#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#include<atcoder/all>
//#include<boost/multiprecision/cpp_int.hpp>
#include<bits/stdc++.h>
#define bgn(a) begin(a)
#define rbg(a) rbegin(a)
#define fin(...) exit(pp(__VA_ARGS__))
#define done(...) rr vo(pp(__VA_ARGS__))
#define ep(...) emplace(__VA_ARGS__)
#define ef(...) emplace_front(__VA_ARGS__)
#define eb(...) emplace_back(__VA_ARGS__)
#define pf(...) pop_front(__VA_ARGS__)
#define pb(...) pop_back(__VA_ARGS__)
#define ei(...) else if(__VA_ARGS__)
#define iif if
#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 rv(...) reverse(al(__VA_ARGS__))
#define rsr(a) srt(a),rv(a)
#define uq(a) srt(a),a.erase(unique(al(a)),end(a))
#define sw(a,b) swap(a,b)
#define sb(...) substr(__VA_ARGS__)
#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 al1(v) bgn(v),end(v)
#define al2(v,b) bgn(v),bgn(v)+b
#define al3(v,a,b) bgn(v)+a,bgn(v)+b
#define al(...) ov3(__VA_ARGS__,al3,al2,al1)(__VA_ARGS__)
#define fo1(b) for(ll ii=0;ii<(ll)(b);ii++)
#define fo2(i,b) for(ll i=0;i<(ll)(b);i++)
#define fo3(i,a,b) for(ll i=(ll)(a);i<(ll)(b);i++)
#define fo4(i,a,b,c) for(ll i=(ll)(a);i<(ll)(b);i+=(ll)(c))
#define fo(...) ov4(__VA_ARGS__,fo4,fo3,fo2,fo1)(__VA_ARGS__)
#define of1(a) fo(a)
#define of2(i,a) for(ll i=(ll)(a)-1;i>=0;i--)
#define of3(i,a,b) for(ll i=(ll)(a)-1;i>=(ll)(b);i--)
#define of4(i,a,b,c) for(ll i=(ll)(a)-1;i>=(ll)(b);i-=(ll)(c))
#define of(...) ov4(__VA_ARGS__,of4,of3,of2,of1)(__VA_ARGS__)
#define fe1(a) fo(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 DD(...) dd __VA_ARGS__;li(__VA_ARGS__)
#define CH(...) chr __VA_ARGS__;li(__VA_ARGS__)
#define ST(...) str __VA_ARGS__;li(__VA_ARGS__)
#define MI(...) mint __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<dd>a(n);li(a)
#define DL3(n,a,b) v1<dd>a(n),b(n);li(a,b)
#define DL4(n,a,b,c) v1<dd>a(n),b(n),c(n);li(a,b,c)
#define DL5(n,a,b,c,d) v1<dd>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<dd>a(n);vi(a)
#define DV3(n,a,b) v1<dd>a(n),b(n);vi(a,b)
#define DV4(n,a,b,c) v1<dd>a(n),b(n),c(n);vi(a,b,c)
#define DV5(n,a,b,c,d) v1<dd>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 U23(h,w,a) u2 a;rs(a,h,w);li(a)
#define U24(h,w,a,b) u2 a,b;rs(a,h,w),rs(b,h,w);li(a,b)
#define U25(h,w,a,b,c) u2 a,b,c;rs(a,h,w),rs(b,h,w),rs(c,h,w);li(a,b,c)
#define U2(...) ov5(__VA_ARGS__,U25,U24,U23)(__VA_ARGS__)
#define S22(h,a) strs a(h);li(a)
#define S23(h,a,b) strs a(h),b(h);li(a,b)
#define S2(...) ov3(__VA_ARGS__,S23,S22)(__VA_ARGS__)
#define AND2(a,b) ((a)&&(b))
#define AND3(a,b,c) ((a)&&(b)&&(c))
#define AND4(a,b,c,d) ((a)&&(b)&&(c)&&(d))
#define AND5(a,b,c,d,e) ((a)&&(b)&&(c)&&(d)&&(e))
#define AND6(a,b,c,d,e,f) ((a)&&(b)&&(c)&&(d)&&(e)&&(f))
#define AND(...) ov6(__VA_ARGS__,AND6,AND5,AND4,AND3,AND2)(__VA_ARGS__)
#define OR2(a,b) ((a)||(b))
#define OR3(a,b,c) ((a)||(b)||(c))
#define OR4(a,b,c,d) ((a)||(b)||(c)||(d))
#define OR5(a,b,c,d,e) ((a)||(b)||(c)||(d)||(e))
#define OR6(a,b,c,d,e,f) ((a)||(b)||(c)||(d)||(e)||(f))
#define OR(...) ov6(__VA_ARGS__,OR6,OR5,OR4,OR3,OR2)(__VA_ARGS__)
#define I template
#define J class
#define O operator
#define rr return
#define ss struct
#define uu using
#define as assert
#define au auto
#define bk break
#define cs const
#define ct continue
#define endl "\n"
#define fd friend
#define sc static
#define th this
namespace my{
uu vo=void;
vo main();
vo solve();
}
int main(){my::main();}
namespace my{
uu namespace std;
