#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#include<atcoder/all>
#include<bits/stdc++.h>
//#include<boost/multiprecision/cpp_int.hpp>
#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 eb(...) emplace_back(__VA_ARGS__)
#define iif if
#define ef(...) 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 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 rs(...) resize(__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 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)(__VA_ARGS__)
#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)(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;li(__VA_ARGS__)
#define DD(...) dd __VA_ARGS__;li(__VA_ARGS__)
#define CH(...) char __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 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)(__VA_ARGS__)
#define U23(h,w,a) u2 a(h,w);li(a)
#define U24(h,w,a,b) u2 a(h,w),b(h,w);li(a,b)
#define U25(h,w,a,b,c) u2 a(h,w),b(h,w),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 I template
#define J typename
#define O operator
#define rr return
#define ss struct
#define uu using
#define au auto
#define bk break
#define cs const
#define ct continue
#define th this
#define endl "\n"
namespace atcoder{}
namespace my{void main();void solve();}
int main(){my::main();}
namespace my{
//uu bint=boost::multiprecision::cpp_int;
uu namespace std;
uu namespace atcoder;
uu vo=void;
uu bo=bool;
uu is=istream;
uu os=ostream;
uu i128=__int128_t;
uu ll=long long;
uu dd=long double;
uu ul=unsigned long long;
uu ml=modint998244353;
is&O>>(is&i,ml&x){ll t;i>>t;x=t;rr i;}
os&O<<(os&o,cs ml&x){rr o<<x.val();}

I<J T>uu v1=vector<T>;
I<J T>ss v2:v1<v1<T>>{uu v1<v1<T>>::v1;v2(ll a,ll b,T x=T{}){th->rs(a,v1<T>(b,x));}};
I<J T>ss v3:v1<v2<T>>{uu v1<v2<T>>::v1;v3(ll a,ll b,ll c,T x=T{}){th->rs(a,v2<T>(b,c,x));}};
I<J T>ss v4:v1<v3<T>>{uu v1<v3<T>>::v1;v4(ll a,ll b,ll c,ll d,T x=T{}){th->rs(a,v3<T>(b,c,d,x));}};
I<J T>ss v5:v1<v4<T>>{uu v1<v4<T>>::v1;v5(ll a,ll b,ll c,ll d,ll e,T x=T{}){th->rs(a,v4<T>(b,c,d,e,x));}};
uu u1=v1<ll>;uu u2=v2<ll>;uu u3=v3<ll>;uu u4=v4<ll>;
uu m1=v1<ml>;uu m2=v2<ml>;uu m3=v3<ml>;uu m4=v4<ml>;uu m5=v5<ml>;
uu str=string;uu strs=v1<str>;
str sp{" "},nc{""},nl{"\n"};

I<J A,J B=A>ss cp{
  A a;B b;
  cp():a(A{}),b(B{}){}
  cp(A a,B b):a(a),b(b){}
  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;}
  I<J T>cp&O+=(cs T&x){a+=x,b+=x;rr*th;}I<J T>cp O+(cs T&x)cs{rr cp{*th}+=x;}
  I<J T>cp&O-=(cs T&x){a-=x;b-=x;rr*th;}I<J T>cp O-(cs T&x)cs{rr cp{*th}-=x;}
  I<J T>cp&O*=(cs T&x){a*=x;b*=x;rr*th;}I<J T>cp O*(cs T&x)cs{rr cp{*th}*=x;}
  au abs()cs{rr std::abs(a)+std::abs(b);}//manhattan
  au abs(cs cp&c)cs{rr(*th-c).abs();}
  bo O==(cs cp&c)cs{rr 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 a!=c.a?a>c.a:b>c.b;}
  friend is&O>>(is&i,cp&c){rr i>>c.a>>c.b;}
  friend os&O<<(os&o,cs cp&c){rr o<<c.a<<sp<<c.b;}
};
I<J A,J B=A,J C=A>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 a==t.a&&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;}
  bo O>(cs tr&t)cs{rr a!=t.a?a>t.a:b!=t.b?b>t.b:c>t.c;}
  friend is&O>>(is&i,tr&t){rr i>>t.a>>t.b>>t.c;}
  friend os&O<<(os&o,cs tr&t){rr o<<t.a<<sp<<t.b<<sp<<t.c;}
};
I<J A,J B=A,J C=A,J D=A>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 a==q.a&&b==q.b&&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;}
  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;}
  friend os&O<<(os&o,cs qu&q){rr o<<q.a<<sp<<q.b<<sp<<q.c<<sp<<q.d;}
};
uu cl=cp<ll>;uu cls=v1<cl>;uu tl=tr<ll>;uu tls=v1<tl>;uu ql=qu<ll>;uu qls=v1<ql>;

