//#define _GLIBCXX_DEBUG

//#pragma GCC target("avx2")
//#pragma GCC optimize("O3")
//#pragma GCC optimize("unroll-loops")

#include<bits/stdc++.h>
using namespace std;


#ifdef LOCAL
#include <debug_print.hpp>
#define OUT(...) debug_print::multi_print(#__VA_ARGS__, __VA_ARGS__)
#else
#define OUT(...) (static_cast<void>(0))
#endif

#define endl '\n'
#define lfs cout<<fixed<<setprecision(15)
#define ALL(a)  (a).begin(),(a).end()
#define ALLR(a)  (a).rbegin(),(a).rend()
#define UNIQUE(a) (a).erase(unique((a).begin(),(a).end()),(a).end())
#define spa << " " <<
#define fi first
#define se second
#define MP make_pair
#define MT make_tuple
#define PB push_back
#define EB emplace_back
#define rep(i,n,m) for(ll i = (n); i < (ll)(m); i++)
#define rrep(i,n,m) for(ll i = (ll)(m) - 1; i >= (ll)(n); i--)
using ll = long long;
using ld = long double;
const ll MOD1 = 1e9+7;
const ll MOD9 = 998244353;
const ll INF = 1e18;
using P = pair<ll, ll>;
template<typename T> using PQ = priority_queue<T>;
template<typename T> using QP = priority_queue<T,vector<T>,greater<T>>;
template<typename T1, typename T2>bool chmin(T1 &a,T2 b){if(a>b){a=b;return true;}else return false;}
template<typename T1, typename T2>bool chmax(T1 &a,T2 b){if(a<b){a=b;return true;}else return false;}
ll median(ll a,ll b, ll c){return a+b+c-max({a,b,c})-min({a,b,c});}
void ans1(bool x){if(x) cout<<"Yes"<<endl;else cout<<"No"<<endl;}
void ans2(bool x){if(x) cout<<"YES"<<endl;else cout<<"NO"<<endl;}
void ans3(bool x){if(x) cout<<"Yay!"<<endl;else cout<<":("<<endl;}
template<typename T1,typename T2>void ans(bool x,T1 y,T2 z){if(x)cout<<y<<endl;else cout<<z<<endl;}  
template<typename T1,typename T2,typename T3>void anss(T1 x,T2 y,T3 z){ans(x!=y,x,z);};  
template<typename T>void debug(const T &v,ll h,ll w,string sv=" "){for(ll i=0;i<h;i++){cout<<v[i][0];for(ll j=1;j<w;j++)cout<<sv<<v[i][j];cout<<endl;}};
template<typename T>void debug(const T &v,ll n,string sv=" "){if(n!=0)cout<<v[0];for(ll i=1;i<n;i++)cout<<sv<<v[i];cout<<endl;};
template<typename T>void debug(const vector<T>&v){debug(v,v.size());}
template<typename T>void debug(const vector<vector<T>>&v){for(auto &vv:v)debug(vv,vv.size());}
template<typename T>void debug(stack<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(queue<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(deque<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop_front();}cout<<endl;}
template<typename T>void debug(PQ<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(QP<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(const set<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
template<typename T>void debug(const multiset<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
template<typename T,size_t size>void debug(const array<T, size> &a){for(auto z:a)cout<<z<<" ";cout<<endl;}
template<typename T,typename V>void debug(const map<T,V>&v){for(auto z:v)cout<<"["<<z.first<<"]="<<z.second<<",";cout<<endl;}
template<typename T>vector<vector<T>>vec(ll x, ll y, T w){vector<vector<T>>v(x,vector<T>(y,w));return v;}
ll gcd(ll x,ll y){ll r;while(y!=0&&(r=x%y)!=0){x=y;y=r;}return y==0?x:y;}
