#include<bits/stdc++.h>
using namespace std;
using ll=long long;
#ifdef DEBUG
template<typename T>
ostream& operator << (ostream &out,vector<T> a){
out<<'[';
for(T x:a)out<<x<<',';
return out<<']';
}
template<typename T>
vector<T> ary(T *a,int l,int r){
return vector<T>{a+l,a+1+r};
}
template<typename T>
void debug(T x){
cerr<<x<<endl;
}
template<typename T,typename...S>
void debug(T x,S...y){
cerr<<x<<' ',debug(y...);
}
#else
#define debug(...) void()
#endif
const int N=1e5+10;
int n,a[N];
namespace SGT1{
int t[N<<2],laz[N<<2];
void pushup(int rt){
t[rt]=min(t[rt<<1],t[rt<<1|1])+laz[rt];
}
void pushadd(int rt,int x){
t[rt]+=x,laz[rt]+=x;
}
void build(int l=1,int r=n,int rt=1){
if(l==r){
t[rt]=a[l];return;
}
int mid=(l+r)>>1;
build(l,mid,rt<<1);
build(mid+1,r,rt<<1|1);
pushup(rt);
}
void update(int L,int R,int x,int l=1,int r=n,int rt=1){
if(L<=l&&r<=R)return pushadd(rt,x);
int mid=(l+r)>>1;
if(L<=mid)update(L,R,x,l,mid,rt<<1);
if(mid<R)update(L,R,x,mid+1,r,rt<<1|1);
pushup(rt);
}
void modify(int x,int y,int l=1,int r=n,int rt=1){
if(l==r)return pushadd(rt,x);
int mid=(l+r)>>1;
if(x<=mid)modify(x,y,l,mid,rt<<1);
else modify(x,y,mid+1,r,rt<<1|1);
pushup(rt);
}
int find(int L,int R,int x,int l=1,int r=n,int rt=1){
if(t[rt]>x)return 0;
if(l==r)return l;
x-=laz[rt];
int mid=(l+r)>>1,s=0;
if(L<=mid)s=find(L,R,x,l,mid,rt<<1);
if(s)return s;
if(mid<R)s=find(L,R,x,mid+1,r,rt<<1|1);
return s;
}
int query(int L,int R,int l=1,int r=n,int rt=1){
if(L<=l&&r<=R)return t[rt];
int mid=(l+r)>>1;
if(R<=mid)return query(L,R,l,mid,rt<<1)+laz[rt];
if(mid<L)return query(L,R,mid+1,r,rt<<1|1)+laz[rt];
return min(query(L,R,l,mid,rt<<1),query(L,R,mid+1,r,rt<<1|1))+laz[rt];
}
}
namespace SGT2{
ll t[N<<2],laz[N<<2];
int siz[N<<2];
void pushup(int rt){
t[rt]=t[rt<<1]+t[rt<<1|1];
}
void pushadd(int rt,ll x){
t[rt]+=1ll*x*siz[rt],laz[rt]+=x;
}
void pushdown(int rt){
if(laz[rt]){
pushadd(rt<<1,laz[rt]);
pushadd(rt<<1|1,laz[rt]);
laz[rt]=0;
}
}
void build(int l=1,int r=n,int rt=1){
if(l==r){
siz[rt]=1,t[rt]=a[l];return;
}
int mid=(l+r)>>1;
build(l,mid,rt<<1);
build(mid+1,r,rt<<1|1);
pushup(rt);
siz[rt]=siz[rt<<1]+siz[rt<<1|1];
}
void update(int L,int R,int x,int l=1,int r=n,int rt=1){
if(L<=l&&r<=R)return pushadd(rt,x);
int mid=(l+r)>>1;
pushdown(rt);
if(L<=mid)update(L,R,x,l,mid,rt<<1);
if(mid<R)update(L,R,x,mid+1,r,rt<<1|1);
pushup(rt);
}
ll query(int L,int R,int l=1,int r=n,int rt=1){
if(L<=l&&r<=R)return t[rt];
int mid=(l+r)>>1;
pushdown(rt);
if(R<=mid)return query(L,R,l,mid,rt<<1);