uu namespace atcoder;
//uu namespace boost::multiprecision;
uu mint=modint998244353;
//uu cint=cpp_int;
uu ll=long long;
uu dd=long double;
uu ui=unsigned int;
uu ul=unsigned long long;
uu chr=char;
uu str=string;
uu bo=bool;
uu is=istream;
uu os=ostream;
str sp=" ",nc="",nl="\n";
au nu=nullptr;
I<J A,J B>ss cp{
  A a;B b;
  cp():a(A()),b(B()){}
  cp(A a,B b):a(a),b(b){}
  cp O+()cs{rr*th;}
  cp O-()cs{rr cp(-a,-b);}
  cp&O++(){a++,b++;rr*th;}cp O++(int){cp r=*th;++*th;rr r;}
  cp&O--(){a--,b--;rr*th;}cp O--(int){cp r=*th;--*th;rr r;}
  cp&O+=(cs cp&c){a+=c.a;b+=c.b;rr*th;}cp O+(cs cp&c)cs{rr cp(*th)+=c;}
  cp&O-=(cs cp&c){a-=c.a;b-=c.b;rr*th;}cp O-(cs cp&c)cs{rr cp(*th)-=c;}
  cp&O*=(cs cp&c){a*=c.a;b*=c.b;rr*th;}cp O*(cs cp&c)cs{rr cp(*th)*=c;}
  cp&O/=(cs cp&c){a/=c.a;b/=c.b;rr*th;}cp O/(cs cp&c)cs{rr cp(*th)/=c;}
  I<J D>cp&O+=(cs D&d){a+=d;b+=d;rr*th;}I<J D>cp O+(cs D&d)cs{rr cp(*th)+=d;}
  I<J D>cp&O-=(cs D&d){a-=d;b-=d;rr*th;}I<J D>cp O-(cs D&d)cs{rr cp(*th)-=d;}
  I<J D>cp&O*=(cs D&d){a*=d;b*=d;rr*th;}I<J D>cp O*(cs D&d)cs{rr cp(*th)*=d;}
  I<J D>cp&O/=(cs D&d){a/=d;b/=d;rr*th;}I<J D>cp O/(cs D&d)cs{rr cp(*th)/=d;}
  bo O==(cs cp&c)cs{rr AND(a==c.a,b==c.b);}bo O!=(cs cp&c)cs{rr!(*th==c);}
  bo O<(cs cp&c)cs{rr a!=c.a?a<c.a:b<c.b;}bo O>=(cs cp&c)cs{rr!(*th<c);}
  bo O>(cs cp&c)cs{rr a!=c.a?a>c.a:b>c.b;}bo O<=(cs cp&c)cs{rr!(*th>c);}
  au ab()cs{rr abs((*th).a)+abs((*th).b);}//manhattan
  au ab(cs cp&c)cs{rr (*th-c).ab();}//manhattan
  fd is&O>>(is&i,cp&c){rr i>>c.a>>c.b;}
  fd os&O<<(os&o,cs cp&c){rr o<<c.a<<sp<<c.b;}
};
I<J A,J B,J C>ss tr{
  A a;B b;C c;
  tr():a(A()),b(B()),c(C()){}
  tr(A a,B b,C c):a(a),b(b),c(c){}
  bo O==(cs tr&t)cs{rr AND(a==t.a,b==t.b,c==t.c);}
  bo O!=(cs tr&t)cs{rr!(*th==t);}
  bo O<(cs tr&t)cs{rr a!=t.a?a<t.a:b!=t.b?b<t.b:c<t.c;}
  bo O>(cs tr&t)cs{rr a!=t.a?a>t.a:b!=t.b?b>t.b:c>t.c;}
  fd is&O>>(is&i,tr&t){rr i>>t.a>>t.b>>t.c;}
  fd os&O<<(os&o,cs tr&t){rr o<<t.a<<sp<<t.b<<sp<<t.c;}
};
I<J A,J B,J C,J D>ss qu{
  A a;B b;C c;D d;
  qu():a(A()),b(B()),c(C()),d(D()){}
  qu(A a,B b,C c,D d):a(a),b(b),c(c),d(d){}
  bo O==(cs qu&q)cs{rr AND(a==q.a,b==q.b,c==q.c,d==q.d);}
  bo O!=(cs qu&q)cs{rr!(*th==q);}
  bo O<(cs qu&q)cs{rr a!=q.a?a<q.a:b!=q.b?b<q.b:c!=q.c?c<q.c:d<q.d;}
  bo O>(cs qu&q)cs{rr a!=q.a?a>q.a:b!=q.b?b>q.b:c!=q.c?c>q.c:d>q.d;}
  fd is&O>>(is&i,qu&q){rr i>>q.a>>q.b>>q.c>>q.d;}
  fd os&O<<(os&o,cs qu&q){rr o<<q.a<<sp<<q.b<<sp<<q.c<<sp<<q.d;}
};
I<J T>uu v1=vector<T>;uu u1=v1<ll>;
I<J T>uu v2=v1<v1<T>>;uu u2=v1<u1>;
I<J T>uu v3=v1<v2<T>>;uu u3=v1<u2>;
I<J T>uu v4=v1<v3<T>>;uu u4=v1<u3>;
I<J T>uu mset=multiset<T>;
I<J T>uu fn=function<T>;
I<J T>uu pqmax=priority_queue<T>;
I<J T>uu pqmin=priority_queue<T,v1<T>,greater<T>>;
uu chrs=v1<chr>;
uu strs=v1<str>;
uu cl=cp<ll,ll>;
uu cd=cp<dd,dd>;
uu tl=tr<ll,ll,ll>;
uu td=tr<dd,dd,dd>;
uu ql=qu<ll,ll,ll,ll>;
uu cls=v1<cl>;
uu cds=v1<cd>;
uu tls=v1<tl>;
uu tds=v1<td>;
uu qls=v1<ql>;
uu dl=deque<ll>;
uu sl=set<ll>;
uu msl=mset<ll>;
uu ml=map<ll,ll>;
uu usl=unordered_set<ll>;
uu umsl=unordered_multiset<ll>;
uu uml=unordered_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;
dd ee=1e-10,pi=acosl(-1);
u1 dx={-1,0,1,0,-1,1,1,-1},dy={0,-1,0,1,-1,-1,1,1};
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 no(){rr yes(0);}
str Yes(bo t=1){rr t?"Yes":"No";}str No(){rr Yes(0);}
str YES(bo t=1){rr t?"YES":"NO";}str NO(){rr YES(0);}
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";}

