#include "thief.h"
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("avx2,bmi,bmi2,lzcnt,popcnt")
#define ll long long
#define ld long double
#define ull unsigned long long
#define ff first
#define ss second
#define pii pair<int,int>
#define pll pair<long long, long long>
#define vi vector<int>
#define vl vector<long long>
#define pb push_back
#define rep(i, b) for(int i = 0; i < (b); ++i)
#define rep2(i,a,b) for(int i = a; i <= (b); ++i)
#define rep3(i,a,b,c) for(int i = a; i <= (b); i+=c)
#define count_bits(x) __builtin_popcountll((x))
#define all(x) (x).begin(),(x).end()
#define siz(x) (int)(x).size()
#define forall(it,x) for(auto& it:(x))
using namespace __gnu_pbds;
using namespace std;
typedef tree<int, null_type, less<int>, rb_tree_tag,tree_order_statistics_node_update> ordered_set;
//mt19937 mt;void random_start(){mt.seed(chrono::time_point_cast<chrono::milliseconds>(chrono::high_resolution_clock::now()).time_since_epoch().count());}
//ll los(ll a, ll b) {return a + (mt() % (b-a+1));}
const int INF = 1e9+50;
const ll INF_L = 1e18+40;
const ll MOD = 1e9+7;
struct elm
{
int e,dir,c;
};
ll dwu[10];
int q_cnt[10001];
vi masks[20];
vector<pii> graph[10001];
vector<pii> centr_tree[10001];
pii edge[15001];
bool odw[10001];
int sub[10001];
vector<vector<elm>> dp[10001];
map<pii,int> edge_ind;
bool is_tree[15001];
int n,m;
vector<elm> oper;
bool del[10001];
struct centr_graph
{
unordered_map<int,vector<pii>> g2;
int c;
void make_oper(int v, int pop, int type)
{
forall(it,g2[v])
{
if(it.ff != pop && !del[it.ff])
{
if(type == 1) oper.pb({it.ss,v,c});
else oper.pb({it.ss,it.ff,c});
make_oper(it.ff,v,type);
}
}
}
};
centr_graph g[10001];
void dfs_sub(int v, int pop)
{
sub[v] = 1;
forall(it,graph[v]) if(it.ff != pop && !odw[it.ff])
{
dfs_sub(it.ff,v);
sub[v] += sub[it.ff];
}
}
void gen_graph(int v, int pop, int c)
{
forall(it,graph[v]) if(it.ff != pop && !odw[it.ff])
{
gen_graph(it.ff,v,c);
g[c].g2[v].pb(it);
g[c].g2[it.ff].pb({v,it.ss});
}
}
int centroid(int v, int n)
{
dfs_sub(v,v);
int pop = v;
while(true)
{
pii best = {-1,-1};
forall(it,graph[v]) if(it.ff != pop && !odw[it.ff]) best = max(best,{sub[it.ff],it.ff});
if(best.ff > n/2)
{
pop = v;
v = best.ss;
}
else break;
}
// cout << v << " " << n << " " << sub[v] << " centr\n";
odw[v] = 1;
g[v].c = v;
dfs_sub(v,v);
gen_graph(v,v,v);
forall(it,graph[v]) if(!odw[it.ff])
{
int c = centroid(it.ff,sub[it.ff]);