if(mid<L)return query(L,R,mid+1,r,rt<<1|1);
return query(L,R,l,mid,rt<<1)+query(L,R,mid+1,r,rt<<1|1);
}
}
namespace SGT3{
const int K=1e5,INF=1e9;
struct tree{
int ls,rs,mn,laz;
tree():mn(INF),laz(0),ls(0),rs(0){}
}t[K];
int k;
void pushup(int rt){
t[rt].mn=min(t[t[rt].ls].mn,t[t[rt].rs].mn)+t[rt].laz;
}
void pushadd(int rt,int x){
if(rt)t[rt].mn+=x,t[rt].laz+=x;
}
void update(int &rt,int L,int R,int x,int l=1,int r=n){
if(!rt)rt=++k;
if(L<=l&&r<=R)return pushadd(rt,x);
int mid=(l+r)>>1;
if(L<=mid)update(t[rt].ls,L,R,l,mid);
if(mid<R)update(t[rt].rs,L,R,mid+1,r);
pushup(rt);
}
void merge(int &x,int y,int l=1,int r=n){
if(!x||!y){
x|=y;return;
}
if(l==r){
t[x].mn+=t[y].mn;return;
}
int mid=(l+r)>>1;
t[x].laz+=t[y].laz;
merge(t[x].ls,t[y].ls,l,mid);
merge(t[x].rs,t[y].rs,mid+1,r);
pushup(x);
}
int query(int rt,int L,int R,int l=1,int r=n){
if(!rt)return INF;
if(L<=l&&r<=R)return t[rt].mn;
int mid=(l+r)>>1,s=INF;
if(L<=mid)s=min(s,query(t[rt].ls,L,R,l,mid));
if(mid<R)s=min(s,query(t[rt].rs,L,R,mid+1,r));
return s+t[rt].laz;
}
}
namespace BIT{
int c[N];
void add(int x,int y){
for(;x<=n;x+=x&-x)c[x]+=y;
}
int get(int x,int y=0){
for(;x;x^=x&-x)y+=c[x];
return y;
}
}
namespace DSU1{
int fa[N];
void init(){
iota(fa,fa+1+n,0);
}
int find(int x){
return fa[x]==x?x:fa[x]=find(fa[x]);
}
bool merge(int x,int y){
x=find(x),y=find(y);
if(x==y)return 0;
return fa[x]=y,1;
}
}
namespace DSU2{
int fa[N],siz[N];
void init(){
iota(fa,fa+1+n,0),fill(siz+1,siz+1+n,1);
}
int find(int x){
return fa[x]==x?x:find(fa[x]);
}
int top,stk[N];
bool merge(int x,int y){
x=find(x),y=find(y);
if(x==y)return 0;
if(siz[x]>siz[y])swap(x,y);
siz[y]+=siz[x],fa[x]=y,stk[++top]=x;
return 1;
}
void undo(int x){
siz[fa[x]]-=siz[x],fa[x]=x;
}
void reduce(int k){
for(;top>k;top--)undo(stk[top--]);
}
}
vector<int>to[N];
int kk,head[N];
struct edges{
int to,nex;
}edge[N*2];
void add(int u,int v){
edge[++kk]={v,head[u]},head[u]=kk;
edge[++kk]={u,head[v]},head[v]=kk;
}
namespace TJ1{
int dft,top,sct,dfn[N],low[N],scc[N],stk[N];
void tarjan(int u){
low[u]=dfn[u]=++dft,stk[++top]=u;
for(int v:to[u]){
if(!dfn[v])tarjan(v),low[u]=min(low[u],low[v]);
else if(!scc[v])low[u]=min(low[u],dfn[v]);
}
if(low[u]==dfn[u]){
sct++;
do scc[stk[top]]=sct;while(stk[top--]^u);
}
}
}
namespace TJ2{
int dft,sct,top,dfn[N],low[N],scc[N],stk[N];
void tarjan(int u,int fa=0){
low[u]=dfn[u]=++dft,stk[++top]=u;
for(int i=head[u];i;i=edge[i].nex)if(i^fa^1){
int v=edge[i].to;
if(!dfn[v])tarjan(v,i),low[u]=min(low[u],low[v]);
else if(!scc[v])low[u]=min(low[u],dfn[v]);
}