vector<ll>dx={1,-1,0,0,1,1,-1,-1};vector<ll>dy={0,0,1,-1,1,-1,1,-1};
template<typename T>vector<T> make_v(size_t a,T b){return vector<T>(a,b);}
template<typename... Ts>auto make_v(size_t a,Ts... ts){return vector<decltype(make_v(ts...))>(a,make_v(ts...));}
template<typename T1, typename T2>ostream &operator<<(ostream &os, const pair<T1, T2>&p){return os << "(" << p.first << "," << p.second << ")";}
template<typename T>ostream &operator<<(ostream &os, const vector<T> &v){os<<"[";for(auto &z:v)os << z << ",";os<<"]"; return os;}
template<typename T>void rearrange(vector<int>&ord, vector<T>&v){
  auto tmp = v;
  for(int i=0;i<tmp.size();i++)v[i] = tmp[ord[i]];
}
template<typename Head, typename... Tail>void rearrange(vector<int>&ord,Head&& head, Tail&&... tail){
  rearrange(ord, head);
  rearrange(ord, tail...);
}
template<typename T> vector<int> ascend(const vector<T>&v){
  vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
  sort(ord.begin(),ord.end(),[&](int i,int j){return make_pair(v[i],i)<make_pair(v[j],j);});
  return ord;
}
template<typename T> vector<int> descend(const vector<T>&v){
  vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
  sort(ord.begin(),ord.end(),[&](int i,int j){return make_pair(v[i],-i)>make_pair(v[j],-j);});
  return ord;
}
template<typename T> vector<T> inv_perm(const vector<T>&ord){
  vector<T>inv(ord.size());
  for(int i=0;i<ord.size();i++)inv[ord[i]] = i;
  return inv;
}
ll FLOOR(ll n,ll div){assert(div>0);return n>=0?n/div:(n-div+1)/div;}
ll CEIL(ll n,ll div){assert(div>0);return n>=0?(n+div-1)/div:n/div;}
ll digitsum(ll n){ll ret=0;while(n){ret+=n%10;n/=10;}return ret;}
ll modulo(ll n,ll d){return (n%d+d)%d;};
template<typename T>T min(const vector<T>&v){return *min_element(v.begin(),v.end());}
template<typename T>T max(const vector<T>&v){return *max_element(v.begin(),v.end());}
template<typename T>T acc(const vector<T>&v){return accumulate(v.begin(),v.end(),T(0));};
template<typename T>T reverse(const T &v){return T(v.rbegin(),v.rend());};
//mt19937 mt(chrono::steady_clock::now().time_since_epoch().count());
int popcount(ll x){return __builtin_popcountll(x);};
int poplow(ll x){return __builtin_ctzll(x);};
int pophigh(ll x){return 63 - __builtin_clzll(x);};
template<typename T>T poll(queue<T> &q){auto ret=q.front();q.pop();return ret;};
template<typename T>T poll(priority_queue<T> &q){auto ret=q.top();q.pop();return ret;};
template<typename T>T poll(QP<T> &q){auto ret=q.top();q.pop();return ret;};
template<typename T>T poll(stack<T> &s){auto ret=s.top();s.pop();return ret;};
ll MULT(ll x,ll y){if(LLONG_MAX/x<=y)return LLONG_MAX;return x*y;}
ll POW2(ll x, ll k){ll ret=1,mul=x;while(k){if(mul==LLONG_MAX)return LLONG_MAX;if(k&1)ret=MULT(ret,mul);mul=MULT(mul,mul);k>>=1;}return ret;}
ll POW(ll x, ll k){ll ret=1;for(int i=0;i<k;i++){if(LLONG_MAX/x<=ret)return LLONG_MAX;ret*=x;}return ret;}
namespace converter{
  int dict[500];
  const string lower="abcdefghijklmnopqrstuvwxyz";
  const string upper="ABCDEFGHIJKLMNOPQRSTUVWXYZ";
  const string digit="0123456789";
  const string digit1="123456789";
  void regi_str(const string &t){
