#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#include<atcoder/all>
#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 eb(...) emplace_back(__VA_ARGS__)
#define ef(...) 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 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 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 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 as assert
#define au auto
#define bk break
#define cs const
#define ct continue
#define th this
#define endl "\n"
namespace my{void main();void solve();}
int main(){my::main();}
namespace my{
uu namespace std;
uu namespace atcoder;
uu vo=void;
uu bo=bool;
uu is=istream;
uu os=ostream;
uu mint=modint998244353;
uu i128=__int128_t;
uu ll=long long;
uu dd=long double;
uu ul=unsigned long long;
uu chr=char;
string 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 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 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 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 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 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;}
};

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){th->rs(a,v1<T>(b));}v2(ll a,ll b,T x){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){th->rs(a,v2<T>(b,c));}v3(ll a,ll b,ll c,T x){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){th->rs(a,v3<T>(b,c,d));}v4(ll a,ll b,ll c,ll d,T x){th->rs(a,v3<T>(b,c,d,x));}};
uu u1=v1<ll>;uu u2=v2<ll>;uu u3=v3<ll>;uu u4=v4<ll>;
uu m1=v1<mint>;uu m2=v2<mint>;uu m3=v3<mint>;uu m4=v4<mint>;

ss str:v1<chr>{
  uu v1<chr>::v1;
  str(cs string&s){fe(c,s)th->eb(c);}
  str(cs chr*s){fe(c,string(s))th->eb(c);}
  str(cs u1&a,cs chr&b='a'){fe(x,a)th->eb(x+b);}
  str(cs u1&a,cs string&b){ll n=a.size();th->rs(n);fo(i,n)fo(j,b.size())if(a[i]==j){(*th)[i]=b[j];bk;}}
  
  str&O+=(cs str&s){fe(c,s)th->eb(c);rr*th;}str O+(cs str&s){rr str{*th}+=s;}
  str&O+=(cs string&s){str t{s};rr*th+=t;}str O+(cs string&s){rr str{*th}+=s;}
  str&O+=(cs chr*s){str t{string(s)};rr*th+=t;}str O+(cs chr*s){rr str{*th}+=s;}
  str&O+=(cs chr&c){th->eb(c);rr*th;}str O+(cs chr&c){rr str{*th}+=c;}
  I<J T>str&O+=(cs T&x){str t{to_string(x)};rr*th+=t;}I<J T>str O+(cs T&x){rr str{*th}+=x;}
  u1 en(cs chr&b='a'){u1 r;fe(c,*th)r.eb(c-b);rr r;}
  u1 en(cs string&b){ll n=th->size();u1 r(n);fo(i,n)fo(j,b.size())if((*th)[i]==b[j]){r[i]=j;bk;}rr r;}
  friend is&O>>(is&i,str&s){string t;i>>t;s=str{t};rr i;}
  friend os&O<<(os&o,cs str&s){fe(c,s)o<<c;rr o;}
};
ss strs:v1<str>{
  uu v1<str>::v1;
  u2 en(cs chr&b='a'){u2 r;fe(e,*th)r.eb(e.en(b));rr r;}
  u2 en(cs string&b){u2 r;fe(e,*th)r.eb(e.en(b));rr r;}
};

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 cl=cp<ll,ll>;uu cd=cp<dd,dd>;uu cls=v1<cl>;
uu tl=tr<ll,ll,ll>;uu td=tr<dd,dd,dd>;uu tls=v1<tl>;
uu ql=qu<ll,ll,ll,ll>;uu qd=qu<dd,dd,dd,dd>;uu qls=v1<ql>;
uu ml=map<ll,ll>;uu uml=unordered_map<ll,ll>;
uu sl=set<ll>;uu usl=unordered_set<ll>;
uu msl=multiset<ll>;
uu dl=deque<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 Yes(bo b=1){rr b?"Yes":"No";}str No(){rr Yes(0);}

ul xx=123456789,yy=362436069,zz=521288629,ww=88675123;
ul rnd(){ul t=(xx^(xx<<11));xx=yy,yy=zz,zz=ww;rr(ww=(ww^(ww>>19))^(t^(t>>8)));}
dd rnd_prb(){rr (dd)rnd()/ULLONG_MAX;}
ll rnd_rng(ll r){rr rnd()%r;}
ll rnd_rng(ll l,ll r){rr rnd()%(r-l)+l;}

bo odd(ll x){rr x&1;}
bo eve(ll x){rr !odd(x);}
ll pm1(ll x){rr 1-2*odd(x);}
ll bi(ll n){rr 1LL<<n;}
ll ppc(ll a,ll l=0,ll r=63){rr __builtin_popcountll((bi(r)-bi(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 l2(ll n){ll r=0;wh(n)n/=2,r++;rr r;}//number of digits in binary
ll l10(ll n){ll r=0;wh(n)n/=10,r++;rr r;}//number of digits in decimal
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(cs T&a){rr a*a;}
I<J T>T cb(cs T&a){rr a*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){as(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){as(y);rr(y<0?flo(-x,-y):(x>0?x/y:x/y-!(x%y)));}
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>vo af(v1<T>&v){v.ep(bgn(v),T{});}
I<J T,J U>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){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>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,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=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...});}
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,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 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(deque<T>q){T r{};wh(yu(q))r+=pop(q);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(qmax<T>q){T r{};wh(yu(q))r+=pop(q);rr r;}
I<J T>T sum(qmin<T>q){T r{};wh(yu(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);}
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;}
I<J T>u1 inv(cs v1<T>&a){ll n=len(a);u1 r(n);fo(i,n)r[a[i]]=i;rr r;}