I<J T>uu fn=function<T>;
I<J T>uu qmax=priority_queue<T>;
I<J T>uu qmin=priority_queue<T,v1<T>,greater<T>>;
uu mp=map<ll,ll>;uu ump=unordered_map<ll,ll>;
I<J T>ss set:std::set<T>{set(){}set(cs v1<T>&a){fe(e,a)th->ep(e);}};
I<J T>ss unordered_set:std::unordered_set<T>{unordered_set(){}unordered_set(cs v1<T>&a){fe(e,a)th->ep(e);}};
I<J T>ss multiset:std::multiset<T>{multiset(){}multiset(cs v1<T>&a){fe(e,a)th->ep(e);}};
uu sl=set<ll>;uu usl=unordered_set<ll>;uu msl=multiset<ll>;
uu dl=deque<ll>;
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 Yes(bo b=1){rr b?"Yes":"No";}str No(){rr Yes(0);}

bo odd(ll x){rr x&1;}
bo eve(ll x){rr !odd(x);}
ll pm1(ll x){rr 1-2*odd(x);}
ll pw2(ll n){rr 1LL<<n;}
ll rng1(ll l,ll r){rr pw2(r)-pw2(l);}//[l,r)
ll rngbit(ll x,ll l,ll r){rr rng1(l,r)&x;}
ll ppc(ll x,ll l=0,ll r=63){rr __builtin_popcountll(rngbit(x,l,r));}
u1 binary(ll x,ll L){u1 r(L);fo(i,L)r[i]=x&1,x>>=1;rr 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 l2(i128 x){x|=1;ll r=0;wh(x)x>>=1,r++;rr r;}//number of digits in binary
ll l10(i128 x){x|=1;ll r=0;wh(x)x/=10,r++;rr r;}//number of digits in decimal
ll msb(ll x){if(x==0)rr-1;rr l2(x)-1;}
ll lsb(ll x){if(x==0)rr-1;rr __builtin_ctzll(x);}
bo in(ll a,ll x,ll b){rr a<=x&&x<b;}//x in [a,b)
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 mu(cs T&a){rr !len(a);}
I<J T>T sq(cs T&a){rr a*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 cei(T x,U y){assert(y);rr(y<0?cei(-x,-y):(x>0?(x+y-1)/y:x/y));}
I<J T,J U>T flo(T x,U y){assert(y);rr(y<0?flo(-x,-y):(x>0?x/y:x/y-!(x%y)));}
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>T rect(cs v1<T>&s,ll l,ll r){assert(0<=l&&l<=r&&r<=len(s));T t{};if(r)t+=s[r-1];if(l)t-=s[l-1];rr t;}
I<J T,J U=T>au as(cs v1<cp<T,U>>&v){v1<T>r;fe(a,b,v)r.eb(a);rr r;}
I<J T,J U=T>au bs(cs v1<cp<T,U>>&v){v1<U>r;fe(a,b,v)r.eb(b);rr r;}