    for(int i=0;i<t.size();i++){
      dict[t[i]]=i;
    }
  }
  void regi_int(const string &t){
    for(int i=0;i<t.size();i++){
      dict[i]=t[i];
    }
  }
  vector<int>to_int(const string &s,const string &t){
    regi_str(t);
    vector<int>ret(s.size());
    for(int i=0;i<s.size();i++){
      ret[i]=dict[s[i]];
    }
    return ret;
  }
  vector<int>to_int(const string &s){
    auto t=s;
    sort(t.begin(),t.end());
    t.erase(unique(t.begin(),t.end()),t.end());
    return to_int(s,t);
  }
  
  vector<vector<int>>to_int(const vector<string>&s,const string &t){
    regi_str(t);
    vector<vector<int>>ret(s.size(),vector<int>(s[0].size()));
    for(int i=0;i<s.size();i++){
      for(int j=0;j<s[0].size();j++){
        ret[i][j]=dict[s[i][j]];
      }
    }
    return ret;
  }
  vector<vector<int>>to_int(const vector<string>&s){
    string t;
    for(int i=0;i<s.size();i++){
      t+=s[i];
    }
    sort(t.begin(),t.end());t.erase(unique(t.begin(),t.end()),t.end());
    return to_int(s,t);
  }
  string to_str(const vector<int>&s,const string &t){
    regi_int(t);
    string ret;
    for(auto z:s)ret+=dict[z];
    return ret;
  }
  vector<string> to_str(const vector<vector<int>>&s,const string &t){
    regi_int(t);
    vector<string>ret(s.size());
    for(int i=0;i<s.size();i++){
      for(auto z:s[i])ret[i]+=dict[z];
    }
    return ret;
  }
}
template< typename T = int >
struct edge {
  int to;
  T cost;
  int id;
  edge():to(-1),id(-1){};
  edge(int to, T cost = 1, int id = -1):to(to), cost(cost), id(id){}
  operator int() const { return to; }
};

template<typename T>
using Graph = vector<vector<edge<T>>>;
template<typename T>
Graph<T>revgraph(const Graph<T> &g){
  Graph<T>ret(g.size());
  for(int i=0;i<g.size();i++){
    for(auto e:g[i]){
      int to = e.to;
      e.to = i;
      ret[to].push_back(e);
    }
  }
  return ret;
}
template<typename T>
Graph<T> readGraph(int n,int m,int indexed=1,bool directed=false,bool weighted=false){
  Graph<T> ret(n);
  for(int es = 0; es < m; es++){
    int u,v;
    T w=1;
    cin>>u>>v;u-=indexed,v-=indexed;
    if(weighted)cin>>w;
    ret[u].emplace_back(v,w,es);
    if(!directed)ret[v].emplace_back(u,w,es);
  }
  return ret;
}
template<typename T>
Graph<T> readParent(int n,int indexed=1,bool directed=true){
  Graph<T>ret(n);
  for(int i=1;i<n;i++){
    int p;cin>>p;
    p-=indexed;
    ret[p].emplace_back(i);
    if(!directed)ret[i].emplace_back(p);
  }
  return ret;
}
template<typename T>
struct HLD{
  using D=long long;
  int n;
  vector<int>sz;//部分木サイズ
  vector<D>dep;
  vector<int>par;
  vector<int>head;
  Graph<T> &g;//隣接リスト
  vector<edge<T>>edges;//データ構造に乗せるedge列
  vector<int>in,out;//[in,out)で部分木、[ina,inb]でa~bのパス(aが上)
  vector<int>comp;//連結成分の根
  //inは頂点のindexを表す。また、edge列の下側の頂点である
  HLD(Graph<T> &g,int r=-1):g(g),n(g.size()){
    hld_build(r);
  }
  void hld_build(int root = -1){
    in.assign(n,-1);out.assign(n,-1);dep.assign(n,0);
    par.assign(n,-1);head.assign(n,-1);sz.assign(n,-1);comp.assign(n,-1);
    edges.assign(n,edge<T>());
    if(root == -1){//根がどこでも良い場合(森でも可)
      for(int i=0;i<n;i++){
        if(sz[i] == -1){