ui rnd(){sc ui x=123456789,y=362436069,z=521288629,w=88675123;ui t=(x^(x<<11));x=y,y=z,z=w;rr(w=(w^(w>>19))^(t^(t>>8)));}
dd rnd_prb(){rr (dd)rnd()/UINT_MAX;}
ll rnd_rng(ll r){rr rnd()%r;}
ll rnd_rng(ll l,ll r){rr rnd()%(r-l)+l;}

bo od(ll x){rr x&1;}
bo ev(ll x){rr !od(x);}
bo eqp(ll x,ll y){rr ev(x-y);}
ll bit(ll a){rr 1LL<<a;}
ll ppc(ll a,ll l=0,ll r=63){rr __builtin_popcountll((bit(r)-bit(l))&a);}//[l,r)
ll sqr(ll x){if(x<=1)rr x;ll r=sqrtl(x)-1;wh(r+1<=x/(r+1))r++;rr r;}
ll cbr(ll x){if(x<=1)rr x;ll r=cbrtl(x)-1;wh(r+1<=x/(r+1)/(r+1))r++;rr r;}
ll lg2(ll n){ll r=0;wh(n){n/=2;r++;}rr r;}
ll lg10(ll n){ll r=0;wh(n)n/=10,r++;rr r;}
bo in(ll a,ll x,ll b){rr a<=x&&x<b;}//x in [a,b)
u1 sub(ll S){u1 r;for(ll b=S;b>0;--b&=S)r.eb(b);srt(r);rr r;}
I<J T,J U,J V>au sum(T a,U d,V n){rr n*(a*2+(n-1)*d)/2;}
I<J T>ll len(cs T&a){rr a.size();}
I<J T>bo yu(cs T&a){rr len(a);}
I<J T>bo mu(cs T&a){rr !yu(a);}
I<J T>T sq(T a){rr a*a;}
I<J T>T cb(T a){rr a*a*a;}
I<J T>T ab(T a){rr abs(a);}
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,J U>T cel(T x,U y){as(y);rr(y<0?cel(-x,-y):(x>0?(x+y-1)/y:x/y));}
I<J T,J U>T flr(T x,U y){as(y);rr(y<0?flr(-x,-y):(x>0?x/y:x/y-!(x%y)));}
str tos(cs chr&a){rr a+nc;}
str tos(cs chr*a){rr a;}
str tos(cs str&a){rr a;}
I<J T>str tos(cs T&a){rr to_string(a);}
I<J T,J...A>str tos(cs T&a,cs A&...b){rr tos(a)+tos(b...);}
u1 tou(cs str&s,chr b='a'){u1 a;fe(c,s)a.eb(c-b);rr a;}
u2 tou(cs strs&s,chr b='a'){u2 a;fe(t,s)a.eb(tou(t,b));rr a;}
u1 tou(cs str&s,cs str&t){ll n=len(s);u1 a(n);fo(i,n)fo(j,len(t))if(s[i]==t[j]){a[i]=j;bk;}rr a;}
u2 tou(cs strs&s,cs str&t){ll h=len(s);u2 a(h);fo(i,h)a[i]=tou(s[i],t);rr a;}
u1 tou(cs v1<chr>&b,cs str&t){ll n=len(b);u1 a(n);fo(i,n)fo(j,len(t))if(b[i]==t[j]){a[i]=j;bk;}rr a;}
u2 tou(cs graph&g){ll n=len(g);u2 a(n);fo(u,n)fe(v,w,g[u])a[u].eb(v);rr a;}
graph tog(cs u2&a){ll n=len(a);graph g(n);fo(u,n)fe(v,a[u])g[u].eb(v);rr g;}
ll rp(ll a,ll x=inf,ll y=-1){rr a==x?y:a;}
u1 rp(u1 a,ll x=inf,ll y=-1){fo(i,len(a))a[i]=rp(a[i],x,y);rr a;}