I<J T>au cut(cs v1<T>&a,ll l,ll r){rr v1<T>(al(a,l,r));}
I<J T,J U=T>vo af(v1<T>&v,U e=U{}){v.ep(bgn(v),e);}
I<J T>vo df(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 pof(deque<T>&q){T r=bg(q);q.pop_front();rr r;}
I<J T>T pob(deque<T>&q){T r=rb(q);q.pop_back();rr r;}
I<J T>T pop(v1<T>&v){T r=rb(v);v.pop_back();rr r;}
I<J T>T pop(qmax<T>&q){T r=q.top();q.pop();rr r;}
I<J T>T pop(qmin<T>&q){T r=q.top();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,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){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){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,int){au r=v;++v;rr r;}
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,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){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){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){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>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*max_element(al(a));}
I<J T>T min(cs v1<T>&a){rr*min_element(al(a));}
I<J T>au max(cs v2<T>&a){T r=bg(bg(a));fe(v,a)amax(r,max(v));rr r;}
I<J T>au min(cs v2<T>&a){T r=bg(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...});}
I<J T>ll argmax(cs v1<T>&a){rr max_element(al(a))-bgn(a);}
I<J T>ll argmin(cs v1<T>&a){rr min_element(al(a))-bgn(a);}
I<J T>T sum(cs v1<T>&a){rr accumulate(al(a),T{});}
I<J T>T sum(cs v2<T>&a){T r{};fe(e,a)r+=sum(e);rr r;}
I<J T>T sum(cs set<T>&s){T r{};fe(x,s)r+=x;rr r;}
I<J T>T sum(deque<T>q){T r{};wh(len(q))r+=pof(q);rr r;}
I<J T>T sum(qmax<T>q){T r{};wh(len(q))r+=pop(q);rr r;}
I<J T>T sum(qmin<T>q){T r{};wh(len(q))r+=pop(q);rr r;}
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);}
u1 io(ll n,ll x=0){u1 a(n);iota(al(a),x);rr a;}

str de(cs u1&a,cs char&b='a'){str r{};fe(x,a)r+=x+b;rr r;}
str de(cs u1&a,cs str&b){ll n=len(a);str r(n,'$');fo(i,n)fo(j,len(b))if(a[i]==j){r[i]=b[j];bk;}rr r;}
u1 en(cs str&s,cs char&b='a'){u1 r;fe(c,s)r.eb(c-b);rr r;}
u1 en(cs str&s,cs string&b){ll n=len(s);u1 r(n,-1);fo(i,n)fo(j,len(b))if(s[i]==b[j]){r[i]=j;bk;}rr r;}
u2 en(cs strs&s,cs char&b='a'){u2 r;fe(e,s)r.eb(en(e,b));rr r;}
u2 en(cs strs&s,cs string&b){u2 r;fe(e,s)r.eb(en(e,b));rr r;}

ss fio{fio(){cin.tie(0);ios::sync_with_stdio(0);cout<<fixed<<setprecision(10);}}fio;
os&O<<(os&o,cs i128&x){if(x<0)rr o<<"-"<<-x;if(x<10)rr o<<(char)(x+'0');rr o<<x/10<<(char)(x%10+'0');}
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,J U>os&O<<(os&o,cs unordered_map<T,U>&m){fe(p,m)o<<p<<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 unordered_set<T>&s){fe(x,s)o<<x<<sp;rr o;}
I<J T>os&O<<(os&o,cs multiset<T>&s){fe(x,s)o<<x<<sp;rr o;}
I<J T>os&O<<(os&o,deque<T>q){wh(len(q))o<<pof(q)<<sp;rr o;}
I<J T>os&O<<(os&o,qmax<T>q){wh(len(q))o<<pop(q)<<sp;rr o;}
I<J T>os&O<<(os&o,qmin<T>q){wh(len(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...);}
str ins(){ST(r);rr r;}
strs ins2(ll n){S2(n,r);rr r;}

