#include "islands.h"
#ifndef LOCAL
#pragma GCC optimize ("Ofast")
#pragma GCC optimize ("unroll-loops")
#endif
#include <bits/stdc++.h>
using namespace std;
using ll=long long;
//#define int ll
#define rng(i,a,b) for(int i=int(a);i<int(b);i++)
#define rep(i,b) rng(i,0,b)
#define gnr(i,a,b) for(int i=int(b)-1;i>=int(a);i--)
#define per(i,b) gnr(i,0,b)
#define pb push_back
#define eb emplace_back
#define a first
#define b second
#define bg begin()
#define ed end()
#define all(x) x.bg,x.ed
#define si(x) int(x.size())
#ifdef LOCAL
#define dmp(x) cerr<<__LINE__<<" "<<#x<<" "<<x<<endl
#else
#define dmp(x) void(0)
#endif
template<class t,class u> bool chmax(t&a,u b){if(a<b){a=b;return true;}else return false;}
template<class t,class u> bool chmin(t&a,u b){if(b<a){a=b;return true;}else return false;}
template<class t> using vc=vector<t>;
template<class t> using vvc=vc<vc<t>>;
using pi=pair<int,int>;
using vi=vc<int>;
template<class t,class u>
ostream& operator<<(ostream& os,const pair<t,u>& p){
return os<<"{"<<p.a<<","<<p.b<<"}";
}
template<class t> ostream& operator<<(ostream& os,const vc<t>& v){
os<<"{";
for(auto e:v)os<<e<<",";
return os<<"}";
}
#define mp make_pair
#define mt make_tuple
#define one(x) memset(x,-1,sizeof(x))
#define zero(x) memset(x,0,sizeof(x))
#ifdef LOCAL
void dmpr(ostream&os){os<<endl;}
template<class T,class... Args>
void dmpr(ostream&os,const T&t,const Args&... args){
os<<t<<" ";
dmpr(os,args...);
}
#define dmp2(...) dmpr(cerr,__LINE__,##__VA_ARGS__)
#else
#define dmp2(...) void(0)
#endif
using uint=unsigned;
using ull=unsigned long long;
template<class t,size_t n>
ostream& operator<<(ostream&os,const array<t,n>&a){
return os<<vc<t>(all(a));
}
template<int i,class T>
void print_tuple(ostream&,const T&){
}
template<int i,class T,class H,class ...Args>
void print_tuple(ostream&os,const T&t){
if(i)os<<",";
os<<get<i>(t);
print_tuple<i+1,T,Args...>(os,t);
}
template<class ...Args>
ostream& operator<<(ostream&os,const tuple<Args...>&t){
os<<"{";
print_tuple<0,tuple<Args...>,Args...>(os,t);
return os<<"}";
}
ll read(){
ll i;
cin>>i;
return i;
}
vi readvi(int n,int off=0){
vi v(n);
rep(i,n)v[i]=read()+off;
return v;
}
pi readpi(int off=0){
int a,b;cin>>a>>b;
return pi(a+off,b+off);
}
template<class t>
void print_single(t x,int suc=1){
cout<<x;
if(suc==1)
cout<<"\n";
if(suc==2)
cout<<" ";
}
template<class t,class u>
void print_single(const pair<t,u>&p,int suc=1){
print_single(p.a,2);
print_single(p.b,suc);
}
template<class T>
void print_single(const vector<T>&v,int suc=1){
rep(i,v.size())
print_single(v[i],i==int(v.size())-1?suc:2);
}
template<class T>
void print_offset(const vector<T>&v,ll off,int suc=1){
rep(i,v.size())
print_single(v[i]+off,i==int(v.size())-1?suc:2);
}
template<class T,size_t N>