I<J T,J U>vo emplace_front(v1<T>&v,U x=U()){v.ep(bgn(v),x);}
I<J T>vo emplace_front(v1<T>&v,T x=T()){v.ep(bgn(v),x);}
I<J T>vo pop_front(v1<T>&v){v.erase(bgn(v));}
I<J T>au&bg(T&a){rr*bgn(a);}
I<J T>au&bg(cs T&a){rr*bgn(a);}
I<J T>au&rb(T&a){rr*rbg(a);}
I<J T>au&rb(cs T&a){rr*rbg(a);}
I<J T>T bg(pqmax<T>&q){rr q.top();}
I<J T>T bg(pqmin<T>&q){rr q.top();}
I<J T>T pop(v1<T>&v){T r=rb(v);v.pb();rr r;}
I<J T>T pop_front(deque<T>&q){T r=bg(q);q.pf();rr r;}
I<J T>T pop_back(deque<T>&q){T r=rb(q);q.pb();rr r;}
I<J T>T pop(deque<T>&q){rr pop_front(q);}
I<J T>T pop(pqmax<T>&q){T r=bg(q);q.pop();rr r;}
I<J T>T pop(pqmin<T>&q){T r=bg(q);q.pop();rr r;}
I<J T>au&O^=(v1<T>&v,cs v1<T>&u){copy(al(u),back_inserter(v));rr v;}
I<J T>au O^(v1<T>v,cs v1<T>&u){rr v^=u;}
I<J T>au O+(v1<T>v){rr v;}
I<J T>au O-(v1<T>v){fe(x,v)x=-x;rr v;}
I<J T>au&O++(v1<T>&v){fe(x,v)x++;rr v;}
I<J T>au&O--(v1<T>&v){fe(x,v)x--;rr v;}
I<J T>au O++(v1<T>&v,int){au r=v;++v;rr r;}
I<J T>au O--(v1<T>&v,int){au r=v;--v;rr r;}
I<J T>au&O+=(v1<T>&v,cs v1<T>&u){fo(i,len(v))v[i]+=u[i];rr v;}
I<J T>au&O-=(v1<T>&v,cs v1<T>&u){fo(i,len(v))v[i]-=u[i];rr v;}
I<J T>au&O*=(v1<T>&v,cs v1<T>&u){fo(i,len(v))v[i]*=u[i];rr v;}
I<J T>au O+(v1<T>v,cs v1<T>&u){rr v+=u;}
I<J T>au O-(v1<T>v,cs v1<T>&u){rr v-=u;}
I<J T>au O*(v1<T>v,cs v1<T>&u){rr v*=u;}
I<J T>au&O+=(v1<T>&v,cs bitset<60>&a){fo(i,len(v))v[i]+=a[i];rr v;}
I<J T>au&O-=(v1<T>&v,cs bitset<60>&a){fo(i,len(v))v[i]-=a[i];rr v;}
I<J T>au&O+=(v1<T>&v,cs bitset<30>&a){fo(i,len(v))v[i]+=a[i];rr v;}
I<J T>au&O-=(v1<T>&v,cs bitset<30>&a){fo(i,len(v))v[i]-=a[i];rr v;}
I<J T,J U>au&O+=(v1<T>&v,cs U&a){fe(x,v)x+=a;rr v;}
I<J T,J U>au&O-=(v1<T>&v,cs U&a){fe(x,v)x-=a;rr v;}
I<J T,J U>au&O*=(v1<T>&v,cs U&a){fe(x,v)x*=a;rr v;}
I<J T,J U>au&O/=(v1<T>&v,cs U&a){fe(x,v)x/=a;rr v;}
I<J T,J U>au&O%=(v1<T>&v,cs U&a){fe(x,v)x%=a;rr v;}
I<J T,J U>au O+(v1<T>v,cs U&a){rr v+=a;}
I<J T,J U>au O-(v1<T>v,cs U&a){rr v-=a;}
I<J T,J U>au O*(v1<T>v,cs U&a){rr v*=a;}
I<J T,J U>au O/(v1<T>v,cs U&a){rr v/=a;}
I<J T,J U>au O%(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){T r=max(bg(a));fe(v,a)amax(r,max(v));rr r;}
I<J T>au min(cs v2<T>&a){T r=min(bg(a));fe(v,a)amin(r,min(v));rr r;}
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...});}
ll mex(cs sl&s){ll r=0;wh(s.count(r))r++;rr r;}
ll mex(cs u1&v){rr mex(sl(al(v)));}
I<J...T>ll mex(cs T&...a){sl s;fe(x,{a...})s.insert(x);rr mex(s);}
I<J T>ll argmax(cs v1<T>&a){rr itmax(a)-bgn(a);}
I<J T>ll argmin(cs v1<T>&a){rr itmin(a)-bgn(a);}
I<J T,J U>au gcd(cs T&a,cs U&b){rr std::gcd(a,b);}
I<J T,J...A>au gcd(cs T&a,cs A&...b){rr gcd(a,gcd(b...));}
I<J T>T sum(cs set<T>&s){T r=T();fe(x,s)r+=x;rr r;}
I<J T>T sum(cs v1<T>&a){rr accumulate(al(a),T());}
I<J T,J U>ll lbs(cs v1<T>&a,cs U&b){rr lb(al(a),b)-bgn(a);}
I<J T,J U>ll ubs(cs v1<T>&a,cs U&b){rr ub(al(a),b)-bgn(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 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,v,m)r.eb(k);rr r;}
I<J T,J U>au vals(cs map<T,U>&m){v1<U>r;fe(k,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&O>>(is&i,mint&x){ll t;i>>t;x=t;rr i;}
os&O<<(os&o,cs mint&x){rr o<<x.val();}
I<J T,J U>is&O>>(is&i,pair<T,U>&p){rr i>>p.first>>p.second;}
I<J T,J U>os&O<<(os&o,cs pair<T,U>&p){rr o<<p.first<<sp<<p.second;}
I<J T,J U>os&O<<(os&o,cs map<T,U>&m){fe(p,m)o<<p<<(&p==&rb(m)?nc:nl);rr o;}
I<J T>os&O<<(os&o,cs set<T>&s){fe(x,s)o<<x<<sp;rr o;}
I<J T>os&O<<(os&o,cs mset<T>&s){fe(x,s)o<<x<<sp;rr o;}
I<J T>os&O<<(os&o,deque<T>q){wh(yu(q))o<<pop(q)<<sp;rr o;}
I<J T>os&O<<(os&o,pqmax<T>q){wh(yu(q))o<<pop(q)<<sp;rr o;}
I<J T>os&O<<(os&o,pqmin<T>q){wh(yu(q))o<<pop(q)<<sp;rr o;}
I<J T>is&O>>(is&i,v1<T>&v){fe(x,v)i>>x;rr i;}
I<J T>os&O<<(os&o,cs v1<T>&v){fe(x,v)o<<x<<(&x==&rb(v)?nc:sp);rr o;}
I<J T>os&O<<(os&o,cs v2<T>&v){fe(x,v)o<<x<<(&x==&rb(v)?nc:nl);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>ll li(T&...a){(cin>>...>>a);rr 0;}
I<J...T>vo vi(ll i,T&...a){(cin>>...>>a[i]);}
I<J T,J...A>vo vi(v1<T>&a,A&...b){fo(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);}
au gi(u2&g,ll m,bo d=1){
  u1 a(m),b(m);
  fo(i,m){
    li(a[i],b[i]);a[i]--,b[i]--;
    g[a[i]].eb(b[i]);
    if(d)g[b[i]].eb(a[i]);
  }
  rr cp(a,b);
}
au gi(graph&g,ll m,bo w=1,bo d=1){
  u1 a(m),b(m),c(m);
  fo(i,m){
    li(a[i],b[i]);a[i]--,b[i]--;
    c[i]=1;if(w)li(c[i]);
    g[a[i]].eb(b[i],c[i]);
    if(d)g[b[i]].eb(a[i],c[i]);
  }
  rr tr(a,b,c);
}
au ti(u2&g,bo d=1){rr gi(g,len(g)-1,d);}
au ti(graph&g,bo w=1,bo d=1){rr gi(g,len(g)-1,w,d);}