          head[i] = i;
          dfs_sz(i, 0, i);
          dfs_hld(i);
        }
      }
    }
    else{
      head[root] = root;
      dfs_sz(root, 0, root);
      dfs_hld(root);
    }
  }
  void dfs_sz(int k, D d,int r){
    sz[k] = 1;
    comp[k] = r;
	dep[k] = d;
    for(auto &e: g[k]){
      if(e.to == par[k])continue;
      par[e.to] = k;
      dfs_sz(e.to, d+e.cost, r);
      sz[k] += sz[e.to];
      if(g[k][0].to==par[k]||sz[e.to] > sz[g[k][0].to])swap(e, g[k][0]);
    }
  }
  int time = 0;
  void dfs_hld(int k){
    in[k] = time++;
    for(auto e:g[k]){
      if(e.to == par[k])continue;
      head[e.to] = (e.to == g[k][0].to ? head[k]: e.to);
      edges[time] = e;
      dfs_hld(e.to);
    }
    out[k] = time;
  }
  int lca(int p,int q){
    while(1){
      if(in[p] < in[q])swap(p,q);
      if(head[p] == head[q])return q;
      p = par[head[p]];
    }
  }
  vector<pair<int,int>>query_path(int p,int q,bool isEdge){
    int r=lca(p,q);
    vector<pair<int,int>>ret;
    for(int i=0;i<2;i++){
      if(i == 1)swap(p,q);
      while(1){
        if(isEdge&&p==r)break;
        if(head[p]==head[r]){
          ret.emplace_back(in[r]+(isEdge?1:i),in[p]+1);
          break;
        }
        ret.emplace_back(in[head[p]],in[p]+1);
        p = par[head[p]];
      }
    }
    return ret;
  }
  vector<vector<pair<int,int>>>query_order_path(int p,int q,bool isEdge){
	//非可換クエリ用、配列0を順番を反転したデータ構造に、配列1を通常のデータ構造に
    vector<vector<pair<int,int>>>ret(2);
    int r=lca(p,q);
    for(int i=0;i<2;i++){
      if(i == 1)swap(p,q);
      while(1){
        if(isEdge&&p==r)break;
        if(head[p]==head[r]){
          if(i==0) ret[i].emplace_back(n-(in[p]+1),n-(in[r]+(isEdge?1:i)));
          else ret[i].emplace_back(in[r]+(isEdge?1:i),in[p]+1);
          break;
        }
        if(i==0) ret[i].emplace_back(n-(in[p]+1),n-(in[head[p]]));
        else ret[i].emplace_back(in[head[p]],in[p]+1);
        p = par[head[p]];
      }
    }
    reverse(ret[1].begin(), ret[1].end());
    return ret;
  }
  pair<int,int>query_subtree(int p,bool isEdge){
    return make_pair(in[p]+isEdge,out[p]);
  }
  //uのv方向の子 子孫関係は前もって確認すること(in,outを見る)
  int child(int u,int v)const{
    while(1){
      if(head[u]==head[v]){
        v=g[u][0].to;
        break;
      }
      v=head[v];
      if(par[v]==u)break;
      v=par[v];
    }
    return v;
  }
  //uをv方向に一つ進めた頂点
  int move(int u,int v)const{
    assert(u!=v);
    if(in[u]<in[v]&&in[v]<out[u])return child(u,v);
    else return par[u];
  }
  D dist(int u,int v){
    return dep[u]+dep[v]-2*dep[lca(u,v)];
  }
  vector<int>rev_in;
  int climb(int u,int k){
    if(rev_in.empty()){
      rev_in.resize(n);
      for(int i=0;i<n;i++)rev_in[in[i]]=i;
    }
    int nd=max<int>(dep[u]-k, 0);
    while(dep[u]>nd){
      if(dep[head[u]]>nd){
        u=par[head[u]];
      }
      else{
        u=rev_in[in[head[u]]+nd-dep[head[u]]];
      }
    }
    return u;
  }
  int jump(int from,int to, int k){
    int r = lca(from, to);
    int d1 = dep[from] - dep[r];
    int d2 = dep[to] - dep[r];
    if(d1 + d2 < k)return -1;
    else if(k <= d1)return climb(from, k);
