();
            Mul(ret,Factorial_List[b]);
            Mul(ret,CalcInv(Factorial_List[b-a]));
            return ret;
        #endif
		for(ll k=b;b-k<a;k--){
			mymath::Mul(ret,k);
		}
		return ret;
	}
}

#if !(defined(Urgent_Mode_On)&&Urgent_Mode_On)
inline ll read(){
	ull x = 0;
	char y = 0,c = gc;
	while(c<'0'||c>'9')y=c,c=gc;
	while(c>='0'&&c<='9')x=x*10-48+c,c=gc;
	return y=='-'?0ll-x:(ll)x;
}
inline void print(ll x){
	ull y;
	if(x<0)pc('-'),y=0ull-x;else y=x;
	if(y>=10)print(y/10);
	pc(y%10+48);
}
/*
inline void print(int x){
	ull y;
	if(x<0)pc('-'),y=0ull-x;else y=x;
	if(y>=10)print(y/10);
	pc(y%10+48);
}
inline void print(ull x){
	ull y;
	if(x<0)pc('-'),y=0ull-x;else y=x;
	if(y>=10)print(y/10);
	pc(y%10+48);
}*/
#endif

#ifndef Prime_Check_List_Cnt
#define Prime_Check_List_Cnt 12
ll Miller_Rabin_Check_List[]={2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};
#endif
bool Miller_Rabin_Check_Prime(ll x){
	if(x<2)return false;
	if(x==2)return true;
	if(x&1==0)return false;
	for(int i=0;i<Prime_Check_List_Cnt&&Miller_Rabin_Check_List[i]<x;i++){
		ll k=x-1ll<<1;
		while((k&1)==0){
			k>>=1;
			ll re=mymath::ksm(Miller_Rabin_Check_List[i],k,x);
			if(re==x-1)break;
			if(re==1)continue;
			return false;
		}
	}
	return true;
}
#endif

#if defined(Init_Prime_List_On)&&Init_Prime_List_On
#define Init_Prime_List_On true
#ifndef Prime_List_Limit
#define Prime_List_Limit 1000005
#endif
int Prime_List[Prime_List_Limit],Cnt_Prime_List;
bool is_Prime[Prime_List_Limit];
void Init_Prime_List(){
	int N=Prime_List_Limit-4,&cnt=Cnt_Prime_List=0;
	for(int i=2;i<Prime_List_Limit;i++)is_Prime[i]=true;
	for(int i=2;i<N;i++){
		if(is_Prime[i])Prime_List[cnt++]=i;
		for(int j=0;j<cnt&&i*Prime_List[j]<N;j++){
			is_Prime[i*Prime_List[j]]=false;
			if(i%Prime_List[j]==0)break;
		}
	}
}
#endif

#if defined(Init_Factorial_List_On)&&Init_Factorial_List_On
void Init_Factorial_List(){
    Has_Calculated_the_Factorial_List=true;
    Factorial_List[0]=Factorial_List[1]=1;
    for(ll i=2;i<Factorial_List_Limit;i++)
        Factorial_List[i]=mymath::mul(Factorial_List[i-1],i);
    Inv_Factorial_List[Factorial_List_Limit-1]=mymath::ksm(Factorial_List[Factorial_List_Limit-1],mo-2);
    for(ll i=Factorial_List_Limit-2;i>=0;i--)
        Inv_Factorial_List[i]=mymath::mul(Inv_Factorial_List[i+1],i+1);
}
#endif