I<J F>ss rcl:private F{
  explicit rcl(F&&f):F(forward<F>(f)){}
  I<J...T>decltype(au)O()(T&&...a)cs{rr F::O()(*th,forward<T>(a)...);}
};
I<J T,J...A>au tzp(A&...a){v1<T>v;fe(e,{a...})v^=e;uq(v);rr v;}
vo pz(cs u1&v,u1&h){fe(x,h)x=lbs(v,x);}
I<J...T>vo pz(cs u1&v,u1&h,T&...t){pz(v,h);pz(v,t...);}
I<J...T>u1 zp(T&...a){u1 v=tzp<ll>(a...);pz(v,a...);rr v;}
I<J T>vo sv(cs u1&){}
I<J T>vo sv(cs u1&o,v1<T>&a){au c=a;fo(i,len(a))a[i]=c[o[i]];}
I<J T,J...A>vo sv(cs u1&o,v1<T>&a,A&...b){sv(o,a);sv(o,b...);}
I<J T>u1 vs(cs fn<bo(ll,ll)>&f,v1<T>&a){u1 o=io(len(a));sort(al(o),f);sv(o,a);rr o;}
I<J T,J...A>u1 vs(cs fn<bo(ll,ll)>&f,v1<T>&a,A&...b){u1 o=io(len(a));sort(al(o),f);sv(o,a);sv(o,b...);rr o;}


ll bsl(cs fn<bo(ll)>&j,ll o,ll n){wh(ab(o-n)>1)(j((o+n)/2)?o:n)=(o+n)/2;rr o;}
dd bsd(cs fn<bo(dd)>&j,dd o,dd n){wh(ab(o-n)>ee)(j((o+n)/2)?o:n)=(o+n)/2;rr o;}
I<J T>au zt(v1<T>a){fo(i,1,len(a))a[i]+=a[i-1];rr a;}
I<J T>au mb(v1<T>a){of(i,len(a),1)a[i]-=a[i-1];rr a;}
I<J T>au zt(v2<T>a){fe(v,a)v=zt(v);fo(i,1,len(a))a[i]+=a[i-1];rr a;}
I<J T>au mb(v2<T>a){fe(v,a)v=mb(v);of(i,len(a),1)a[i]-=a[i-1];rr a;}
I<J T>au ztdiv(v1<T>a){ll n=len(a);fo(i,1,n)fo(j,i*2,n,i)a[i]+=a[j];rr a;}
I<J T>au mbdiv(v1<T>a){ll n=len(a);of(i,n,1)fo(j,i*2,n,i)a[i]-=a[j];rr a;}
au rle(u1 a){v1<cl>r;fe(x,a)yu(r)&&x==rb(r).a?rb(r).b++,0:(r.eb(x,1),0);rr r;}
au rce(u1 a){v1<cl>r;ml m;fe(x,a)m[x]++;fe(k,v,m)r.eb(k,v);rr r;}
I<J T>T esq(T x,ll n,T e){wh(n){if(n&1)e*=x;x*=x;n/=2;}rr e;}
vo wafl(u2&d){ll n=len(d);fo(k,n)fo(i,n)fo(j,n)amin(d[i][j],d[i][k]+d[k][j]);}
u1 divs(ll n){ml m;fo(i,1,n/i+1)if(n%i==0)m[i]++,m[n/i]++;rr keys(m);}
ml fact(ll n){ml m;fo(i,2,n/i+1)wh(n%i==0)m[i]++,n/=i;if(n>1)m[n]++;rr m;}
bo isp(ll n){ml fa=fact(n);rr len(fa)==1&&bg(fa).second==1;}
ll inv(ll a,ll m){a=(a%m+m)%m;ll b=m,u=1,v=0;wh(b)u-=a/b*v,sw(u,v),a-=a/b*b,sw(a,b);rr(u%m+m)%m;}
ll pow(ll x,ll n,ll m=inf){if(n<0)n=-n,x=inv(x,m);ll r=1;wh(n){if(n&1)r*=x,r%=m;x*=x,x%=m,n/=2;}rr r;}


