#include<bits/stdc++.h>
#ifdef ONLINE_JUDGE
#include<atcoder/all>
#include<boost/multiprecision/cpp_int.hpp>
#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#endif
namespace eQe{
#define fin(...) exit(pp(__VA_ARGS__))
#define ep(...) emplace(__VA_ARGS__)
#define eb(...) emplace_back(__VA_ARGS__)
#define pb() pop_back()
#define ef(...) emplace_front(__VA_ARGS__)
#define pf(a) pop_front(a)
#define ei(...) else if(__VA_ARGS__)
#define el else
#define wh(...) while(__VA_ARGS__)
#define lb(...) lower_bound(__VA_ARGS__)
#define ub(...) upper_bound(__VA_ARGS__)
#define itmax(...) max_element(al(__VA_ARGS__))
#define itmin(...) min_element(al(__VA_ARGS__))
#define srt(...) sort(al(__VA_ARGS__))
#define rev(...) reverse(al(__VA_ARGS__))
#define uq(a) srt(a);a.erase(unique(al(a)),end(a))
#define jo(a,b) copy(al(b),back_inserter(a))
#define ov3(a,b,c,d,...) d
#define ov4(a,b,c,d,e,...) e
#define ov5(a,b,c,d,e,f,...) f
#define ov6(a,b,c,d,e,f,g,...) g
#define bg begin
#define rbg rbegin
#define al1(v) bg(v),end(v)
#define al2(v,b) bg(v),bg(v)+b
#define al3(v,a,b) bg(v)+a,bg(v)+b
#define al(...) ov3(__VA_ARGS__,al3,al2,al1)(__VA_ARGS__)
#define rl1(v) rbg(v),rend(v)
#define rl2(v,b) rbg(v),rbg(v)+b
#define rl3(v,a,b) rbg(v)+a,rbg(v)+b
#define rl(...) ov3(__VA_ARGS__,rl3,rl2,rl1)(__VA_ARGS__)
#define rp1(b) for(ll ii=0;ii<(ll)(b);ii++)
#define rp2(i,b) for(ll i=0;i<(ll)(b);i++)
#define rp3(i,a,b) for(ll i=(ll)(a);i<(ll)(b);i++)
#define rp4(i,a,b,c) for(ll i=(ll)(a);i<(ll)(b);i+=(ll)(c))
#define rp(...) ov4(__VA_ARGS__,rp4,rp3,rp2,rp1)(__VA_ARGS__)
#define pr1(a) rp(a)
#define pr2(i,a) for(ll i=(ll)(a)-1;i>=0;i--)
#define pr3(i,a,b) for(ll i=(ll)(a)-1;i>=(ll)(b);i--)
#define pr4(i,a,b,c) for(ll i=(ll)(a)-1;i>=(ll)(b);i-=(ll)(c))
#define pr(...) ov4(__VA_ARGS__,pr4,pr3,pr2,pr1)(__VA_ARGS__)
#define fe1(a) rp(len(a))
#define fe2(a,v) for(au&&a:v)
#define fe3(a,b,v) for(au&&[a,b]:v)
#define fe4(a,b,c,v) for(au&&[a,b,c]:v)
#define fe5(a,b,c,d,v) for(au&&[a,b,c,d]:v)
#define fe(...) ov5(__VA_ARGS__,fe5,fe4,fe3,fe2,fe1)(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;li(__VA_ARGS__)
#define LD(...) ld __VA_ARGS__;li(__VA_ARGS__)
#define ST(...) str __VA_ARGS__;li(__VA_ARGS__)
#define UL2(n,a) u1 a(n);li(a)
#define UL3(n,a,b) u1 a(n),b(n);li(a,b)
#define UL4(n,a,b,c) u1 a(n),b(n),c(n);li(a,b,c)
#define UL5(n,a,b,c,d) u1 a(n),b(n),c(n),d(n);li(a,b,c,d)
#define UL6(n,a,b,c,d,e) u1 a(n),b(n),c(n),d(n),e(n);li(a,b,c,d,e)
#define UL(...) ov6(__VA_ARGS__,UL6,UL5,UL4,UL3,UL2)(__VA_ARGS__)
#define UV2(n,a) u1 a(n);vi(a)
#define UV3(n,a,b) u1 a(n),b(n);vi(a,b)
#define UV4(n,a,b,c) u1 a(n),b(n),c(n);vi(a,b,c)
#define UV5(n,a,b,c,d) u1 a(n),b(n),c(n),d(n);vi(a,b,c,d)
#define UV6(n,a,b,c,d,e) u1 a(n),b(n),c(n),d(n),e(n);vi(a,b,c,d,e)