ss fio{fio(){cin.tie(0);ios::sync_with_stdio(0);cout<<fixed<<setprecision(10);}}fio;
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();}
os&O<<(os&o,cs i128&x){if(x<0)rr o<<"-"<<-x;if(x<10)rr o<<(chr)(x+'0');rr o<<x/10<<(chr)(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<<(&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 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(yu(q))o<<pop(q)<<sp;rr o;}
I<J T>os&O<<(os&o,qmax<T>q){wh(yu(q))o<<pop(q)<<sp;rr o;}
I<J T>os&O<<(os&o,qmin<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...);}
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);}
str ins(){ST(r);rr r;}
strs ins2(ll n){S2(n,r);rr r;}

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;}
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;}
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(cs u1&a){cls r;fe(x,a)yu(r)&&x==rb(r).a?rb(r).b++,0:(r.eb(x,1),0);rr r;}
au rce(cs u1&a){cls 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){if(n<=1)rr 0;fo(i,2,n/i+1)if(n%i==0)rr 0;rr 1;}
au inv(i128 a,ll 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;}
au pow(i128 x,ll n,ll m=inf){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/=2;}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);of(i,l+1,1)r.eb(i,n/(i+1)+1,n/i);rr r;}

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=bi(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){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(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/=2,r/=2;}
    rr op(sml,smr);
  }
  friend os&O<<(os&o,cs sgt&s){fo(i,len(s))o<<s.get(i)<<sp;rr o;}
};

ss ufin{
  ll n;u1 d;
  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){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 kazoeage{
  v1<T>fa,rf,bs;
  kazoeage(ll n):fa(n+1,1),rf(n+1,1),bs(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);
    fo(i,1,n+1)bs[i]=bs[i-1]+pm1(i)*rf[i];
  }
  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 kazo=kazoeage<mint>;


ss spf{
  ll n;
  u1 sp;//sp[i]:iを割り切る最小の素数
  u1 ps;
  spf(ll n):n(n){//前処理
    sp=io(n+1);
    fo(i,2,n/i+1)if(sp[i]==i)fo(j,i+i,n+1,i)sp[j]=i;//O(nloglog(n))
    fo(i,2,n+1)if(sp[i]==i)ps.eb(i);
  }
  
  bo ispr(ll x){as(x<=n);rr sp[x]==x;}
  
  cls qv(ll x){//素因数分解クエリ
    as(x<=n);
    v1<cl>r;
    wh(x>1){//O(log(x))
      if(yu(r)&&rb(r).a==sp[x])rb(r).b++;
      el r.eb(sp[x],1);
      x/=sp[x];
    }
    rr r;
  }
  ml qm(ll x){//素因数分解クエリ
    as(x<=n);
    ml r;
    wh(x>1)r[sp[x]]++,x/=sp[x];//O(log(x))
    rr r;
  }
};


vo main(){
  ll T=1;
  //li(T);
  fo(T)solve();
}
vo solve(){
  LL(N);
  UL(N,a);
  
  ll Amax=max(a);
  spf sp(Amax);
  u1 primes=sp.ps;
  
  u1 m(Amax+1);
  fe(x,a)m[x]++;
  
  u1 cnt(Amax+1);
  fe(p,primes){
    fo(j,p,Amax+1,p){
      cnt[p]+=m[j];
    }
  }
  
  u1 small,large;
  fe(p,primes){
    if(p<=sqr(Amax))small.eb(p);
    el large.eb(p);
  }
  
  ll n=len(small);
  u2 dp(n+1,Amax+1);
  dp[0][1]=1;
  fo(i,n){
    fo(j,Amax+1){
      dp[i+1][j]|=dp[i][j];
      if(j*small[i]<=Amax)dp[i+1][j*small[i]]|=dp[i][j];
    }
  }
  
  m1 su(Amax);
  fe(p,large)su[p]=mint(2).pow(cnt[p])-1-cnt[p];
  su=zt(su);
  
  u1 V;
  fo(j,2,Amax+1)if(dp[n][j])V.eb(j);
  
  mint ans=0;
  fe(S,V){
    u1 v;
    fe(p,small)if(S%p==0)v.eb(p);
    
    mint t=1;
    fe(p,v)t*=mint(2).pow(cnt[p])-1-cnt[p];
    
    ans+=-pm1(len(v))*t;
    ans+=-pm1(len(v)+1)*(t*su[Amax/S]);
  }
  pp(ans+N);
}}