I<J T>ss tww{
  v1<T>fa,rf,bs;
  v2<T>m;
  v2<bo>u;
  tww(ll n){
    n++;
    rs(fa,n);rs(rf,n);rs(bs,n);
    fa[0]=rf[0]=bs[0]=1;
    fo(i,1,n)fa[i]=fa[i-1]*i;
    rf[n-1]=fa[n-1].inv();
    of(i,n-1)rf[i]=rf[i+1]*(i+1);
    
    fo(i,1,n)bs[i]=bs[i-1]+rf[i]*pow(-1,i&1);
    if(n<=5000)rs(m,n,n),rs(u,n,n);
  }
  T c(ll n,ll k){
    if(k<0)rr 0;
    if(n<0)rr h(-n,k)*(ev(k)*2-1);
    rr n-k<0?0:fa[n]*rf[k]*rf[n-k];
  }
  T p(ll n,ll k){rr c(n,k)*fa[k];}
  T h(ll n,ll r){rr c(n+r-1,r);}//n種類の選択肢から重複を許してr個選ぶ．
  T s(ll n,ll k){T r=0;fo(i,k+1)r+=c(k,i)*T(i).pow(n)*pow(-1,k-i);rr r*rf[k];}
  T b(ll n,ll k){T r=0;fo(i,k+1)r+=T(i).pow(n)*rf[i]*bs[k-i];rr r;}
  T par(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&&k==0)rr 1;
    if(k==0)rr 0;
    if(u[n][k])rr m[n][k];
    if(n-k>=0)rr u[n][k]=1,m[n][k]=p(n,k-1)+p(n-k,k);
    rr u[n][k]=1,m[n][k]=p(n,k-1);
  }
};


ss ufin{
  ll n;
  u1 d;
  ufin():n(0){}
  ufin(ll n):n(n),d(n,-1){}
  ll ldr(ll a){as(in(0,a,n));if(d[a]<0)rr a;rr d[a]=ldr(d[a]);}
  bo same(ll a,ll b){as(in(0,a,n)&&in(0,b,n));rr ldr(a)==ldr(b);}
  ll size(ll a){as(in(0,a,n));rr-d[ldr(a)];}
  ll mrg(ll a,ll b){
    as(in(0,a,n)&&in(0,b,n));
    ll x=ldr(a),y=ldr(b);
    if(x==y)rr x;
    if(-d[x]<-d[y])sw(x,y);
    d[x]+=d[y];
    d[y]=x;
    rr x;
  }
  u2 groups(){
    u1 l(n);u2 g(n);
    fo(i,n)l[i]=ldr(i);fo(i,n)g[l[i]].eb(i);fo(i,n)srt(g[i]);srt(g);
    g.erase(remove_if(al(g),[&](cs u1&v){rr mu(v);}),end(g));
    rr g;
  }
};


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());
    fo(i,n)d[i+A]=v[i];
    of(i,A,1)upd(i);
  }
  ll size()cs{rr n;}
  vo upd(ll k){d[k]=op(d[k*2],d[k*2+1]);}
  vo set(ll i,S x){as(in(0,i,n));i+=A;d[i]=x;fo(j,1,log+1)upd(i>>j);}
  vo add(ll i,S x){set(i,get(i)+x);}
  S get(ll i)cs{as(in(0,i,n));rr d[i+A];}
  S aprd(){rr d[1];}
  S prd(ll l,ll r){
    as(AND(0<=l,l<=r,r<=n));
    S sml=e(),smr=e();
    l+=A,r+=A;
    wh(l<r){if(od(l))sml=op(sml,d[l++]);if(od(r))smr=op(d[--r],smr);l/=2,r/=2;}
    rr op(sml,smr);
  }
  fd os&O<<(os&o,cs sgt&s){fo(i,len(s))o<<s.get(i)<<sp;rr o;}
};


namespace fft{
  uu real=double;
  ss C{
    real x,y;
    C():x(0),y(0){}
    C(real x,real y):x(x),y(y){}
    inline C operator+(cs C &c)cs{rr C(x+c.x,y+c.y);}
    inline C operator-(cs C &c)cs{rr C(x-c.x,y-c.y);}
    inline C operator*(cs C &c)cs{rr C(x*c.x-y*c.y,x*c.y+y*c.x);}
    inline C conj()cs{rr C(x,-y);}
  };

  cs real PI=acosl(-1);
  ll base=1;
  v1<C>rts={{0,0},{1,0}};
  v1<int>rev={0,1};

  vo ensure_base(int nbase){
    if(nbase<=base)rr;
    rev.resize(1<<nbase);
    rts.resize(1<<nbase);
    fo(i,1<<nbase)rev[i]=(rev[i>>1]>>1)+((i&1)<<(nbase-1));
    wh(base<nbase){
      real angle=PI*2.0/(1<<(base+1));
      fo(i,1<<(base-1),1<<base){
        rts[i<<1]=rts[i];
        real angle_i=angle*(2*i+1-(1<<base));
        rts[(i<<1)+1]=C(cos(angle_i),sin(angle_i));
      }
      ++base;
    }
  }