#define UV(...) ov6(__VA_ARGS__,UV6,UV5,UV4,UV3,UV2)(__VA_ARGS__)
#define DL2(n,a) v1<ld>a(n);li(a)
#define DL3(n,a,b) v1<ld>a(n),b(n);li(a,b)
#define DL4(n,a,b,c) v1<ld>a(n),b(n),c(n);li(a,b,c)
#define DL5(n,a,b,c,d) v1<ld>a(n),b(n),c(n),d(n);li(a,b,c,d)
#define DL(...) ov5(__VA_ARGS__,DL5,DL4,DL3,DL2)(__VA_ARGS__)
#define DV2(n,a) v1<ld>a(n);vi(a)
#define DV3(n,a,b) v1<ld>a(n),b(n);vi(a,b)
#define DV4(n,a,b,c) v1<ld>a(n),b(n),c(n);vi(a,b,c)
#define DV5(n,a,b,c,d) v1<ld>a(n),b(n),c(n),d(n);vi(a,b,c,d)
#define DV(...) ov5(__VA_ARGS__,DV5,DV4,DV3,DV2)(__VA_ARGS__)
#define I template
#define J class
#define oo operator
#define rr return
#define ss struct
#define uu using
#define au auto
#define stc static
#define bk break
#define cs const
#define ct continue
#define fr friend
#define endl "\n"
uu namespace std;
#ifdef ONLINE_JUDGE
uu namespace atcoder;
uu namespace boost::multiprecision;
uu mint=modint1000000007;
#endif
uu ll=long long;
uu ld=long double;
uu ul=unsigned long long;
uu str=string;
uu bo=bool;
uu vo=void;
uu is=istream;
uu os=ostream;
str sp=" ",nc="",nl="\n";
au np=nullptr;
I<J T,J U>ss cp{
  T a;U b;
  cp():a(T()),b(U()){}
  cp(T a,U b):a(a),b(b){}
  cp oo+()cs{rr*this;}
  cp oo-()cs{rr cp(-a,-b);}
  cp&oo++(){a++,b++;rr*this;}
  cp&oo--(){a--,b--;rr*this;}
  cp oo++(int){cp r=*this;++*this;rr r;}
  cp oo--(int){cp r=*this;--*this;rr r;}
  cp&oo+=(cs cp&c){a+=c.a;b+=c.b;rr*this;}
  cp&oo-=(cs cp&c){a-=c.a;b-=c.b;rr*this;}
  cp&oo*=(cs cp&c){a*=c.a;b*=c.b;rr*this;}
  cp oo+(cs cp&c)cs{rr cp(*this)+=c;}
  cp oo-(cs cp&c)cs{rr cp(*this)-=c;}
  cp oo*(cs cp&c)cs{rr cp(*this)*=c;}
  cp&oo+=(cs T&d){a+=d;b+=d;rr*this;}
  cp&oo-=(cs T&d){a-=d;b-=d;rr*this;}
  cp&oo*=(cs T&d){a*=d;b*=d;rr*this;}
  cp oo+(cs T&d)cs{rr cp(*this)+=d;}
  cp oo-(cs T&d)cs{rr cp(*this)-=d;}
  cp oo*(cs T&d)cs{rr cp(*this)*=d;}
  bo oo==(cs cp&c)cs{rr a==c.a and b==c.b;}
  bo oo!=(cs cp&c)cs{rr!(*this==c);}
  bo oo<(cs cp&c)cs{rr a!=c.a?a<c.a:b<c.b;}
  bo oo>(cs cp&c)cs{rr a!=c.a?a>c.a:b>c.b;}
  bo oo<=(cs cp&c)cs{rr!(*this>c);}
  bo oo>=(cs cp&c)cs{rr!(*this<c);}
  friend is&oo>>(is&i,cp&c){rr i>>c.a>>c.b;}
  friend os&oo<<(os&o,cs cp&c){rr o<<c.a<<sp<<c.b;}
};
I<J T>uu v1=vector<T>;
I<J T>uu v2=v1<v1<T>>;
I<J T>uu v3=v1<v2<T>>;
I<J T>uu v4=v1<v3<T>>;
I<J T>uu mset=multiset<T>;
I<J T>uu fn=function<T>;
uu u1=v1<ll>;
uu u2=v1<u1>;
uu u3=v1<u2>;
uu u4=v1<u3>;
uu cl=cp<ll,ll>;
uu ql=queue<ll>;
uu sl=set<ll>;
uu msl=mset<ll>;
uu ml=map<ll,ll>;
ss edg{ll t,w;edg(){}edg(ll t,ll w=1):t(t),w(w){}};
uu graph=v2<edg>;
ll inf=3e18;
ld pi=acosl(-1);
ld ee=1e-10;
str first(bo t=1){rr t?"first":"second";}
str First(bo t=1){rr t?"First":"Second";}
str yes(bo t=1){rr t?"yes":"no";}
str Yes(bo t=1){rr t?"Yes":"No";}
str YES(bo t=1){rr t?"YES":"NO";}