void print_single(const array<T,N>&v,int suc=1){
rep(i,N)
print_single(v[i],i==int(N)-1?suc:2);
}
template<class T>
void print(const T&t){
print_single(t);
}
template<class T,class ...Args>
void print(const T&t,const Args&...args){
print_single(t,2);
print(args...);
}
string readString(){
string s;
cin>>s;
return s;
}
template<class T>
T sq(const T& t){
return t*t;
}
void YES(bool ex=true){
cout<<"YES\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void NO(bool ex=true){
cout<<"NO\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void Yes(bool ex=true){
cout<<"Yes\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void No(bool ex=true){
cout<<"No\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
//#define CAPITAL
/*
void yes(bool ex=true){
#ifdef CAPITAL
cout<<"YES"<<"\n";
#else
cout<<"Yes"<<"\n";
#endif
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void no(bool ex=true){
#ifdef CAPITAL
cout<<"NO"<<"\n";
#else
cout<<"No"<<"\n";
#endif
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}*/
void possible(bool ex=true){
#ifdef CAPITAL
cout<<"POSSIBLE"<<"\n";
#else
cout<<"Possible"<<"\n";
#endif
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void impossible(bool ex=true){
#ifdef CAPITAL
cout<<"IMPOSSIBLE"<<"\n";
#else
cout<<"Impossible"<<"\n";
#endif
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
constexpr ll ten(int n){
return n==0?1:ten(n-1)*10;
}
const ll infLL=LLONG_MAX/3;
#ifdef int
const int inf=infLL;
#else
const int inf=INT_MAX/2-100;
#endif
int topbit(signed t){
return t==0?-1:31-__builtin_clz(t);
}
int topbit(ll t){
return t==0?-1:63-__builtin_clzll(t);
}
int topbit(ull t){
return t==0?-1:63-__builtin_clzll(t);
}
int botbit(signed a){
return a==0?32:__builtin_ctz(a);
}
int botbit(ll a){
return a==0?64:__builtin_ctzll(a);
}
int botbit(ull a){
return a==0?64:__builtin_ctzll(a);
}
int popcount(signed t){
return __builtin_popcount(t);
}
int popcount(ll t){
return __builtin_popcountll(t);
}
int popcount(ull t){
return __builtin_popcountll(t);
}
int bitparity(ll t){
return __builtin_parityll(t);
}
bool ispow2(int i){
return i&&(i&-i)==i;
}
ll mask(int i){
return (ll(1)<<i)-1;
}
ull umask(int i){
return (ull(1)<<i)-1;
}
ll minp2(ll n){
if(n<=1)return 1;
else return ll(1)<<(topbit(n-1)+1);
}
bool inc(int a,int b,int c){
return a<=b&&b<=c;
}
template<class t> void mkuni(vc<t>&v){
sort(all(v));
v.erase(unique(all(v)),v.ed);
}
ll rand_int(ll l, ll r) { //[l, r]
//#ifdef LOCAL
static mt19937_64 gen;
/*#else
static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count());
#endif*/
return uniform_int_distribution<ll>(l, r)(gen);
}
ll rand_int(ll k){ //[0,k)
return rand_int(0,k-1);
}
template<class t>
void myshuffle(vc<t>&a){
rep(i,si(a))swap(a[i],a[rand_int(0,i)]);
}