    else return climb(to, d1 + d2 - k); 
  }
  template<typename I>
  Graph<T>lca_tree(vector<I>&v){
    auto compare=[&](int x,int y){return in[x]<in[y];};
    sort(v.begin(),v.end(),compare);
    int sz1=v.size();
    for(int i=0;i<sz1-1;i++)v.push_back(lca(v[i],v[i+1]));
    sort(v.begin(),v.end(),compare);
    v.erase(unique(v.begin(),v.end()),v.end());
    int sz2=v.size();
    Graph<T>ret(sz2);
    stack<int>st;
    for(int i=0;i<sz2;i++){
      while(!st.empty()&&out[v[st.top()]]<=in[v[i]])st.pop();
      if(!st.empty()){
        ret[st.top()].emplace_back(i,dep[v[i]]-dep[v[st.top()]]);
        ret[i].emplace_back(st.top(),dep[v[i]]-dep[v[st.top()]]);
      }
      st.push(i);
    }
    return ret;
  }
};

const int VERTEX = 0;
const int HEAVY_LIGHT = -1;
const int LIGHT = -2;
const int HEAVY = -3;
template<typename T>
struct StaticToptree{
  const HLD<T> &hld;
  int count;
  int root;
  vector<int>l,r,p;
  vector<int>aff;
  T edge_identity;
  StaticToptree(const HLD<T> &hld):hld(hld),edge_identity(edge_identity),count(hld.n),l(hld.n*2-1,-1),r(hld.n*2-1,-1),p(hld.n*2-1,-1),aff(hld.n*2-1,VERTEX){
    int hld_root=min_element(hld.in.begin(),hld.in.end())-hld.in.begin();
    root = dfs(hld_root,-1).second;
  }
  //sz,vid
  pair<int,int>dfs(int v,int par){
    int sub=0;
    vector<int>vs;
    vector<int>sz;
    int now=v,pre=par;
    while(1){
      int sum_sz=1;
      //vid,eid
      priority_queue<pair<int,int>,vector<pair<int,int>>,greater<pair<int,int>>>pq;
      for(int i=1;i<hld.g[now].size();i++){
        if(hld.g[now][i].to==pre)continue;
        auto [sz, vid] = dfs(hld.g[now][i].to, now);
        pq.emplace(sz,vid);
        sum_sz+=sz;
      }
      sub+=sum_sz;
      while(pq.size()>=2){
        auto [sx,x]=pq.top();pq.pop();
        auto [sy,y]=pq.top();pq.pop();
        int nv=count++;
        aff[nv]=LIGHT;
        l[nv]=x;
        p[x]=nv;
        r[nv]=y;
        p[y]=nv;
        pq.emplace(sx+sy,nv);
      }
      int last=now;
      if(!pq.empty()){
        auto [sx,x]=pq.top();pq.pop();
        int nv=count++;
        aff[nv]=HEAVY_LIGHT;
        l[nv]=now;
        p[x]=nv;
        r[nv]=x;
        p[now]=nv;
        last=nv;
      }
      vs.push_back(last);
      sz.push_back(sum_sz);
      if(hld.g[now].empty()||hld.g[now][0]==pre)break;
      pre=now;
      now=hld.g[now][0];
    }
    vector<int>csz(sz.size()+1);
    for(int i=0;i<sz.size();i++){
      csz[i+1]=csz[i]+sz[i];
    }
    auto rec=[&](auto &&rec,int ls,int rs)->int {
      if(ls+1==rs){
        return vs[ls];
      }
      int sum=csz[rs]-csz[ls];
      int lv=-1,rv=-1;
      if(sz[ls]*2>=sum){
        lv=rec(rec,ls,ls+1);
        rv=rec(rec,ls+1,rs);
      }
      else if(sz[rs-1]*2>=sum){
        lv=rec(rec,ls,rs-1);
        rv=rec(rec,rs-1,rs);
      }
      else{
        int ri=min<int>(rs-1,lower_bound(csz.begin()+ls,csz.begin()+rs,csz[ls]+(sum+1)/2)-csz.begin());
        int li=max(ls+1,ri-1);
        if(abs((csz[li]-csz[ls])*2-sum)<abs((csz[ri]-csz[ls])*2-sum)){
          ri=li;
        }
        lv=rec(rec,ls,ri);
        rv=rec(rec,ri,rs);
      }
      int nv=count++;
      aff[nv]=HEAVY;