  vo fft(v1<C>&a,int n){
    as((n&(n-1))==0);
    int zeros=__builtin_ctz(n);
    ensure_base(zeros);
    int shift=base-zeros;
    fo(i,n)if(i<(rev[i]>>shift))sw(a[i],a[rev[i]>>shift]);
    
    for(int k=1;k<n;k<<=1){
      for(int i=0;i<n;i+=2*k){
        for(int j=0;j<k;j++){
          C z=a[i+j+k]*rts[j+k];
          a[i+j+k]=a[i+j]-z;
          a[i+j]=a[i+j]+z;
        }
      }
    }
  }
};


I<J T >ss arbitrary_mod_convolution{
  using real=fft::real;
  using C=fft::C;
  arbitrary_mod_convolution()=default;

  v1<T>multiply(cs v1<T>&a,cs v1<T>&b,int need=-1){
    if(need==-1)need=len(a)+len(b)-1;
    int nbase=0;
    wh((1<<nbase)<need)nbase++;
    fft::ensure_base(nbase);
    int sz=1<<nbase;
    v1<C>fa(sz);
    fo(i,len(a))fa[i]=C(a[i].val()&((1<<15)-1),a[i].val()>>15);
    
    fft::fft(fa,sz);
    v1<C>fb(sz);
    if(a==b){
      fb=fa;
    }el{
      fo(i,len(b))fb[i]=C(b[i].val()&((1<<15)-1),b[i].val()>>15);
      fft::fft(fb,sz);
    }
    real ratio=0.25/sz;
    C r2(0,-1),r3(ratio,0),r4(0,-ratio),r5(0,1);
    for(int i=0;i<=(sz>>1);i++){
      int j=(sz-i)&(sz-1);
      C a1=(fa[i]+fa[j].conj());
      C a2=(fa[i]-fa[j].conj())*r2;
      C b1=(fb[i]+fb[j].conj())*r3;
      C b2=(fb[i]-fb[j].conj())*r4;
      if(i!=j){
        C c1=(fa[j]+fa[i].conj());
        C c2=(fa[j]-fa[i].conj())*r2;
        C d1=(fb[j]+fb[i].conj())*r3;
        C d2=(fb[j]-fb[i].conj())*r4;
        fa[i]=c1*d1+c2*d2*r5;
        fb[i]=c1*d2+c2*d1;
      }
      fa[j]=a1*b1+a2*b2*r5;
      fb[j]=a1*b2+a2*b1;
    }
    fft::fft(fa,sz);
    fft::fft(fb,sz);
    v1<T>ret(need);
    fo(i,need){
      int64_t aa=llround(fa[i].x);
      int64_t bb=llround(fb[i].x);
      int64_t cc=llround(fa[i].y);
      aa=T(aa).val(),bb=T(bb).val(),cc=T(cc).val();
      ret[i]=aa+(bb<<15)+(cc<<30);
    }
    rr ret;
  }
};
uu amc=arbitrary_mod_convolution<mint>;


I<J T>ss formal_power_series:v1<T>{
  uu v1<T>::v1;
  uu P=formal_power_series;
  
  uu MULT=fn<P(P,P)>;

  sc MULT&get_mult(){sc MULT mult=nu;rr mult;}
  sc vo set_fft(MULT f){get_mult()=f;}
  
  P&O+=(cs P&f){
    if(len(f)>len(*th))th->resize(len(f));
    fo(i,len(f))(*th)[i]+=f[i];
    rr*th;
  }
  P&O-=(cs P&f){
    if(len(f)>len(*th))th->resize(len(f));
    fo(i,len(f))(*th)[i]-=f[i];
    rr*th;
  }
  
  P&O+=(cs T&t){if(mu(*th))th->resize(1);(*th)[0]+=t;rr*th;}
  P&O-=(cs T&t){if(mu(*th))th->resize(1);(*th)[0]-=t;rr*th;}
  P&O*=(cs T&t){fo(i,len(*th))(*th)[i]*=t;rr*th;}
  
  P&O*=(cs P&f){
    if(mu(*th)||mu(f))rr*th=P();
    as(get_mult());
    rr*th=get_mult()(*th,f);
  }
  P&O%=(cs P&f){*th-=*th/f*f;shrink();rr*th;}
  
  P&O/=(cs P&f){
    if(len(*th)<len(f))rr*th=P();
    ll n=len(*th)-len(f)+1;
    rr*th=(rev().pre(n)*f.rev().inv(n)).pre(n).rev(n);
  }
  
  P O+(cs P&f)cs{rr P(*th)+=f;}
  P O-(cs P&f)cs{rr P(*th)-=f;}
  P O*(cs P&f)cs{rr P(*th)*=f;}
  P O/(cs P&f)cs{rr P(*th)/=f;}
  P O%(cs P&f)cs{rr P(*th)%=f;}
  P O+(cs T&t)cs{rr P(*th)+=t;}
  P O-(cs T&t)cs{rr P(*th)-=t;}
  P O*(cs T&t)cs{rr P(*th)*=t;}
  P O-()cs{P r=*th;fe(x,r)x=-x;rr r;}
  P O>>(ll sz)cs{if(len(*th)<=sz)rr P();P r(*th);r.erase(al(r,sz));rr r;}
  P O<<(ll sz)cs{P r(*th);r.insert(bgn(r),sz,T(0));rr r;}
  vo shrink(){wh(yu(*th)&&rb(*th)==T(0))pop(*th);}
  P pre(ll deg)cs{rr P(al(*th,min(len(*th),deg)));}//mod x^deg
  P rev(ll deg=-1)cs{P r(*th);if(deg!=-1)r.resize(deg,T(0));rv(r);rr r;}
  T eval(T x)cs{T r=0,w=1;fe(v,*th)r+=w*v,w*=x;rr r;}
  