template<class t>
int lwb(const vc<t>&v,const t&a){
return lower_bound(all(v),a)-v.bg;
}
template<class t>
bool bis(const vc<t>&v,const t&a){
return binary_search(all(v),a);
}
vvc<int> readGraph(int n,int m){
vvc<int> g(n);
rep(i,m){
int a,b;
cin>>a>>b;
//sc.read(a,b);
a--;b--;
g[a].pb(b);
g[b].pb(a);
}
return g;
}
vvc<int> readTree(int n){
return readGraph(n,n-1);
}
template<class t>
vc<t> presum(const vc<t>&a){
vc<t> s(si(a)+1);
rep(i,si(a))s[i+1]=s[i]+a[i];
return s;
}
vc<ll> presum(const vi&a){
vc<ll> s(si(a)+1);
rep(i,si(a))s[i+1]=s[i]+a[i];
return s;
}
//BIT で数列を管理するときに使う (CF850C)
template<class t>
vc<t> predif(vc<t> a){
gnr(i,1,si(a))a[i]-=a[i-1];
return a;
}
template<class t>
vvc<ll> imos(const vvc<t>&a){
int n=si(a),m=si(a[0]);
vvc<ll> b(n+1,vc<ll>(m+1));
rep(i,n)rep(j,m)
b[i+1][j+1]=b[i+1][j]+b[i][j+1]-b[i][j]+a[i][j];
return b;
}
//verify してないや
void transvvc(int&n,int&m){
swap(n,m);
}
template<class t,class... Args>
void transvvc(int&n,int&m,vvc<t>&a,Args&...args){
assert(si(a)==n);
vvc<t> b(m,vi(n));
rep(i,n){
assert(si(a[i])==m);
rep(j,m)b[j][i]=a[i][j];
}
a.swap(b);
transvvc(n,m,args...);
}
//CF854E
void rotvvc(int&n,int&m){
swap(n,m);
}
template<class t,class... Args>
void rotvvc(int&n,int&m,vvc<t>&a,Args&...args){
assert(si(a)==n);
vvc<t> b(m,vi(n));
rep(i,n){
assert(si(a[i])==m);
rep(j,m)b[m-1-j][i]=a[i][j];
}
a.swap(b);
rotvvc(n,m,args...);
}
//ソートして i 番目が idx[i]
//CF850C
template<class t>
vi sortidx(const vc<t>&a){
int n=si(a);
vi idx(n);iota(all(idx),0);
sort(all(idx),[&](int i,int j){return a[i]<a[j];});
return idx;
}
//vs[i]=a[idx[i]]
//例えば sortidx で得た idx を使えば単にソート列になって返ってくる
//CF850C
template<class t>
vc<t> a_idx(const vc<t>&a,const vi&idx){
int n=si(a);
assert(si(idx)==n);
vc<t> vs(n);
rep(i,n)vs[i]=a[idx[i]];
return vs;
}
//CF850C
vi invperm(const vi&p){
int n=si(p);
vi q(n);
rep(i,n)q[p[i]]=i;
return q;
}
template<class t,class s=t>
s SUM(const vc<t>&a){
return accumulate(all(a),s(0));
}
template<class t>
t MAX(const vc<t>&a){
return *max_element(all(a));
}
template<class t>
pair<t,int> MAXi(const vc<t>&a){
auto itr=max_element(all(a));
return mp(*itr,itr-a.bg);
}
template<class t>
t MIN(const vc<t>&a){
return *min_element(all(a));
}
template<class t>
pair<t,int> MINi(const vc<t>&a){
auto itr=min_element(all(a));
return mp(*itr,itr-a.bg);
}
vi vid(int n){
vi res(n);iota(all(res),0);
return res;
}
template<class S>
void soin(S&s){
sort(all(s));
}
template<class S>
S soout(S s){
soin(s);
return s;
}
template<class S>
void rein(S&s){
reverse(all(s));
}
template<class S>
S reout(S s){
rein(s);
return s;
}
template<class t,class u>
pair<t,u>&operator+=(pair<t,u>&a,pair<t,u> b){
a.a+=b.a;a.b+=b.b;return a;}
template<class t,class u>
pair<t,u>&operator-=(pair<t,u>&a,pair<t,u> b){