//转的，有空研读下↓
namespace fastIO{
    #define BUF_SIZE 100000
    #define OUT_SIZE 100000
    //fread->read
    bool IOerror=0;
    inline char nc(){
        static char buf[BUF_SIZE],*p1=buf+BUF_SIZE,*pend=buf+BUF_SIZE;
        if (p1==pend){
            p1=buf; pend=buf+fread(buf,1,BUF_SIZE,stdin);
            if (pend==p1){IOerror=1;return -1;}
            //{printf("IO error!
");system("pause");for (;;);exit(0);}
        }
        return *p1++;
    }
    inline bool blank(char ch){return ch==' '||ch=='
'||ch=='
'||ch=='	';}
    inline void read(int &x){
        bool sign=0; char ch=nc(); x=0;
        for (;blank(ch);ch=nc());
        if (IOerror)return;
        if (ch=='-')sign=1,ch=nc();
        for (;ch>='0'&&ch<='9';ch=nc())x=x*10+ch-'0';
        if (sign)x=-x;
    }
    inline ll read(){
		ll x=0; bool sign=0; char ch=nc();
        for (;blank(ch);ch=nc());
        if (IOerror)return -1;
        if (ch=='-')sign=1,ch=nc();
        for (;ch>='0'&&ch<='9';ch=nc())x=x*10+ch-'0';
        if (sign)x=-x;
		return x;
    }
    inline void read(ll &x){
        bool sign=0; char ch=nc(); x=0;
        for (;blank(ch);ch=nc());
        if (IOerror)return;
        if (ch=='-')sign=1,ch=nc();
        for (;ch>='0'&&ch<='9';ch=nc())x=x*10+ch-'0';
        if (sign)x=-x;
    }
    inline void read(double &x){
        bool sign=0; char ch=nc(); x=0;
        for (;blank(ch);ch=nc());
        if (IOerror)return;
        if (ch=='-')sign=1,ch=nc();
        for (;ch>='0'&&ch<='9';ch=nc())x=x*10+ch-'0';
        if (ch=='.'){
            double tmp=1; ch=nc();
            for (;ch>='0'&&ch<='9';ch=nc())tmp/=10.0,x+=tmp*(ch-'0');
        }
        if (sign)x=-x;
    }
    inline void read(char *s){
        char ch=nc();
        for (;blank(ch);ch=nc());
        if (IOerror)return;
        for (;!blank(ch)&&!IOerror;ch=nc())*s++=ch;
        *s=0;
    }
    inline void read(char &c){
        for (c=nc();blank(c);c=nc());
        if (IOerror){c=-1;return;}
    }
    //getchar->read
    inline void read1(int &x){
        char ch;int bo=0;x=0;
        for (ch=getchar();ch<'0'||ch>'9';ch=getchar())if (ch=='-')bo=1;
        for (;ch>='0'&&ch<='9';x=x*10+ch-'0',ch=getchar());
        if (bo)x=-x;
    }
    inline void read1(ll &x){
        char ch;int bo=0;x=0;
        for (ch=getchar();ch<'0'||ch>'9';ch=getchar())if (ch=='-')bo=1;
        for (;ch>='0'&&ch<='9';x=x*10+ch-'0',ch=getchar());
        if (bo)x=-x;
    }
    inline void read1(double &x){
        char ch;int bo=0;x=0;
        for (ch=getchar();ch<'0'||ch>'9';ch=getchar())if (ch=='-')bo=1;
        for (;ch>='0'&&ch<='9';x=x*10+ch-'0',ch=getchar());
        if (ch=='.'){
            double tmp=1;
            for (ch=getchar();ch>='0'&&ch<='9';tmp/=10.0,x+=tmp*(ch-'0'),ch=getchar());
        }
        if (bo)x=-x;
    }
    inline void read1(char *s){
        char ch=getchar();
        for (;blank(ch);ch=getchar());
        for (;!blank(ch);ch=getchar())*s++=ch;
        *s=0;
    }
    inline void read1(char &c){for (c=getchar();blank(c);c=getchar());}
    //scanf->read
    inline void read2(int &x){scanf("%d",&x);}
    inline void read2(ll &x){
        #ifdef _WIN32
            scanf("%I64d",&x);
        #else
        #ifdef __linux
            scanf("%lld",&x);
        #else
            puts("error:can't recognize the system!");
        #endif
        #endif
    }
    inline void read2(double &x){scanf("%lf",&x);}
    inline void read2(char *s){scanf("%s",s);}
    inline void read2(char &c){scanf(" %c",&c);}
    inline void readln2(char *s){gets(s);}
    //fwrite->write
    struct Ostream_fwrite{
        char *buf,*p1,*pend;
        Ostream_fwrite(){buf=new char[BUF_SIZE];p1=buf;pend=buf+BUF_SIZE;}
        void out(char ch){
            if (p1==pend){
                fwrite(buf,1,BUF_SIZE,stdout);p1=buf;
            }
            *p1++=ch;
        }
        void print(int x){
            static char s[15],*s1;s1=s;
            if (!x)*s1++='0';if (x<0)out('-'),x=-x;
            while(x)*s1++=x%10+'0',x/=10;
            while(s1--!=s)out(*s1);
        }
        void println(int x){
            static char s[15],*s1;s1=s;
            if (!x)*s1++='0';if (x<0)out('-'),x=-x;
            while(x)*s1++=x%10+'0',x/=10;
            while(s1--!=s)out(*s1); out('
');
        }
        void print(ll x){
            static char s[25],*s1;s1=s;
            if (!x)*s1++='0';if (x<0)out('-'),x=-x;
            while(x)*s1++=x%10+'0',x/=10;
            while(s1--!=s)out(*s1);
        }
        void println(ll x){
            static char s[25],*s1;s1=s;