if(dfn[u]==low[u]){
sct++;
do scc[stk[top]]=sct;while(stk[top--]^u);
}
}
}
namespace TJ3{
vector<int>A[N];
int k,dft,top,dfn[N],low[N],stk[N];
void add(int u,int v){
A[u].push_back(v),A[v].push_back(u);
}
void tarjan(int u,int fa=0){
dfn[u]=low[u]=++dft,stk[++top]=u;
for(int v:to[u])if(v^fa){
if(!dfn[v]){
tarjan(v),low[u]=min(low[u],low[v]);
if(low[v]>=dfn[u]){
add(++k,u);
do add(k,stk[top]);while(stk[top--]^v);
}
}else low[u]=min(low[u],dfn[v]);
}
}
}
namespace FHQ{
struct tree{
int ls,rs,rnd,val,siz,laz;
}t[N];
void pushup(int rt){
t[rt].siz=t[t[rt].ls].siz+t[t[rt].rs].siz+1;
}
void pushadd(int rt,int x){
if(rt)t[rt].val+=x,t[rt].laz+=x;
}
void pushdown(int rt){
if(t[rt].laz){
pushadd(t[rt].ls,t[rt].laz);
pushadd(t[rt].rs,t[rt].laz);
}
}
void split(int rt,int val,int &x,int &y){
if(!rt){
x=y=0;return;
}
pushdown(rt);
if(val<t[rt].val)y=rt,split(t[rt].ls,val,x,t[rt].ls);
else x=rt,split(t[rt].rs,val,t[rt].rs,y);
pushup(rt);
}
int merge(int x,int y){
if(!x||!y)return x|y;
if(t[x].rnd<t[y].rnd){
pushdown(x);
t[x].rs=merge(t[x].rs,y);
return pushup(x),x;
}else{
pushdown(y);
t[y].ls=merge(x,t[y].ls);
return pushup(y),y;
}
}
}
const int INF=1e9;
namespace Floyd{
const int N=1e2+10;
int a[N][N],f[N][N];
void solve(){
for(int i=1;i<=n;i++){
copy(a[i]+1,a[i]+1+n,f[i]+1);
f[i][i]=0;
}
for(int k=1;k<=n;k++){
for(int i=1;i<=n;i++){
for(int j=1;j<=n;j++){
f[i][j]=min(f[i][j],f[i][k]+f[k][j]);
}
}
}
}
}
namespace DIJ{
vector<pair<int,int> >to[N];
#define v e.first
#define w e.second
ll dis[N];
const ll INF=1e18;
int vis[N];
void dij(int s){
fill(dis+1,dis+1+n,INF),dis[s]=0;
priority_queue<pair<ll,int> >q;
q.push({0,s});
for(int u;!q.empty();){
u=q.top().second,q.pop();
if(vis[u])continue;
vis[u]=1;
for(auto e:to[u]){
if(dis[v]>dis[u]+w){
dis[v]=dis[u]+w,q.push({-dis[v],v});
}
}
}
}
#undef v
#undef w
}
namespace Flow1{
const int V=1e5,E=1e5;
int s,t,kk,head[V],cur[V],d[V];
struct edges{
int to,c,nex;
}edge[E];
void init(int x,int y){
s=x,t=y,kk=1;
fill(head+s,head+1+t,0);
}
int add(int u,int v,int c){
edge[++kk]={v,c,head[u]},head[u]=kk;
edge[++kk]={u,0,head[v]},head[v]=kk;
return kk-1;
}
bool bfs(){
fill(d+s,d+1+t,-1),d[s]=0;
copy(head+s,head+1+t,cur+s);
queue<int>q;
q.push(s);
for(int u;!q.empty();){
u=q.front(),q.pop();
for(int i=head[u];i;i=edge[i].nex){
int v=edge[i].to,c=edge[i].c;
if(c&&!~d[v])d[v]=d[u]+1,q.push(v);
}
}
return !~d[t];
}
int dfs(int u,int lim=INF){
if(u==t)return lim;
int flow=0;
for(int i=cur[u];i&&flow<lim;i=edge[i].nex){
cur[u]=i;
int v=edge[i].to,c=edge[i].c;