a.a-=b.a;a.b-=b.b;return a;}
template<class t,class u>
pair<t,u> operator+(pair<t,u> a,pair<t,u> b){return mp(a.a+b.a,a.b+b.b);}
template<class t,class u>
pair<t,u> operator-(pair<t,u> a,pair<t,u> b){return mp(a.a-b.a,a.b-b.b);}
template<class t>
t gpp(vc<t>&vs){
assert(si(vs));
t res=move(vs.back());
vs.pop_back();
return res;
}
template<class t>
void pb(vc<t>&a,const vc<t>&b){
a.insert(a.ed,all(b));
}
template<class t>
vc<t> cat(vc<t> a,const vc<t>&b){
pb(a,b);
return a;
}
template<class t,class u>
vc<t>& operator+=(vc<t>&a,u x){
for(auto&v:a)v+=x;
return a;
}
template<class t,class u>
vc<t> operator+(vc<t> a,u x){
return a+=x;
}
template<class t,class u>
vc<t>& operator-=(vc<t>&a,u x){
for(auto&v:a)v-=x;
return a;
}
template<class t,class u>
vc<t>& operator-(vc<t> a,u x){
return a-=x;
}
//CCPC Finals 2021 H
//Multiuni2023-3 E
struct dominator_tree{
int n;
vvc<int> g,rG,bct;
vi idom,semi,us,id,rId,par,mn,anc;
dominator_tree(int nn):n(nn),
g(n),rG(n),bct(n),
idom(n,-1),semi(n),us(n),id(n,-1),rId(n),par(n,-1),mn(n),anc(n,-1){
iota(all(semi),0);
iota(all(mn),0);
}
void ae(int a,int b){
if(a==b)return;
g[a].pb(b);
rG[b].pb(a);
}
int find(int v){
if(anc[v]==-1)
return v;
int a=find(anc[v]);
if(id[semi[mn[anc[v]]]]<id[semi[mn[v]]])
mn[v]=mn[anc[v]];
return anc[v]=a;
}
void link(int c,int p){
anc[c]=p;
}
void dfs(int v,int p,int& i){
if(id[v]!=-1)
return;
id[v]=i;
rId[i++]=v;
par[v]=p;
for(int c:g[v])
dfs(c,v,i);
}
//return idom or -1 if unreachable from the root
vi calc(int root){
int sz=0;
dfs(root,-1,sz);
for(int i=sz-1;i>0;i--){
int w=rId[i];
for(int v:rG[w])if(id[v]!=-1){
find(v);
if(id[semi[mn[v]]]<id[semi[w]])
semi[w]=semi[mn[v]];
}
bct[semi[w]].pb(w);
for(int v:bct[par[w]]){
find(v);
us[v]=mn[v];
}
bct[par[w]].clear();
link(w,par[w]);
}
rng(i,1,sz){
int w=rId[i];
if(semi[w]==semi[us[w]])
idom[w]=semi[w];
else
idom[w]=idom[us[w]];
}
return idom;
}
};
//dep,par は not verified
//yukicoder no.1024
//https://www.codechef.com/LTIME110A/problems/TREEQUER
//IOI2022 Islands
template<class E>
struct treedfs{
const vvc<E>&g;
const int n;
int cnt;
vi par,dep,in,out,ni;
void dfs(int v,int p,int d){
par[v]=p;
dep[v]=d;
in[v]=cnt++;
for(auto e:g[v])if(e!=p)
dfs(e,v,d+1);
out[v]=cnt;
}
treedfs(const vvc<E>&gg,int r):g(gg),n(si(g)),cnt(0),par(n),dep(n,-1),in(n,-1),out(n,-1){
dfs(r,-1,0);
//not necessarily connected
ni.resize(cnt);
rep(i,n)if(in[i]!=-1)ni[in[i]]=i;
}
bool asde(int a,int b){
return in[a]<=in[b]&&out[b]<=out[a];
}
};
//CF869B
//複数始点
template<class E>
tuple<vi,vi,vc<E>> bfs(const vvc<E>&g,vi rs){
int n=si(g);
vi dist(n,-1),from(n,-1);
vc<E> use(n);
vi q(n);
int b=0,e=0;
auto reach=[&](int v,int d,int f,E u){
if(dist[v]!=-1)return;
dist[v]=d;
from[v]=f;
use[v]=u;
q[e++]=v;
};
for(auto r:rs)reach(r,0,-1,E());
while(b<e){