            if (!x)*s1++='0';if (x<0)out('-'),x=-x;
            while(x)*s1++=x%10+'0',x/=10;
            while(s1--!=s)out(*s1); out('
');
        }
        void print(double x,int y){
            static ll mul[]={1,10,100,1000,10000,100000,1000000,10000000,100000000,
                1000000000,10000000000LL,100000000000LL,1000000000000LL,10000000000000LL,
                100000000000000LL,1000000000000000LL,10000000000000000LL,100000000000000000LL};
            if (x<-1e-12)out('-'),x=-x;x*=mul[y];
            ll x1=(ll)floor(x); if (x-floor(x)>=0.5)++x1;
            ll x2=x1/mul[y],x3=x1-x2*mul[y]; print(x2);
            if (y>0){out('.'); for (size_t i=1;i<y&&x3*mul[i]<mul[y];out('0'),++i); print(x3);}
        }
        void println(double x,int y){print(x,y);out('
');}
        void print(char *s){while (*s)out(*s++);}
        void println(char *s){while (*s)out(*s++);out('
');}
        void flush(){if (p1!=buf){fwrite(buf,1,p1-buf,stdout);p1=buf;}}
        ~Ostream_fwrite(){flush();}
    }Ostream;
    inline void print(int x){Ostream.print(x);}
    inline void println(int x){Ostream.println(x);}
    inline void print(char x){Ostream.out(x);}
    inline void println(char x){Ostream.out(x);Ostream.out('
');}
    inline void print(ll x){Ostream.print(x);}
    inline void println(ll x){Ostream.println(x);}
    inline void print(double x,int y){Ostream.print(x,y);}
    inline void println(double x,int y){Ostream.println(x,y);}
    inline void print(char *s){Ostream.print(s);}
    inline void println(char *s){Ostream.println(s);}
    inline void println(){Ostream.out('
');}
    inline void flush(){Ostream.flush();}
    //puts->write
    char Out[OUT_SIZE],*o=Out;
    inline void print1(int x){
        static char buf[15];
        char *p1=buf;if (!x)*p1++='0';if (x<0)*o++='-',x=-x;
        while(x)*p1++=x%10+'0',x/=10;
        while(p1--!=buf)*o++=*p1;
    }
    inline void println1(int x){print1(x);*o++='
';}
    inline void print1(ll x){
        static char buf[25];
        char *p1=buf;if (!x)*p1++='0';if (x<0)*o++='-',x=-x;
        while(x)*p1++=x%10+'0',x/=10;
        while(p1--!=buf)*o++=*p1;
    }
    inline void println1(ll x){print1(x);*o++='
';}
    inline void print1(char c){*o++=c;}
    inline void println1(char c){*o++=c;*o++='
';}
    inline void print1(char *s){while (*s)*o++=*s++;}
    inline void println1(char *s){print1(s);*o++='
';}
    inline void println1(){*o++='
';}
    inline void flush1(){if (o!=Out){if (*(o-1)=='
')*--o=0;puts(Out);}}
    struct puts_write{
        ~puts_write(){flush1();}
    }_puts;
    inline void print2(int x){printf("%d",x);}
    inline void println2(int x){printf("%d
",x);}
    inline void print2(char x){printf("%c",x);}
    inline void println2(char x){printf("%c
",x);}
    inline void print2(ll x){
        #ifdef _WIN32
            printf("%I64d",x);
        #else
        #ifdef __linux
            printf("%lld",x);
        #else
            puts("error:can't recognize the system!");
        #endif
        #endif
    }
    inline void println2(ll x){print2(x);printf("
");}
    inline void println2(){printf("
");}
    #undef ll
    #undef OUT_SIZE
    #undef BUF_SIZE
};
namespace FileIO{
    void fileio(char s[]){
        char Input[500],Output[500];
        sprintf(Input,"%s.in",s);
        sprintf(Output,"%s.out",s);
        freopen(Input,"r",stdin);
        freopen(Output,"w",stdout);
    }
    #ifdef FileName
    void fileio(){
        char s[]=FileName;
        char Input[500],Output[500];
        sprintf(Input,"%s.in",s);
        sprintf(Output,"%s.out",s);
        freopen(Input,"r",stdin);
        freopen(Output,"w",stdout);
    }
    #endif
}
#if defined N
#ifndef M
#define M N*2
#endif
struct A_List{
	int top[N],nxt[M],to[M],cnt;
	#if defined(Memory_Unlimited_Mode_On)&&Memory_Unlimited_Mode_On
	ll val[M];
	#else
    int val[M];
    #endif
	void add2(int x,int y,int v=1){
		nxt[++cnt]=top[x];top[x]=cnt;to[cnt]=y;val[cnt]=v;
		nxt[++cnt]=top[y];top[y]=cnt;to[cnt]=x;val[cnt]=v;
	}
	void add1(int x,int y,int v=1){
		nxt[++cnt]=top[x];top[x]=cnt;to[cnt]=y;val[cnt]=v;
	}
    #ifdef _STL_PAIR_H
    void add2(pii a,int v=1){
        add2(a.first,a.second,v);
    }
    #endif
};
struct A_List_No_Edge_Weight{
	int top[N],nxt[M],to[M],cnt;
	void add2(int x,int y){
		nxt[++cnt]=top[x];top[x]=cnt;to[cnt]=y;
		nxt[++cnt]=top[y];top[y]=cnt;to[cnt]=x;
	}
	void add1(int x,int y){
		nxt[++cnt]=top[x];top[x]=cnt;to[cnt]=y;
	}
    #ifdef _STL_PAIR_H
    void add2(pii a){
        add2(a.first,a.second);
    }
    #endif
};
#endif