if(!c||d[v]!=d[u]+1)continue;
int f=dfs(v,min(c,lim-flow));
if(!f)d[v]=-1;
flow+=f,edge[i].c+=f,edge[i^1].c-=f;
}
return flow;
}
static const int M=2;
int dinic(){
int sum=0;
for(;bfs();)sum+=dfs(s);
return sum;
}
}
namespace Flow2{
const int V=1e5,E=1e5;
int s,t,kk,head[V],dis[V],cur[V];
struct edges{
int to,c,w,nex;
}edge[E];
void init(int x,int y){
s=x,t=y,kk=1;
fill(head+s,head+1+t,0);
}
int add(int u,int v,int c,int w){
edge[++kk]={v,c,w,head[u]},head[u]=kk;
edge[++kk]={u,0,-w,head[v]},head[v]=kk;
return kk-1;
}
bool spfa(){
fill(dis+s,dis+1+t,INF),dis[s]=0;
queue<int>q;
q.push(s);
for(int u;!q.empty();){
u=q.front(),q.pop();
for(int i=head[u];i;i=edge[i].nex){
int v=edge[i].to,c=edge[i].c,w=edge[i].w;
if(c&&dis[v]>dis[u]+w){
dis[v]=dis[u]+w,q.push(v);
}
}
}
return dis[t]<INF;
}
int cost,vis[N];
int dfs(int u,int lim=INF){
if(u==t)return lim;
vis[u]=1;
int flow=0;
for(int i=cur[u];i&&flow<lim;i=edge[i].nex){
cur[u]=i;
int v=edge[i].to,c=edge[i].c,w=edge[i].w;
if(vis[v]||!c||dis[v]!=dis[u]+w)continue;
int f=dfs(v,min(lim-flow,c));
edge[i].c-=f,edge[i^1].c+=f;
flow+=c,cost+=c*w;
}
vis[u]=0;
return flow;
}
int dinic(){
int sum=0;
for(;spfa();)sum+=dfs(s);
return sum;
}
}
const int mod=998244353;
namespace Matrix{
struct matrix{
static const int M=2;
int a[M][M];
matrix(){
memset(a,0,sizeof a);
}
matrix operator * (const matrix &x)const{
matrix b;
for(int k=0;k<M;k++){
for(int i=0;i<M;i++){
for(int j=0;j<M;j++){
b.a[i][j]=(b.a[i][j]+1ll*a[i][k]*x.a[k][j])%mod;
}
}
}
return b;
}
};
struct vec{
static const int M=2;
int a[M];
vec(){
memset(a,0,sizeof a);
}
vec operator * (const matrix &x)const{
vec b;
for(int i=0;i<M;i++){
for(int j=0;j<M;j++){
b.a[j]=(b.a[j]+1ll*a[i]*x.a[i][j])%mod;
}
}
return b;
}
};
}
namespace binom{
int fac[N],ifac[N];
ll qpow(ll x,ll y=mod-2,ll ans=1){
for(;y;(x*=x)%=mod,y>>=1)if(y&1)(ans*=x)%=mod;
return ans;
}
void init(int n){
for(int i=fac[0]=1;i<=n;i++)fac[i]=1ll*fac[i-1]*i%mod;
ifac[n]=qpow(fac[n]);
for(int i=n;i>=1;i--)ifac[i-1]=1ll*ifac[i]*i%mod;
}
int C(int n,int m){
if(0>m||m>n)return 0;
return 1ll*fac[n]*ifac[m]%mod*ifac[n-m]%mod;
}
}
namespace Path{
int fa[N],siz[N],dep[N],top[N],son[N];
void dfs1(int u){
siz[u]=1,son[u]=0;
for(int v:to[u])if(v^fa[u]){
fa[v]=u,dfs1(v);
siz[u]+=siz[v];
if(siz[v]>siz[son[u]])son[u]=v;
}
}
int dft,dfn[N],pos[N];
void dfs2(int u,int t){
top[u]=t,pos[dfn[u]=++dft]=u;
if(son[u])dfs2(son[u],t);
for(int v:to[u])if(v^fa[u]&&v^son[u])dfs2(v,v);
}
int LCA(int u,int v){
for(;top[u]^top[v];u=fa[top[u]]){
if(dep[top[u]]<dep[top[v]])swap(u,v);