int v=q[b++];
for(auto to:g[v])
reach(to,dist[v]+1,v,to);
}
return mt(move(dist),move(from),move(use));
}
bool dbg=false;
struct E{
int to,id;
operator int()const{return to;}
};
std::variant<bool, std::vector<int>> find_journey(
int n, int m, std::vector<int> U, std::vector<int> V) {
vvc<E> g(n);
rep(i,m)g[U[i]].pb({V[i],i});
int twocyc[2]{-1,-1};
vi pv(n,-1),pe(n,-1),dep(n,-1),ord;
{
vi vis(n,0),bw(n,-1);
vi ls;
auto dfs=[&](auto self,int v,int p,int e,int d)->void{
assert(vis[v]==0);
assert(dep[v]==-1);
vis[v]=1;
ls.pb(v);
pv[v]=p;
pe[v]=e;
dep[v]=d;
ord.pb(v);
int x[2]{-1,-1};
for(auto [to,id]:g[v]){
if(dep[to]==-1){
self(self,to,v,id,d+1);
int y=bw[to];
rep(k,2)if(x[k]<y)swap(x[k],y);
}else if(vis[to]==1){
bw[ls[dep[to]+1]]=id;
}else{
//do nothing
}
}
ls.pop_back();
rep(k,2)chmax(bw[v],x[k]);
if(x[0]!=-1&&x[1]!=-1){
rep(k,2)twocyc[k]=x[k];
}
vis[v]=2;
};
dfs(dfs,0,-1,-1,0);
}
auto getcycle=[&](int c){
vi res{c};
for(int i=U[c];i!=V[c];i=pv[i]){
res.pb(pe[i]);
}
return reout(res);
};
vc<bool> avail(m,true);
auto getwalk=[&](vi cyc){
vi pos(n,-1);
rep(i,si(cyc))pos[U[cyc[i]]]=i;
vc<bool> vis(n);
vi cur;
auto dfs=[&](auto self,int v)->bool{
assert(!vis[v]);
vis[v]=true;
int step=rand_int(0,si(g[v]));
myshuffle(g[v]);
rep(_,si(g[v])+1){
if(_==step){
if(pos[v]!=-1){
rep(i,si(cyc))cur.pb(cyc[(pos[v]+i)%si(cyc)]);
return true;
}
}
if(_<si(g[v])){
auto [to,id]=g[v][_];
if(avail[id]){
if(!vis[to]){
cur.pb(id);
if(self(self,to)){
cur.pb(id);
return true;
}else{
cur.pop_back();
}
}
}
}
}
return false;
};
bool res=dfs(dfs,0);
assert(res);
return cur;
};
auto mg=[&](vi a,vi b){
assert(a!=b);
for(auto v:b){
if(si(a)&&a.back()==v)a.pop_back();
else a.pb(v);
}
return a;
};
if(twocyc[0]!=-1&&twocyc[1]!=-1){
vi cyc[2];
rep(k,2)cyc[k]=getcycle(twocyc[k]);
rep(k,2)for(auto i:cyc[k])avail[i]=false;
vi x=getwalk(cyc[0]);
vi y=getwalk(cyc[1]);
return mg(mg(x,y),mg(reout(x),reout(y)));
}
dominator_tree dt(n*2);
vi bef(n,-1);
vc<pi> arc;
int s=n;
{
vi vis(n);
vi ls;
auto dfs=[&](auto self,int v,int d)->void{
assert(!vis[v]);
vis[v]=1;
for(auto [to,id]:g[v]){
if(!vis[to]){
int e=s++;
dt.ae(v,e);
dt.ae(e,to);
ls.pb(e);
bef[to]=e;
self(self,to,d+1);
ls.pop_back();
}else{
dt.ae(v,to);
if(vis[to]==1){
arc.eb(id,ls[dep[to]]);
}
}
}
vis[v]=2;
};
dfs(dfs,0,0);
}
vi idom=dt.calc(0);
vvc<int> t(s);
rep(i,s)if(idom[i]!=-1)t[idom[i]].pb(i);
treedfs<int> w(t,0);
int singlecyc=-1;
vc<bool> uni(n,true);
for(auto i:ord)if(i){
uni[i]=uni[pv[i]];
if(idom[i]!=bef[i])uni[i]=false;
}
for(auto [i,pre]:arc){
if(!w.asde(pre,U[i]))singlecyc=i;
if(!uni[V[i]])singlecyc=i;
}
if(singlecyc==-1){
return false;
}
vi cyc=getcycle(singlecyc);
for(auto i:cyc)avail[i]=false;
vi x=getwalk(cyc);
vi y;
do{y=getwalk(cyc);}while(x==y);
return mg(x,reout(y));
}