str no(){rr "no";}
str No(){rr "No";}
str NO(){rr "NO";}
str possible(bo t=1){rr t?"possible":"impossible";}
str Possible(bo t=1){rr t?"Possible":"Impossible";}
str POSSIBLE(bo t=1){rr t?"POSSIBLE":"IMPOSSIBLE";}
vo as(bo t){assert(t);}
ll bit(ll a){rr 1LL<<a;}
ll ppc(ll a){rr __builtin_popcountll(a);}
str to_str(char a){rr str()+a;}
I<J T>str to_str(T a){rr to_string(a);}
I<J T>ll len(cs T&a){rr a.size();}
I<J T>T sq(T a){rr a*a;}
I<J T>uu hpmax=std::priority_queue<T>;
I<J T>uu hpmin=std::priority_queue<T,v1<T>,greater<T>>;
ll at(ll S,ll i){rr S>>i&1;}
I<J T>T at(cs v1<T>&v,ll i){ll n=len(v);rr v[(i%n+n)%n];}
I<J T>ll sgn(cs T&a){rr(a>ee)-(a<-ee);}
I<J T,J U>ll sgn(cs T&a,cs U&b){rr sgn(a-b);}
I<J T>au&oo+(v1<T>&v){rr v;}
I<J T>au&oo-(v1<T>&v){fe(x,v)x=-x;rr v;}
I<J T>au&oo++(v1<T>&v){fe(x,v)x++;rr v;}
I<J T>au&oo--(v1<T>&v){fe(x,v)x--;rr v;}
I<J T>au oo++(v1<T>&v,int){au r=v;++v;rr r;}
I<J T>au oo--(v1<T>&v,int){au r=v;--v;rr r;}
I<J T>au&oo+=(v1<T>&v,cs v1<T>&u){rp(i,len(v))v[i]+=u[i];rr v;}
I<J T>au&oo-=(v1<T>&v,cs v1<T>&u){rp(i,len(v))v[i]-=u[i];rr v;}
I<J T>au&oo*=(v1<T>&v,cs v1<T>&u){rp(i,len(v))v[i]*=u[i];rr v;}
I<J T>au oo+(v1<T>v,cs v1<T>&u){rr v+=u;}
I<J T>au oo-(v1<T>v,cs v1<T>&u){rr v-=u;}
I<J T>au oo*(v1<T>v,cs v1<T>&u){rr v*=u;}
I<J T,J U>au&oo+=(v1<T>&v,cs U&a){fe(x,v)x+=a;rr v;}
I<J T,J U>au&oo-=(v1<T>&v,cs U&a){fe(x,v)x-=a;rr v;}
I<J T,J U>au&oo*=(v1<T>&v,cs U&a){fe(x,v)x*=a;rr v;}
I<J T,J U>au oo+(v1<T>v,cs U&a){rr v+=a;}
I<J T,J U>au oo-(v1<T>v,cs U&a){rr v-=a;}
I<J T,J U>au oo*(v1<T>v,cs U&a){rr v*=a;}
I<J T,J U>bo amax(T&a,cs U&b){rr a<b?a=b,1:0;}
I<J T,J U>bo amin(T&a,cs U&b){rr a>b?a=b,1:0;}
I<J T>T max(cs v1<T>&a){rr *itmax(a);}
I<J T>T min(cs v1<T>&a){rr *itmin(a);}
I<J T>au max(cs v2<T>&a){rr max(*itmax(a));}
I<J T>au min(cs v2<T>&a){rr min(*itmin(a));}
I<J...T>au max(T...a){rr max(initializer_list<common_type_t<T...>>{a...});}
I<J...T>au min(T...a){rr min(initializer_list<common_type_t<T...>>{a...});}
I<J T>ll argmax(cs v1<T>&a){rr itmax(a)-bg(a);}
I<J T>ll argmin(cs v1<T>&a){rr itmin(a)-bg(a);}
I<J T,J...A>T gcd(cs T&a,cs A&...b){rr gcd(a,gcd(b...));}
I<J T>T sum(cs v1<T>&a){rr accumulate(al(a),T(0));}
I<J T,J U>ll lbs(cs v1<T>&a,cs U&b){rr lb(al(a),b)-bg(a);}
I<J T,J U>ll ubs(cs v1<T>&a,cs U&b){rr ub(al(a),b)-bg(a);}
I<J T,J U>au minmax(cs T&a,cs U&b){rr cp(min(a,b),max(a,b));}
I<J T,J U>au minmax(cs cp<T,U>&p){rr minmax(p.a,p.b);}
I<J T>vo emplace_front(v1<T>&v,T x=0){v.ep(bg(v),x);}
I<J T>vo pop_front(v1<T>&v){v.erase(bg(v));}
I<J X,J Y>au mp(cs fn<Y(X)>&f,v1<X>x){v1<Y>y;fe(a,x)y.eb(f(a));rr y;}
I<J T,J U>au keys(cs map<T,U>&m){v1<T>r;fe(k,z,m)r.eb(k);rr r;}
I<J T,J U>au vals(cs map<T,U>&m){v1<U>r;fe(z,v,m)r.eb(v);rr r;}
u1 io(ll n,ll a=0){u1 i(n);iota(al(i),a);rr i;}