}
return dep[u]<dep[v]?u:v;
}
int jump(int u,int k){
for(;k>dfn[u]-dfn[top[u]];u=fa[top[u]]){
k-=dfn[u]-dfn[top[u]]+1;
}
return pos[dfn[u]-k];
}
}
namespace Euler_LCA{
const int M=N*2,K=__lg(M)+2;
int dft,dfn[N],f[K][M],dep[N];
bool cmp(int x,int y){
return dep[x]<dep[y];
}
void make(int u,int fa=0){
f[0][dfn[u]=++dft]=u,dep[u]=dep[fa]+1;
for(int v:to[u])if(v^fa){
make(v,u);
f[0][++dft]=u;
}
}
void init(){
dft=0,make(1);
for(int i=1;(1<<i)<=dft;i++){
for(int j=1;j+(1<<i)-1<=dft;j++){
f[i][j]=min(f[i-1][j],f[i-1][j+(1<<i-1)],cmp);
}
}
}
int LCA(int l,int r){
if(l=dfn[l],r=dfn[r],l>r)swap(l,r);
int k=__lg(r-l+1);
return min(f[k][l],f[k][r-(1<<k)+1],cmp);
}
}
namespace AC{
const int K=26;
int k,ch[N][K],fail[N];
int insert(char *a){
int n=strlen(a),u=0;
for(int i=0;i<n;i++){
int &v=ch[u][a[i]-'a'];
if(!v)v=++k;
u=v;
}
return u;
}
void getfail(){
queue<int>q;
for(int i=0;i<K;i++){
int v=ch[0][i];
if(v)q.push(v),fail[v]=0;
}
for(int u;!q.empty();){
u=q.front(),q.pop();
for(int i=0;i<K;i++){
int &v=ch[u][i],x=ch[fail[u]][i];
if(v)fail[v]=x,q.push(v);
else v=x;
}
}
}
}
namespace KMP{
char a[N];
int nex[N];
void kmp(){
int n=strlen(a+1);
for(int i=2,j=0;i<=n;i++){
for(;j&&a[j+1]!=a[i];j=nex[j]);
nex[i]=j+=a[j+1]==a[i];
}
}
}
namespace Manacher{
int n,m;
char a[N],b[N*2];
int len[N*2];
void init(){
n=strlen(a+1),m=0,b[++m]='$';
for(int i=1;i<=n;i++){
b[++m]=a[i],b[++m]='$';
}
for(int i=1,mx=0,id=0;i<=n;i++){
len[i]=i<=mx?min(len[id*2-i],mx-i+1):0;
for(;i+len[i]<=m&&i-len[i]>=1&&b[i+len[i]]==b[i-len[i]];len[i]++);
if(i+len[i]>mx)mx=i+len[i]-1,id=i;
}
}
}
namespace Z{
char a[N];
int n,z[N];
void init(){
n=strlen(a+1);
fill(z+1,if(fi[rt<<1]>=fi[rt<<1|1])pushcov(rt<<1,tag[rt]);
if(fi[rt<<1|1]>=fi[rt<<1])pushcov(rt<<1|1,tag[rt]);z+1+n,0);
for(int i=2,gap=0,mx=0;i<=n;i++){
z[i]=i<=mx?min(z[i-gap],mx-i+1):0;
for(;i+z[i]<=n&&a[z[i]]==a[i+z[i]-1];z[i]++);
if(i+z[i]>mx)mx=i+z[i]-1,gap=i-1;
}
}
}
namespace MinCircle{
char a[N];
int get(){
int n=strlen(a),i=0,j=1,k=0;
for(;i<n&&j<n&&k<n;){
if(a[(i+k)%n]==a[(j+k)%n])k++;
else{
if(a[(i+k)%n]<a[(j+k)%n])j+=k+1;
else i+=k+1;
if(i==j)i++;
k=0;
}
}
return min(i,j);
}
}
namespace Hash{
char a[N];
int n,f[N],pw[N];
const int base=19260817,mod=1e9+7;
void init(){
n=strlen(a+1);
for(int i=pw[0]=1;i<=n;i++)pw[i]=1ll*pw[i-1]*base%mod;
for(int i=1;i<=n;i++)f[i]=(1ll*f[i-1]*base+a[i])%mod;
}
int Hash(int l,int r){
return (f[r]+1ll*(mod-f[l-1])*pw[r-l+1])%mod;
}
}
int main(){
freopen(".in","r",stdin);
// freopen(".out","w",stdout);
return 0;
}