centr_tree[v].pb({c,it.ff});
}
return v;
}
void merge_dp(vector<elm>& d1, vector<elm>& d2)
{
if(siz(d1) < siz(d2)) swap(d1,d2);
forall(it,d2) d1.pb(it);
}
void dfs_dp(int v)
{
//cout << v << " dfs_dp\n";
forall(it,centr_tree[v]) dfs_dp(it.ff);
if(siz(centr_tree[v]) == 0)
{
dp[v] = {};
return;
}
vector<pair<int,pii>> dps;
forall(it,centr_tree[v]) dps.pb({siz(dp[it.ff]),{it.ff,it.ss}});
sort(all(dps));
reverse(all(dps));
pii best = {q_cnt[siz(centr_tree[v])]+dps[0].ff,0};
int cur_max = 0;
rep2(pref,1,siz(centr_tree[v]))
{
cur_max = max(cur_max,pref*2+dps[pref-1].ff);
best = min(best,{max(cur_max,pref*2+q_cnt[siz(centr_tree[v])-pref]+(pref != siz(centr_tree[v]) ? dps[pref].ff : 0)),pref});
}
int pref = best.ss;
dp[v].resize(best.ff);
rep(i,pref)
{
rep(j,siz(dp[dps[i].ss.ff])) merge_dp(dp[v][(i+1)*2+j],dp[dps[i].ss.ff][j]);
}
rep2(i,pref,siz(dps)-1)
{
rep(j,siz(dp[dps[i].ss.ff])) merge_dp(dp[v][pref*2+q_cnt[siz(dps)-pref]+j],dp[dps[i].ss.ff][j]);
}
rep(i,pref)
{
oper = {};
g[v].make_oper(v,dps[i].ss.ss,1);
merge_dp(dp[v][i*2],oper);
oper = {};
g[v].make_oper(dps[i].ss.ss,v,0);
merge_dp(dp[v][i*2],oper);
dp[v][i*2].pb({edge_ind[{v,dps[i].ss.ss}],dps[i].ss.ss,v});
oper = {};
g[v].make_oper(v,dps[i].ss.ss,0);
merge_dp(dp[v][i*2+1],oper);
oper = {};
g[v].make_oper(dps[i].ss.ss,v,1);
merge_dp(dp[v][i*2+1],oper);
dp[v][i*2+1].pb({edge_ind[{v,dps[i].ss.ss}],v,v});
del[dps[i].ss.ss] = 1;
}
vi rest;
rep2(i,pref,siz(dps)-1) rest.pb(dps[i].ss.ss);
int k = q_cnt[siz(dps)-pref];
rep(o,k)
{
rep(i,siz(rest))
{
if(masks[k][i]&(1<<o))
{
oper = {};
g[v].make_oper(rest[i],v,0);
merge_dp(dp[v][pref*2+o],oper);
dp[v][pref*2+o].pb({edge_ind[{v,rest[i]}],rest[i],v});
}
else
{
oper = {};
g[v].make_oper(rest[i],v,1);
merge_dp(dp[v][pref*2+o],oper);
dp[v][pref*2+o].pb({edge_ind[{v,rest[i]}],v,v});
}
}
}
forall(it,dps) del[it.ss.ss] = 0;
}
vi dir;
int ord[10001];
vector<pii> dag[10001];
int deg[10001];
vector<pii> dag2[10001];
int deg2[10001];
void make_ord(vector<elm> q)
{
forall(it,q)
{
if(it.dir == edge[it.e].ff) dag[edge[it.e].ss].pb({it.dir,it.e});
else dag[edge[it.e].ff].pb({it.dir,it.e});
deg[it.dir]++;
}
queue<int> q2;
rep(i,n) if(deg[i] == 0) q2.push(i);
int cur = 0;
while(!q2.empty())
{
int t = q2.front();
q2.pop();
ord[t] = cur++;
forall(it,dag[t])
{
deg[it.ff]--;
if(deg[it.ff] == 0) q2.push(it.ff);
}
}
rep(i,n) deg[i] = 0;
rep(i,n) dag[i] = {};
}
bool ask(vector<elm> q, bool is = 1)