      l[nv]=lv,p[lv]=nv;
      r[nv]=rv,p[rv]=nv;
      return nv;
    };
    int last=rec(rec,0,sz.size());
    return make_pair(sub,last);
  }
};

template<typename T,typename V,typename F1,typename F2,typename F3>
struct DynamicReRooting{
  const StaticToptree<T> &t;
  vector<V>seg,ges;
  const F1 &fh;
  const F2 &fl;
  const F3 &fhl;
  const V M1;
  vector<int>path;
  DynamicReRooting(const StaticToptree<T> &t,const F1 &heavy_merge,const F2 &light_merge,const F3 &heavy_light_merge,const V M1):
  t(t),fh(heavy_merge),fl(light_merge),fhl(heavy_light_merge),M1(M1){
    seg.assign(t.aff.size(),M1);
    ges.assign(t.aff.size(),M1);
  }
  void set(int k,V val){
    seg[k]=val;
    ges[k]=val;
  }
  void recalc(int k){
    if(t.aff[k]==LIGHT){
      seg[k]=fl(seg[t.l[k]],seg[t.r[k]]);
    }
    else if(t.aff[k]==HEAVY){
      seg[k]=fh(seg[t.l[k]],seg[t.r[k]]);
      ges[k]=fh(ges[t.r[k]],ges[t.l[k]]);
    }
    else if(t.aff[k]==HEAVY_LIGHT){
      seg[k]=fhl(seg[t.l[k]],seg[t.r[k]]);
      ges[k]=seg[k];
    }
  }
  void build(){
    for(int i=0;i<t.aff.size();i++){
      recalc(i);
    }
  }
  void update(int k,V val){
    seg[k]=val;
    ges[k]=val;
    while(t.p[k]!=-1){
      k=t.p[k];
      recalc(k);
    }
  }
  int root_path(int k){
    int cnt=0;
    while(1){
      if(cnt>=path.size())path.push_back(0);
      path[cnt++]=k;
      if(t.p[k]==-1)break;
      k=t.p[k];
    }
    return cnt;
  }
  //head,tail,light
  tuple<V,V,V>get(int k){
    int m=root_path(k);
    V head=M1,tail=M1,light=M1;
    int hl=-1;
    for(int i=m-1;i>=1;i--){
      if(t.aff[path[i]]==HEAVY){
        if(t.l[path[i]]==path[i-1]){
          tail=fh(seg[t.r[path[i]]],tail);
        }
        else{
          auto lt=ges[t.l[path[i]]];
          head=fh(ges[t.l[path[i]]],head);
        }
      }
      else if(t.aff[path[i]]==HEAVY_LIGHT){
        if(i==1&&k==t.l[path[i]])light=seg[t.r[path[i]]];
        else hl=t.l[path[i]];
      }
      else{
        //LIGHT
        if(t.l[path[i]]==path[i-1]){
          light=fl(seg[t.r[path[i]]],light);
        }
        else{
          light=fl(seg[t.l[path[i]]],light);
        }
      }
      if(hl!=-1&&t.aff[path[i-1]]!=LIGHT){
        head=fhl(seg[hl],fl(light,fl(head,tail)));
        tail=M1;
        light=M1;
        hl=-1;
      }
    }
    return make_tuple(head,tail,light);
  }
  V reroot(int r){
    auto [head,tail,light]=get(r);
    return fhl(seg[r],fl(light,fl(head,tail)));
  }
  V reroot(int r,int k){
    if(r==k)return reroot(r);
    r=t.hld.move(k,r);
    if(r==t.hld.par[k]){
      auto [head,tail,light]=get(k);
      return fhl(seg[k],fl(light,tail));
    }
    else{
      auto [head,tail,light]=get(r);
      return head;
    }
  }
  V prod_root(){
    return seg.back();
  }
  template<typename PF>
  void print(PF printer){
    vector<vector<int>>vst(t.aff.size());
    for(int i=0;i<t.aff.size();i++){
      if(t.aff[i]==VERTEX){
        vst[i].push_back(i);
        cout<<"[VERTEX]: "<<i<<endl;
      }
      else if(t.aff[i]==HEAVY){
        vst[i]=vst[t.l[i]];
        vst[i].insert(vst[i].end(),vst[t.r[i]].begin(),vst[t.r[i]].end());