  
  P dif()cs{
    ll n=len(*th);
    P r(max(n-1,0));
    fo(i,1,n)r[i-1]=(*th)[i]*i;
    rr r;
  }
  P integral()cs{
    ll n=len(*th);
    P r(n+1);
    r[0]=T(0);
    fo(i,n)r[i+1]=(*th)[i]/T(i+1);
    rr r;
  }
  
  P inv(ll deg=-1)cs{
    as((*th)[0]!=T(0));
    ll n=len(*th);
    if(deg==-1)deg=n;
    P r{T(1)/bg(*th)};
    
    for(ll i=1;i<deg;i<<=1)r=(r+r-sq(r)*pre(i<<1)).pre(i<<1);
    rr r.pre(deg);
  }
  
  P log(ll deg=-1)cs{
    as((*th)[0]==T(1));
    ll n=len(*th);
    if(deg==-1)deg=n;
    rr(th->dif()*th->inv(deg)).pre(deg-1).integral();
  }
  
  P exp(ll deg=-1)cs{
    as((*th)[0]==T(0));
    ll n=len(*th);
    if(deg==-1)deg=n;
    P r{1};
    for(ll i=1;i<deg;i<<=1)r=(r*(pre(i<<1)+T(1)-r.log(i<<1))).pre(i<<1);
    rr r.pre(deg);
  }
  
  P sqrt(ll deg=-1)cs{
    ll n=len(*th);
    if(deg==-1)deg=n;
    
    if((*th)[0]==T(0)){
      fo(i,1,n){
        if((*th)[i]!=T(0)){
          if(i&1)rr P();
          if(deg-i/2<=0)bk;
          au r=(*th>>i).sqrt(deg-i/2);
          if(mu(r))rr P();
          r=r<<(i/2);
          if(len(r)<deg)r.resize(deg,T(0));
          rr r;
        }
      }
      rr P(deg,T(0));
    }
    
    au s=msqr((*th)[0].val(),mint::mod());
    if(sq(s)!=(*th)[0])rr P();
    
    P r{s};
    T inv2=T(1)/T(2);
    for(ll i=1;i<deg;i<<=1)r=(r+pre(i<<1)*r.inv(i<<1))*inv2;
    rr r.pre(deg);
  }
  
  P pow(ll k,ll deg=-1)cs{
    ll n=len(*th);
    if(deg==-1)deg=n;
    
    fo(i,n){
      if((*th)[i]!=T(0)){
        T rev=T(1)/(*th)[i];
        P c(*th*rev),d(n-i);
        fo(j,i,n)d[j-i]=c[j];
        d=(d.log(deg)*T(k)).exp(deg)*(*th)[i].pow(k);
        P e(deg);
        if(i*k>deg)rr e;
        for(ll j=0;j<min(deg-i*k,len(d));j++)e[j+i*k]=d[j];
        rr e;
      }
    }
    rr*th;
  }
};
uu fps=formal_power_series<mint>;


mint kitamasa(ll n,fps p,fps q){//[x^n]\frac{p(x)}{q(x)}
  q.shrink();
  mint ret=0;
  if(len(p)>=len(q)){
    au r=p/q;
    p-=r*q;
    p.shrink();
    if(n<len(r))ret+=r[n];
  }
  if(mu(p))rr ret;

  p.resize(len(q)-1);
  wh(n){
    au q2=q;
    fo(i,1,len(q2),2)q2[i]=-q2[i];
    au s=p*q2;
    au t=q*q2;
    if(n&1){
      fo(i,1,len(s),2)p[i>>1]=s[i];
      fo(i,0,len(t),2)q[i>>1]=t[i];
    }el{
      fo(i,0,len(s),2)p[i>>1]=s[i];
      fo(i,0,len(t),2)q[i>>1]=t[i];
    }
    n>>=1;
  }
  rr ret+p[0];
}


vo main(){
  ll T=1;
  //li(T);
  fo(T)solve();
}
vo solve(){
  amc fft;
  au mul=[&](cs fps&a,cs fps&b){
    au r=convolution(a,b);
    rr fps(al(r));
  };
  fps::set_fft(mul);
  
  LL(N,M);
  UL(N,a);
  
  uu T=tr<ll,fps,fps>;
  au cmp=[](T a,T b){rr a.a<b.a;};
  
  priority_queue<T,v1<T>,decltype(cmp)>q{cmp};
  fo(i,N)q.ep(2,fps{1},fps{1,-a[i]});
  
  wh(len(q)>1){
    au[n1,g1,f1]=q.top();q.pop();
    au[n2,g2,f2]=q.top();q.pop();
    q.ep(n1+n2,f1*g2+f2*g1,f1*f2);
  }
  
  au ret=q.top().c.inv(M+1)*q.top().b;
  pp(fps(al(ret,1,M+1)));
}}