{
make_ord(q);
dir.resize(m);
rep(i,m) dir[i] = 0;
forall(it,q) if(edge[it.e].ss != it.dir) dir[it.e] = 1;
map<int,int> mp;
forall(it,q) mp[it.e] = 1;
rep(i,m) if(!mp[i])
{
if(ord[edge[i].ff] < ord[edge[i].ss]) dir[i] = 0;
else dir[i] = 1;
}
if(is) return query(dir);
return 0;
}
vector<pii> graph2[10001];
bool odw2[10001];
void dfs_tree(int v)
{
odw2[v] = 1;
forall(it,graph2[v])
{
if(!odw2[it.ff])
{
dfs_tree(it.ff);
is_tree[it.ss] = 1;
}
}
}
void solve(int N, int M, vi U, vi V)
{
n = N;
m = M;
rep2(i,1,9)
{
dwu[i] = 1;
rep2(j,1,i*2) dwu[i] *= j;
rep2(j,1,i) dwu[i] /= j*j;
rep(mask,(1<<(i*2)))
{
if(count_bits(mask) == i) masks[i*2].pb(mask);
}
}
rep(i,m)
{
graph2[U[i]].pb({V[i],i});
graph2[V[i]].pb({U[i],i});
}
dfs_tree(0);
rep2(i,1,10000) rep2(k,1,9) if(dwu[k] >= i)
{
q_cnt[i] = k*2;
break;
}
int kk = 0;
rep(i,m)
{
edge_ind[{U[i],V[i]}] = i;
edge_ind[{V[i],U[i]}] = i;
edge[i] = {U[i],V[i]};
if(is_tree[i])
{
graph[U[i]].pb({V[i],i});
graph[V[i]].pb({U[i],i});
kk++;
}
}
int root = centroid(0,n);
dfs_dp(root);
vector<elm> ok_graph = dp[root].back();
rep(i,siz(dp[root])-1)
{
if(ask(dp[root][i]))
{
ok_graph = dp[root][i];
break;
}
}
ask(ok_graph,0);
rep(i,m)
{
if(dir[i] == 1)
{
dag[edge[i].ss].pb({edge[i].ff,i});
deg[edge[i].ff]++;
dag2[edge[i].ff].pb({edge[i].ss,i});
deg2[edge[i].ss]++;
}
else
{
dag[edge[i].ff].pb({edge[i].ss,i});
deg[edge[i].ss]++;
dag2[edge[i].ss].pb({edge[i].ff,i});
deg2[edge[i].ff]++;
}
}
queue<int> q;
vi vs;
vi vs2;
rep(i,n) if(deg[i] == 0) q.push(i);
while(!q.empty())
{
int t = q.front();
q.pop();
forall(it,dag[t])
{
vs.pb(it.ss);
deg[it.ff]--;
if(deg[it.ff] == 0) q.push(it.ff);
}
}
rep(i,n) if(deg2[i] == 0) q.push(i);
while(!q.empty())
{
int t = q.front();
q.pop();
forall(it,dag2[t])
{
vs2.pb(it.ss);
deg2[it.ff]--;
if(deg2[it.ff] == 0) q.push(it.ff);
}
};
int l = 0;
int r = siz(vs)-2;
int ans1 = siz(vs)-1;
while(l <= r)
{
int mid = (l+r)/2;
rep(i,mid+1) dir[vs[i]] ^= 1;
int a = query(dir);
rep(i,mid+1) dir[vs[i]] ^= 1;
if(a == 0)
{
r = mid-1;
ans1 = mid;
}
else
{
l = mid+1;
}
}
int v1 = vs[ans1];
if(dir[v1] == 0) v1 = U[v1];
else v1 = V[v1];
l = 0;
r = siz(vs2)-2;
int ans2 = siz(vs2)-1;
while(l <= r)
{
int mid = (l+r)/2;
rep(i,mid+1) dir[vs2[i]] ^= 1;
int a = query(dir);
rep(i,mid+1) dir[vs2[i]] ^= 1;
if(a == 0)
{
r = mid-1;
ans2 = mid;
}
else
{
l = mid+1;
}
}
int v2 = vs2[ans2];
if(dir[v2] == 0) v2 = V[v2];
else v2 = U[v2];
answer(v1,v2);
}