        cout<<"[HEAVY]: ";for(auto v:vst[i])cout<<v<<" ";cout<<endl;
      }
      else if(t.aff[i]==LIGHT){
        for(auto v:{t.l[i],t.r[i]}){
          if(t.aff[v]==LIGHT)vst[i].insert(vst[i].end(),vst[v].begin(),vst[v].end());
          else vst[i].push_back(vst[v][0]);
        }
        cout<<"[LIGHT]: ";for(auto v:vst[i])cout<<v<<" ";cout<<endl;
      }
      else{
        vst[i].push_back(vst[t.l[i]][0]);
        cout<<"[HL]: "<<vst[i][0]<<" exclude: "<<t.hld.g[vst[i][0]][0].to<<endl;
      }
      printer(seg[i]);
    }
  }
};

struct UnionFind {
  vector<ll> data;
  ll num;
  UnionFind(ll size) : data(size, -1) ,num(size){ }
  bool unite(int x, int y) {
    x = root(x); y = root(y);
    if (x != y) {
      if (data[y] < data[x]) swap(x, y);
      data[x] += data[y]; data[y] = x;
    num--;
    }
    return x != y;
  }
  bool find(int x, int y) {
    return root(x) == root(y);
  }
  int root(int x) {
    return data[x] < 0 ? x : data[x] = root(data[x]);
  }
  ll size(int x) {
    return -data[root(x)];
  }
  vector<vector<int>>groups(){
    vector<vector<int>>ret(data.size());
    for(int i=0;i<data.size();i++){
      ret[root(i)].push_back(i);
    }
    return ret;
  }
  void print(){
    auto tmp=groups();
    for(int i=0;i<data.size();i++){
      if(!tmp[i].empty()){
        for(auto z:tmp[i])cout<<z<<",";
        cout<<endl;
      }
    }
  }
};
struct FastSet{
  using ull = unsigned long long;
  static constexpr unsigned int B = 64;
  int n,lg;
  vector<vector<ull>> seg;
  FastSet(int _n) : n(_n){
    do{
      seg.push_back(vector<ull>((_n + B - 1) / B));
      _n = (_n + B - 1) / B;
    } while (_n > 1);
    lg = seg.size();
  }
  bool operator[](int i) const {
    return (seg[0][i / B] >> (i % B) & 1) != 0;
  }
  void set(int i)
  {
    for(int h = 0; h < lg; h++){
      seg[h][i / B] |= 1ULL << (i % B);
      i /= B;
    }
  }
  void reset(int i)
  {
    for(int h = 0; h < lg; h++){
      seg[h][i / B] &= ~(1ULL << (i % B));
      if (seg[h][i / B])
        break;
      i /= B;
    }
  }
  // x以上最小の要素
  int next(int i){
    for(int h = 0;h < lg; h++){
      if (i / B == seg[h].size())
        break;
      ull d = seg[h][i / B] >> (i % B);
      if (!d){
        i = i / B + 1;
        continue;
      }
      // find
      i += __builtin_ctzll(d);
      for (int g = h - 1;g >= 0;g--){
        i *= B;
        i += __builtin_ctzll(seg[g][i / B]);
      }
      return i;
    }
    return n;
  }
  // x未満最大の要素
  int prev(int i){
    i--;
    for (int h = 0;h < lg; h++){
      if (i == -1)
        break;
      ull d = seg[h][i / B] << (63 - i % 64);
      if (!d){
        i = i / B -1;
        continue;
      }
      // find
      i += (63 - __builtin_clzll(d)) - (B - 1);
      for(int g = h - 1;g >= 0;g--){
        i *= B;
        i += (63 - __builtin_clzll(seg[g][i / B]));
      }
      return i;
    }
    return -1;
  }
};
int main(){
  cin.tie(nullptr);
  ios_base::sync_with_stdio(false);
  ll res=0,buf=0;
  bool judge = true;
  int n=200010;
  //cout<<"naose"<<endl;int n=6;
  int off=n/2;
  UnionFind uf(n);
  ll q;cin>>q;
  vector<ll>x(q),y(q),e(q,-1);