ss edg{ll t,w;edg(){}edg(ll t,ll w=1):t(t),w(w){}};
uu graph=v2<edg>;
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;}
au mgi(u2&g,ll m){UV(m,a,b);a--,b--;fo(i,m)g[a[i]].eb(b[i]),g[b[i]].eb(a[i]);rr cp(a,b);}
au ygi(u2&g,ll m){UV(m,a,b);a--,b--;fo(i,m)g[a[i]].eb(b[i]);rr cp(a,b);}
au mgi(graph&g,ll m){UV(m,a,b,c);a--,b--;fo(i,m)g[a[i]].eb(b[i],c[i]),g[b[i]].eb(a[i],c[i]);rr tr(a,b,c);}
au ygi(graph&g,ll m){UV(m,a,b,c);a--,b--;fo(i,m)g[a[i]].eb(b[i],c[i]);rr tr(a,b,c);}
au ti(u2&g){rr mgi(g,len(g)-1);}
au ti(graph&g){rr mgi(g,len(g)-1);}

I<J F>ss rec:private F{explicit rec(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;}
u1 zp(u2&a){u1 v;fe(e,a)v^=e;uq(v);fe(e,a)fe(x,e)x=lbs(v,x);rr v;}
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(abs(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(abs(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;}
au rle(cs u1&a){cls r;fe(x,a)len(r)&&x==rb(r).a?rb(r).b++,0:(r.eb(x,1),0);rr r;}
au rce(cs u1&a){cls r;ump m;fe(x,a)m[x]++;fe(k,v,m)r.eb(k,v);rr uq(r),r;}
u1 divs(ll n){u1 r;fo(i,1,n/i+1)if(n%i==0)r.eb(i),r.eb(n/i);rr uq(r),r;}
mp fact(ll n){ump m;fo(i,2,n/i+1)wh(n%i==0)m[i]++,n/=i;if(n>1)m[n]++;rr mp(al(m));}
i128 inv(i128 a,i128 m){a=(a%m+m)%m;i128 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;}
i128 pow(i128 x,i128 n){assert(n>=0);i128 r=1;wh(n){if(n&1)r*=x;x*=x,n>>=1;}rr r;}
i128 pow(i128 x,i128 n,i128 m){if(n<0)n=-n,x=inv(x,m);i128 r=1;wh(n){if(n&1)r*=x,r%=m;x*=x,x%=m,n>>=1;}rr r;}
au quot_rng(ll n){tls r;ll m=sqr(n),l=n/(m+1);fo(i,1,m+1)r.eb(n/i,i,i+1);of(i,l+1,1)r.eb(i,n/(i+1)+1,n/i+1);rr r;}//[a,b)

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=l2(n*2-1)-1,A=pw2(log),d.rs(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 i){d[i]=op(d[i*2],d[i*2+1]);}
  vo set(ll i,S x){assert(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{assert(in(0,i,n));rr d[i+A];}
  S O[](ll i)cs{assert(in(0,i,n));rr d[i+A];}
  S aprd(){rr d[1];}
  S prd(ll l,ll r){
    assert(0<=l&&l<=r&&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>>=1,r>>=1;}
    rr op(sml,smr);
  }
  friend os&O<<(os&o,cs sgt&s){fo(i,len(s))o<<s.get(i)<<(i==len(s)-1?nc:sp);rr o;}
};

ss ufin{
  ll n;u1 d;
  ufin(ll n):n(n),d(n,-1){}
  ll ldr(ll a){assert(in(0,a,n));if(d[a]<0)rr a;rr d[a]=ldr(d[a]);}
  bo same(ll a,ll b){rr ldr(a)==ldr(b);}
  ll size(ll a){rr-d[ldr(a)];}
  u2 groups(){u2 g(n);fo(i,n)g[ldr(i)].eb(i);fe(e,g)srt(e);uq(g);if(mu(bg(g)))df(g);rr g;}
  ll mrg(ll a,ll b){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;}
};