ss z{z(){cin.tie(0);ios::sync_with_stdio(0);cout<<fixed<<setprecision(10);}}z;
is&oo>>(is&i,mint&x){ll t;i>>t;x=t;rr i;}
os&oo<<(os&o,cs mint&x){rr o<<x.val();}
I<J T,J U>is&oo>>(is&i,pair<T,U>&p){rr i>>p.first>>p.second;}
I<J T,J U>os&oo<<(os&o,cs pair<T,U>&p){rr o<<p.first<<sp<<p.second;}
I<J T>is&oo>>(is&i,v1<T>&v){fe(x,v)i>>x;rr i;}
I<J T>os&oo<<(os&o,cs v1<T>&v){fe(x,v)o<<x<<(&x==&*rbg(v)?nc:sp);rr o;}
I<J T>os&oo<<(os&o,cs v2<T>&v){fe(x,v)o<<x<<(&x==&*rbg(v)?nc:nl);rr o;}
I<J T>os&oo<<(os&o,cs set<T>&s){fe(x,s)o<<x<<(&x==&*rbg(s)?nc:sp);rr o;}
I<J T>os&oo<<(os&o,cs mset<T>&s){fe(x,s)o<<x<<(&x==&*rbg(s)?nc:sp);rr o;}
I<J T,J U>os&oo<<(os&o,cs map<T,U>&m){fe(p,m)o<<p<<(&p==&*rbg(m)?nc:nl);rr o;}
I<J T>os&oo<<(os&o,queue<T>q){wh(len(q))o<<q.front()<<sp,q.pop();rr o;}
I<J T>os&oo<<(os&o,deque<T>q){wh(len(q))o<<q.front()<<sp,q.pop_front();rr o;}
I<J T>os&oo<<(os&o,hpmax<T>q){wh(len(q))o<<q.top()<<sp,q.pop();rr o;}
I<J T>os&oo<<(os&o,hpmin<T>q){wh(len(q))o<<q.top()<<sp,q.pop();rr o;}
ll pp(){cout<<endl;rr 0;}
I<J T,J...A>ll pp(cs T&a,cs A&...b){cout<<a;((cout<<sp<<b),...);rr pp();}
I<J...T>vo li(T&...a){(cin>>...>>a);}
I<J...T>vo vi(ll i,T&...a){(cin>>...>>a[i]);}
I<J T,J...A>vo vi(v1<T>&a,A&...b){rp(i,len(a))vi(i,a,b...);}
I<J T>vo rs(v1<T>&v,ll n){v.resize(n);}
I<J T,J...A>vo rs(v1<T>&V,ll n,cs A&...a){rs(V,n);fe(v,V)rs(v,a...);}
I<J T,J U>vo fl(T&x,cs U&a){x=a;}
I<J T,J U>vo fl(v1<T>&V,cs U&a){fe(v,V)fl(v,a);}
vo gi(u2&g,ll m,bo d=1){rp(i,m){LL(a,b);a--;b--;g[a].eb(b);if(d)g[b].eb(a);}}
vo gi(graph&g,ll m,bo w=1,bo d=1){
  rp(i,m){
    LL(a,b);a--,b--;ll c=1;if(w)li(c);
    g[a].eb(b,c);if(d)g[b].eb(a,c);
  }
}
vo ti(u2&g,bo d=1){gi(g,len(g)-1,d);}
vo ti(graph&g,bo w=1,bo d=1){gi(g,len(g)-1,w,d);}