  Graph<int>g(2*n-1);
  int ei=n;
  rep(i,0,q){
    cin>>x[i]>>y[i];
    y[i]+=off;
    if(uf.unite(x[i],y[i])){
      e[i]=ei++;
      g[x[i]].emplace_back(e[i],1);
      g[y[i]].emplace_back(e[i],1);
      g[e[i]].emplace_back(x[i],1);
      g[e[i]].emplace_back(y[i],1);
      //OUT(x[i],y[i],e[i]);
    }
  }
  rep(i,1,n){
    if(uf.unite(0,i)){
      g[0].EB(ei,1);
      g[ei].EB(0,1);
      g[ei].EB(i,1);
      g[i].EB(ei,1);
      ei++;
    }
  }
  HLD hld(g);
  StaticToptree stp(hld);
  struct S{
    bool cut;
    int odd=0;
    int sum0,sum1;
  };
  auto hm=[&](S h,S t){
    if(h.cut)return h;
    if(h.odd)swap(t.sum0,t.sum1);
    auto ret=S{t.cut,h.odd^t.odd,h.sum0+t.sum0,h.sum1+t.sum1};
    return ret;
  };
  auto lm=[&](S x,S y){
    return S{false,0,x.sum0+y.sum0,x.sum1+y.sum1};
  };
  auto hlm=[&](S h,S t){
    if(h.cut)return h;
    return S{false,h.odd,h.sum0+t.sum0,h.sum1+t.sum1};
  };
  S iden{false,0,0,0};
  DynamicReRooting seg(stp,hm,lm,hlm,iden);
  rep(i,0,n)seg.set(i,S{false,0,0,0});
  rep(i,n,2*n-1)seg.set(i,S{true,0,0,0});
  seg.build();
  uf=UnionFind(n);
  vector<P>dia(n);
  rep(i,0,n){
    dia[i]=P(i,i);
  }
  FastSet fs(2*n);
  rep(i,n,ei)fs.set(hld.in[i]);
  auto inv=inv_perm(hld.in);
  vector<P>sum(n);
  P ret={0,0};
  auto recalc=[&](ll pos){
    pos=uf.root(pos);
    sum[pos]=P(0,0);
    ll l=dia[pos].fi,r=dia[pos].se;
    ll diam=hld.dist(l,r);
    ll ce=hld.jump(l,r,diam/2);
    if(ce>=n){
      if(l<off)l=hld.jump(l,r,2);
      else r=hld.jump(r,l,2);
      ll diam=hld.dist(l,r);
      ce=hld.jump(l,r,diam/2);
      sum[0].fi++;
      //OUT(l,r);
    }
    auto tmp=seg.reroot(ce);
    if(ce<off)swap(tmp.sum0,tmp.sum1);
    sum[pos].fi+=tmp.sum0;
    sum[pos].se+=tmp.sum1;
    //OUT(ce,tmp.sum0,tmp.sum1);
    
  };
  rep(i,0,q){
    if(uf.find(x[i],y[i])){
      auto tmp=hld.query_path(x[i],y[i],false);
      OUT(tmp);
      bool ch=false;
      ll pos=uf.root(x[i]);
      for(auto z:tmp){
        while(1){
          int nxt=fs.next(z.fi);
          if(nxt>=z.se)break;
          fs.reset(nxt);
          nxt=inv[nxt];
          seg.update(nxt,S{true,0,0,0});
          OUT(i,nxt);
          ch=true;
        }
      }
      if(ch){
        ret.fi-=sum[pos].fi,ret.se-=sum[pos].se;
        recalc(pos);
        ret.fi+=sum[pos].fi,ret.se+=sum[pos].se;
      }
    }
    else{
      ll posl=uf.root(x[i]),posr=uf.root(y[i]);
      ret.fi-=sum[posl].fi+sum[posr].fi,ret.se-=sum[posl].se+sum[posr].se;
      ll ndl=-1,ndr=-1,ndist=-1;
      for(auto lv:{dia[posl].fi,dia[posl].se}){
        for(auto rv:{dia[posr].fi,dia[posr].se}){
          ll dist=hld.dist(lv,rv);
          if(chmax(ndist,dist)){
            ndl=lv;
            ndr=rv;
          }
        }
      }
      uf.unite(x[i],y[i]);
      dia[uf.root(x[i])]=P(ndl,ndr);
      seg.update(e[i],S{false,1,1,0});
      recalc(uf.root(x[i]));
      ret.fi+=sum[uf.root(x[i])].fi,ret.se+=sum[uf.root(x[i])].se;
    }
    // OUT(sum);
    // OUT(dia);
    // seg.print([&](S x){
    //   cout<<x.cut<<" "<<x.odd<<" "<<x.sum0<<" "<<x.sum1<<endl;
    // });
    cout<<ret.fi<<" "<<ret.se<<endl;
  }
  return 0;
}