I<J T>ss twelvefold{
  v1<T>fa,rf;
  twelvefold(ll n):fa(n+1,1),rf(n+1,1){fo(i,1,n+1)fa[i]=fa[i-1]*i;rf[n]=fa[n].inv();of(i,n)rf[i]=rf[i+1]*(i+1);}
  T c(ll n,ll k){rr n<0?pm1(k)*c(-n+k-1,k):k<0||n<k?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-1+r,r);}
};
uu twf=twelvefold<ml>;

vo main(){
  ll T=1;
  //li(T);
  fo(T)solve();
}


//https://ei1333.github.io/luzhiled/snippets/math/fast-fourier-transform.html
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 O+(cs C &c)cs{rr C(x+c.x,y+c.y);}
    inline C O-(cs C &c)cs{rr C(x-c.x,y-c.y);}
    inline C O*(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.rs(1<<nbase);
    rts.rs(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){
    assert((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;
        }
      }
    }
  }
};


//https://ei1333.github.io/luzhiled/snippets/math/arbitrary-mod-convolution.html
I<J T>ss arbitrary_mod_convolution{
  uu real=fft::real;
  uu 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<ml>;


//https://ei1333.github.io/luzhiled/snippets/math/formal-power-series.html
I<J T>ss formal_power_series:v1<T>{
  uu v1<T>::v1;
  uu P=formal_power_series;
  
  uu MULT=fn<P(P,P)>;

  static MULT&get_mult(){static MULT mult=nullptr;rr mult;}
  static vo set_fft(MULT f){get_mult()=f;}
  
  I<J U>formal_power_series(cs v1<U>&a){
    ll n=len(a);
    th->rs(n);
    fo(i,n)(*th)[i]=a[i];
  }
  
  bo O<(cs P&f)cs{rr len(*th)<len(f);}
  bo O>(cs P&f)cs{rr len(*th)>len(f);}
  
  P&O+=(cs P&f){
    if(len(f)>len(*th))th->rs(len(f));
    fo(i,len(f))(*th)[i]+=f[i];
    rr*th;
  }
  P&O-=(cs P&f){
    if(len(f)>len(*th))th->rs(len(f));
    fo(i,len(f))(*th)[i]-=f[i];
    rr*th;
  }
  
  P&O+=(cs T&t){if(mu(*th))th->rs(1);(*th)[0]+=t;rr*th;}
  P&O-=(cs T&t){if(mu(*th))th->rs(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();
    assert(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;}
  I<J U=T>P O+(cs U&t)cs{rr P(*th)+=t;}
  I<J U=T>P O-(cs U&t)cs{rr P(*th)-=t;}
  I<J U=T>P O*(cs U&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{});rr r;}
  vo shrink(){wh(len(*th)&&rb(*th)==T{})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.rs(deg,T{});rv(r);rr r;}
  T O()(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]*T{i};
    rr r;
  }
  P integral()cs{
    ll n=len(*th);
    P r(n+1);
    r[0]=T{};
    fo(i,n)r[i+1]=(*th)[i]/T{i+1};
    rr r;
  }
  
  //https://judge.yosupo.jp/problem/inv_of_formal_power_series
  P inv(ll deg=-1)cs{
    assert((*th)[0]!=T{});
    ll n=len(*th);
    if(deg==-1)deg=n;
    P r{T{1}/(*th)[0]};
    
    for(ll i=1;i<deg;i<<=1)r=(r+r-sq(r)*pre(i<<1)).pre(i<<1);
    rr r.pre(deg);
  }
  
  //https://judge.yosupo.jp/problem/log_of_formal_power_series
  P log(ll deg=-1)cs{
    assert((*th)[0]==T{1});
    ll n=len(*th);
    if(deg==-1)deg=n;
    rr(th->dif()*th->inv(deg)).pre(deg-1).integral();
  }
  
  //https://judge.yosupo.jp/problem/exp_of_formal_power_series
  P exp(ll deg=-1)cs{
    assert((*th)[0]==T{});
    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);
  }
  