vo zzp(cs u1&v,u1&h){fe(x,h)x=lbs(v,x);}
I<J...T>vo zzp(cs u1&v,u1&h,T&...t){zzp(v,h);zzp(v,t...);}
I<J...T>u1 zp(T&...a){u1 v;fe(e,{a...})jo(v,e);uq(v);zzp(v,a...);rr v;}
I<J T>vo zvs(cs u1&o,v1<T>&a){au c=a;rp(i,len(a))a[i]=c[o[i]];}
I<J T,J...A>vo zvs(cs u1&o,v1<T>&a,A&...b){zvs(o,a);zvs(o,b...);}
I<J...T>au vs(cs fn<bo(ll,ll)>&f,T&...a){
  ll n=0;fe(e,{a...})n=len(e);
  u1 o=io(n),p(n);
  sort(al(o),f);
  rp(i,n)p[o[i]]=i;
  zvs(o,a...);
  rr cp(o,p);
}
I<J T>T bs(cs fn<bo(T)>&j,T o,T n){
  T d=(T)ee?ee:1;
  wh(abs(o-n)>d)(j((o+n)/2)?o:n)=(o+n)/2;
  rr o;
}
I<J T>au zet(v1<T>a){rp(i,1,len(a))a[i]+=a[i-1];rr a;}
I<J T>au moe(v1<T>a){pr(i,len(a),1)a[i]-=a[i-1];rr a;}
I<J T>au zet(v2<T>a){fe(v,a)v=zet(v);rp(i,1,len(a))a[i]+=a[i-1];rr a;}
I<J T>au moe(v2<T>a){fe(v,a)v=moe(v);pr(i,len(a),1)a[i]-=a[i-1];rr a;}
vo wafl(u2&d){ll n=len(d);rp(k,n)rp(i,n)rp(j,n)amin(d[i][j],d[i][k]+d[k][j]);}
I<J T>T exq(ll n,T r,T x){wh(n){if(n&1)r*=x;x*=x;n/=2;}rr r;}
u1 divs(ll n){ml m;rp(i,1,n/i+1)if(n%i==0)m[i]++,m[n/i]++;rr keys(m);}
ml fact(ll n){ml m;rp(i,2,n/i+1)wh(n%i==0)m[i]++,n/=i;if(n>1)m[n]++;rr m;}
ll mpw(ll x,ll n,ll m=inf){ll r=1;wh(n){if(n&1)r*=x,r%=m;x*=x,x%=m,n/=2;}rr r;}
ll minv(ll a,ll m){
  ll b=m,u=1,v=0;
  wh(b)u-=a/b*v,swap(u,v),a-=a/b*b,swap(a,b);
  rr(u%m+m)%m;
}
ll sqr(ll x){ll r=sqrtl(x)-1;wh(r+1<=x/(r+1))r++;rr r;}
ll lg2(ll n){ll r=0;wh(n){n/=2;r++;}rr r;}
I<J T,J U>au ceil(T a,U b){rr (a+b-1)/b;}
#ifdef ONLINE_JUDGE
ll cinv(u1 a){
  ll m=len(zp(a));
  fenwick_tree<ll>fw(m);
  ll c=0;
  fe(x,a)c+=fw.sum(x+1,m),fw.add(x,1);
  rr c;
}
#endif
u1 dx={-1,0,1,0,-1,1,1,-1};
u1 dy={0,-1,0,1,-1,-1,1,1};