  //https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
  P sqr(ll deg=-1)cs{
    ll n=len(*th);
    if(deg==-1)deg=n;
    
    if((*th)[0]==T{}){
      fo(i,1,n){
        if((*th)[i]!=T{}){
          if(i&1)rr{};//no solution
          if(deg-i/2<=0)bk;
          au r=(*th>>i).sqr(deg-i/2);
          if(mu(r))rr{};//no solution
          r=r<<(i/2);
          if(len(r)<deg)r.rs(deg,T{});
          rr r;
        }
      }
      rr P(deg,T{});
    }
    
    au s=msqr((*th)[0].val(),ml::mod());
    if(sq(s)!=(*th)[0])rr{};//no solution
    
    P r{s};
    T iv2=T{1}/T{2};
    for(ll i=1;i<deg;i<<=1)r=(r+pre(i<<1)*r.inv(i<<1))*iv2;
    rr r.pre(deg);
  }
  
  //https://judge.yosupo.jp/problem/pow_of_formal_power_series
  P pow(i128 k,ll deg=-1)cs{
    ll n=len(*th);
    if(deg==-1)deg=n;
    if(k==0){P r(deg);r[0]=1;rr r;}
    
    fo(i,n){
      if((*th)[i]!=T{}){
        T rv=T{1}/(*th)[i];
        P r=(((*th*rv)>>i).log()*k).exp()*((*th)[i].pow(k));
        if(i*k>deg)rr P(deg,T{});
        r=(r<<(i*k)).pre(deg);
        if(len(r)<deg)r.rs(deg,T{});
        rr r;
      }
    }
    rr*th;
  }

  //https://judge.yosupo.jp/problem/polynomial_taylor_shift
  P taylor_shift(ll C,ll deg=-1){//f(x+C)
    ll n=len(*th);
    if(deg==-1)deg=n;
    
    m1 c(n,1);
    fo(i,1,n)c[i]=c[i-1]*C;
    
    twf tw(n);
    
    P f(n),g(n);
    fo(i,n){
      f[i]=(*th)[i]*tw.fa[i];
      g[n-1-i]=c[i]*tw.rf[i];
    }
    f*=g;
    fo(i,n)f[i+n-1]*=tw.rf[i];
    rr cut(f,n-1,n*2-1);
  }
  
};
uu fps=formal_power_series<ml>;


//https://nyaannyaan.github.io/library/fps/kitamasa.hpp.html
ml kitamasa(ll n,fps p,fps q){//[x^n]\frac{p(x)}{q(x)}
  q.shrink();
  ml 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.rs(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];
}


//k項間漸化式 a_n=\sum_{i=1}^k c_i a_{n-i}のn項目
//a_0,a_1,...,a_{k-1}が分かっている
ml kitamasa(ll n,cs m1&a,cs m1&c){
  assert(len(c)==len(a));
  ll k=len(c);
  au t=m1{1}+(-c);
  fps q(al(t));
  fps p=(q*fps(al(a))).pre(k);
  rr kitamasa(n,p,q);
}


fps prod(cs v1<fps>&fs){
  qmin<fps>q;//マージテク
  fe(f,fs)q.ep(f);
  wh(len(q)>1){
    au f=pop(q),g=pop(q);
    q.ep(f*g);
  }
  rr pop(q);
}


vo solve(){
  amc fft;
  au mul=[&](cs fps&a,cs fps&b){
    au r=fft.multiply(a,b);
    //au r=convolution(a,b);
    rr fps(r);
  };
  fps::set_fft(mul);
  
  
  LL(N);
  UL(N,a);
  ll S=sum(a);
  ll P=999630629;
  ll M=S-P+1;
  
  if(M<=0)fin(ml(2).pow(N-1)*S);
  
  m1 c(M);
  fe(e,a)if(e<M)c[e]++;
  
  m1 iv(M);
  iv[1]=1;
  ll p=ml::mod();
  fo(i,2,M)iv[i]=-iv[p%i]*(p/i);
  
  fps f(M);
  fo(i,1,M){
    fo(n,M/i+1){
      f[i*n]+=pm1(n-1)*iv[n]*c[i];
    }
  }
  f=f.exp();
  f/=fps{1,-1};
  pp(S*ml(2).pow(N-1)-P*f[M-1]);
}}