namespace Rnd{
ul rnd(){
  stc ul x=123456789,y=362436069,z=521288629,w=88675123;
  ul t=(x^(x<<11));
  x=y,y=z,z=w;
  rr(w=(w^(w>>19))^(t^(t>>8)));
}
ld rnd_prob(){rr (ld)rnd()/UINT_MAX;}
ll rnd_rng(ll r){rr rnd()%r;}
ll rnd_rng(ll l,ll r){rr rnd()%(r-l)+l;}
}uu namespace Rnd;


u1 djks(ll s,graph&g){
  ll n=len(g);
  u1 d(n,inf);
  hpmin<cl>q;
  d[s]=0;
  q.ep(0,s);
  wh(len(q)){
    au[du,u]=q.top();q.pop();
    if(d[u]<du)ct;
    fe(v,w,g[u])if(amin(d[v],du+w))q.ep(du+w,v);
  }
  rr d;
}


I<J S,cs fn<S(S,S)>&op,cs fn<S()>&e>ss sgt{
  ll n,A,log;
  v1<S>d;
  sgt(ll n):sgt(v1<S>(n,e())){}
  sgt(cs v1<S>&v):n(len(v)){
    log=lg2(n*2-1)-1;
    A=bit(log);
    d=v1<S>(A*2,e());
    rp(i,n)d[i+A]=v[i];
    pr(i,A,1)upd(i);
  }
  vo upd(ll k){d[k]=op(d[k*2],d[k*2+1]);}
  vo set(ll p,S x){as(0<=p and p<n);p+=A;d[p]=x;rp(i,1,log+1)upd(p>>i);}
  vo add(ll p,S x){as(0<=p and p<n);set(p,get(p)+x);}
  S get(ll p)cs{as(0<=p and p<n);rr d[p+A];}
  S aprod(){rr d[1];}
  S prod(ll l,ll r){
    as(0<=l and l<=r and r<=n);
    S sml=e(),smr=e();
    l+=A,r+=A;
    wh(l<r){if(l&1)sml=op(sml,d[l++]);if(r&1)smr=op(d[--r],smr);l/=2,r/=2;}
    rr op(sml,smr);
  }
  fr os&oo<<(os&o,cs sgt&s){rp(i,s.n)o<<s.get(i)<<(i==s.n-1?nc:sp);rr o;}
};


I<J T>ss comb{
  v1<T>fa,ra,bs;
  v2<T>m;
  v2<bo>u;
  comb(ll n){
    n++;
    rs(fa,n);
    rs(ra,n);
    rs(bs,n);
    fa[0]=ra[0]=bs[0]=1;
    rp(i,1,n)fa[i]=fa[i-1]*i;
    rp(i,1,n)ra[i]=ra[i-1]/i;
    rp(i,1,n)bs[i]=bs[i-1]+T(-1).pow(i&1)*ra[i];
    if(n<=5e3)rs(m,n,n),rs(u,n,n);
  }
  T C(ll n,ll k){rr n<k or n<0 or k<0?0:fa[n]*ra[k]*ra[n-k];}
  T P(ll n,ll k){rr C(n,k)*fa[k];}
  T H(ll n,ll k){rr C(n+k-1,n);}//n個の区別できない球をk個の区別できる箱に配る
  T S(ll n,ll k){T r=0;rp(i,k+1)r+=C(k,i)*T(i).pow(n)*mpw(-1,k-i);rr r*ra[k];}
  T B(ll n,ll k){T r=0;rp(i,k+1)r+=T(i).pow(n)*ra[i]*bs[k-i];rr r;}
  T p(ll n,ll k){//自然数nをk個の0以上の整数に分割する
    if(n==0 and k==0)rr 1;
    ei(k==0)rr 0;
    ei(u[n][k])rr m[n][k];
    ei(n-k>=0)rr u[n][k]=1,m[n][k]=p(n,k-1)+p(n-k,k);
    el rr u[n][k]=1,m[n][k]=p(n,k-1);
  }
};


I<J T>ss mrbst{
  ss node{
    T val,sum;
    ll sz;
    node*l,*r;
    node():val(T()),sum(T()),sz(1),l(np),r(np){}
    node(T val):val(val),sum(val),sz(1),l(np),r(np){}
    vo upd(){}
    vo push(){}
  };
  
  node*ro;
  mrbst():ro(np){}
  mrbst(node*ro):ro(ro){}
  
  ll size(node*no)cs{rr !no?0:no->sz;}
  ll size()cs{rr size(ro);}
  T sum(node*no)cs{rr !no?T():no->sum;}
  T sum()cs{rr sum(ro);}
  
  node* upd(node*no){
    no->sz=size(no->l)+size(no->r)+1;
    no->sum=sum(no->l)+sum(no->r)+no->val;
    no->upd();
    rr no;
  }
  vo push(node*no){if(no)no->push();}
  
  ll lb(node*no,T val){
    push(no);
    if(!no)rr 0;
    if(val<=no->val)rr lb(no->l,val);
    el rr size(no->l)+lb(no->r,val)+1;
  }
  ll lb(T val){rr lb(ro,val);}
  
  ll ub(node*no,T val){
    push(no);
    if(!no)rr 0;
    if(no->val<=val)rr size(no->l)+ub(no->r,val)+1;
    el rr ub(no->l,val);
  }
  ll ub(T val){rr ub(ro,val);}
  
  ll count(T val){rr ub(val)-lb(val);}
  
  T get(node*no,ll k){
    push(no);
    if(k==size(no->l))rr no->val;
    if(k<size(no->l))rr get(no->l,k);
    el rr get(no->r,k-size(no->l)-1);
  }
  T get(ll k){as(k<size(ro));rr get(ro,k);}
  
  node*merge(node*l,node*r){
    push(l),push(r);
    if(!l)rr r;
    if(!r)rr l;
    
    if(rnd()%(l->sz+r->sz)<l->sz){
      l->r=merge(l->r,r);
      rr upd(l);
    }el{
      r->l=merge(l,r->l);
      rr upd(r);
    }
  }
  vo merge(mrbst add){ro=merge(ro,add.ro);}
  
  au split(node*no,ll k){
    push(no);
    if(!no)rr cp(no,no);
    if(k<=size(no->l)){
      au sb=split(no->l,k);
      no->l=sb.b;
      rr cp(sb.a,upd(no));
    }el{
      au sb=split(no->r,k-size(no->l)-1);
      no->r=sb.a;
      rr cp(upd(no),sb.b);
    }
  }
  mrbst split(ll k){
    au sb=split(ro,k);
    ro=sb.a;
    rr mrbst(sb.b);
  }
  
  vo insert(cs T val){
    au sb=split(ro,lb(val));
    ro=merge(merge(sb.a,new node(val)),sb.b);
  }
  
  vo erase(cs T val){
    if(!count(val))rr;
    au sb=split(ro,lb(val));
    ro=merge(sb.a,split(sb.b,1).b);
  }
  
  fr os&oo<<(os&o,mrbst s){rp(i,len(s))o<<s.get(i)<<sp;rr o;}
};
I<J T>ss rbst:mrbst<T>{
  uu super=mrbst<T>;
  uu super::mrbst;
  vo insert(cs T val){if(super::count(val)==0)super::insert(val);}
};


I<J T>ss mat{
  v2<T>m;
  ll h,w;
  mat(ll n):h(n),w(n){rs(m,n,n);}
  mat(ll h,ll w):h(h),w(w){rs(m,h,w);}
  mat(ll h,ll w,T x):h(h),w(w){rs(m,h,w);fl(m,x);}
  mat(v2<T>a):h(len(a)),w(len(a[0])),m(a){}
  
  v1<T>oo[](cs ll&i)cs{rr m[i];}
  v1<T>&oo[](cs ll&i){rr m[i];}
  
  mat&oo+=(cs mat&a){as(h==a.h and w==a.w);rp(i,h)m[i]+=a[i];rr*this;}
  mat&oo-=(cs mat&a){as(h==a.h and w==a.w);rp(i,h)m[i]-=a[i];rr*this;}
  mat&oo*=(cs mat&a){
    as(w==a.h);
    mat m2(h,a.w);
    rp(i,h)rp(k,w)rp(j,a.w)m2[i][j]+=m[i][k]*a[k][j];
    rr*this=m2;
  }
  
  //行列
  mat oo+(cs mat&a)cs{rr mat(*this)+=a;}
  mat oo-(cs mat&a)cs{rr mat(*this)-=a;}
  mat oo*(cs mat&a)cs{rr mat(*this)*=a;}
  bo oo==(cs mat&a){rr m==a.m;}
  //スカラ
  mat&oo+=(cs T&a){fe(v,m)v+=a;rr*this;}
  mat&oo-=(cs T&a){fe(v,m)v-=a;rr*this;}
  mat&oo*=(cs T&a){fe(v,m)v*=a;rr*this;}
  mat&oo/=(cs T&a){fe(v,m)v/=a;rr*this;}
  mat oo+(cs T&a)cs{rr mat(*this)+=a;}
  mat oo-(cs T&a)cs{rr mat(*this)-=a;}
  mat oo*(cs T&a)cs{rr mat(*this)*=a;}
  mat oo/(cs T&a)cs{rr mat(*this)/=a;}
  
  mat pow(ll n){mat x=*this;mat r(h);rp(i,h)r[i][i]=1;rr exq(n,r,x);}
  vo trs(){mat m2(w,h);rp(i,h)rp(j,w)m2[j][i]=m[i][j];*this=m2;}
  
  fr is&oo>>(is&i,mat&m){rr i>>m.m;}
  fr os&oo<<(os&o,cs mat&m){rr o<<m.m;}
};


vo solve(){
  LL(N);
  
  mat<mint>a(
    {{1,0,1,0,0},
    {1,1,0,1,0},
    {0,1,0,0,0},
    {0,0,1,0,0},
    {0,0,0,1,2}});
  
  mat<mint>v({{1,0,0,0,0}});
  v.trs();
  a=a.pow(N);
  a*=v;
  pp(a[4][0]);
}


}int main(){eQe::solve();}