//#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;}
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;}
std::ostream &operator<<(std::ostream &dest, __int128_t value) {
  std::ostream::sentry s(dest);
  if (s) {
    __uint128_t tmp = value < 0 ? -value : value;
    char buffer[128];
    char *d = std::end(buffer);
    do {
      --d;
      *d = "0123456789"[tmp % 10];
      tmp /= 10;
    } while (tmp != 0);
    if (value < 0) {
      --d;
      *d = '-';
    }
    int len = std::end(buffer) - d;
    if (dest.rdbuf()->sputn(d, len) != len) {
      dest.setstate(std::ios_base::badbit);
    }
  }
  return dest;
}
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),r(hld.n*2,-1),p(hld.n*2,-1),aff(hld.n*2,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){
        //OUT("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){
        //OUT("HL");
        if(i==1&&k==t.l[path[i]])light=seg[t.r[path[i]]];
        else hl=t.l[path[i]];
      }
      else{
        //OUT("LIGHT");
        //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;
      }
      //OUT(path[i],head.ans,head.num,head.mul,tail.ans,tail.num,tail.mul,light.ans);
    }
    return make_tuple(head,tail,light);
  }
  V reroot(int r){
    auto [head,tail,light]=get(r);
    //OUT(seg[r].idx,seg[r].ans,seg[r].mul,fl(light,fl(head,tail)).ans);
    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;
    }
  }
};
template< class T , int maxsize>
struct SquareMatrix {
  using arr = array <T, maxsize>;
  using sqmat = array <arr, maxsize>;
  sqmat A;
  ll n;
  const T iden = 1;//mulの単位元
  const T zero = 0;//addの単位元
  const T add(T x, T y){
    return x + y;
  }
  const T mul(T x, T y){
    return x * y;
  }
  sqmat zeros(){
    sqmat mat;
    for(ll i=0;i<maxsize;i++)for(ll j=0;j<maxsize;j++)mat[i][j]=zero;
    return mat;
  }
  SquareMatrix() {
    n = maxsize;
    A = zeros();
  }
  SquareMatrix(size_t N) : n(N){
    A = zeros();
  };
  SquareMatrix(vector<vector<T>>v):n(v.size()){
    for(ll i=0;i<n;i++)for(ll j=0;j<n;j++)A[i][j]=v[i][j];
  };
  inline const arr &operator[](int k) const {
    return (A[k]);
  }

  inline arr &operator[](int k) {
    return (A[k]);
  }
  static SquareMatrix I(size_t N){
    SquareMatrix mat(N);
    for(int i = 0; i < N; i++) mat[i][i] = mat.iden;
    return (mat);
  }
  SquareMatrix &operator+=(const SquareMatrix &B) {
    for(int i = 0; i < n; i++)
      for(int j = 0; j < n; j++)
        (*this)[i][j] = add((*this)[i][j], B[i][j]);
    return (*this);
  }

  SquareMatrix &operator*=(const SquareMatrix &B) {
    sqmat C = zeros();
    for(int i = 0; i < n; i++)
      for(int j = 0; j < n; j++)
        for(int k = 0; k < n; k++)
          C[i][j] = add(C[i][j],mul((*this)[i][k] , B[k][j]));
    A.swap(C);
    return (*this);
  }
  SquareMatrix &operator^=(long long k) {
    SquareMatrix B = SquareMatrix::I(n);
    while(k > 0) {
      if(k & 1) B *= *this;
      *this *= *this;
      k >>= 1LL;
    }
    A.swap(B.A);
    return (*this);
  }

  SquareMatrix operator+(const SquareMatrix &B) const {
    return (SquareMatrix(*this) += B);
  }

  SquareMatrix operator*(const SquareMatrix &B) const {
    return (SquareMatrix(*this) *= B);
  }

  SquareMatrix operator^(const long long k) const {
    return (SquareMatrix(*this) ^= k);
  }
  bool operator==(const SquareMatrix &B)const{
    return this->A == B.A;
  }
  bool operator!=(const SquareMatrix &B)const{
    return this->A != B.A;
  }
  SquareMatrix& operator=(const SquareMatrix &b){
    this->n=b.n;
    this->A=b.A;
    return *this;
  }
  friend ostream &operator<<(ostream &os, SquareMatrix p) {
    for(int i = 0; i < p.n; i++) {
      os << "[";
      for(int j = 0; j < p.n; j++) {
        os << p[i][j] << (j + 1 == p.n ? "]\n" : ",");
      }
    }
    return (os);
  }
};
template< int mod >
struct ModInt {
  int x;

  ModInt() : x(0) {}

  ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

  ModInt &operator+=(const ModInt &p) {
    if((x += p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator-=(const ModInt &p) {
    if((x += mod - p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator*=(const ModInt &p) {
    x = (int) (1LL * x * p.x % mod);
    return *this;
  }

  ModInt &operator/=(const ModInt &p) {
    *this *= p.inverse();
    return *this;
  }

  ModInt operator-() const { return ModInt(-x); }

  friend ModInt operator+(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) += rhs;
  }
  friend ModInt operator-(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) -= rhs;
  }
  friend ModInt operator*(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) *= rhs;
  }
  friend ModInt operator/(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) /= rhs;
  }

  bool operator==(const ModInt &p) const { return x == p.x; }

  bool operator!=(const ModInt &p) const { return x != p.x; }

  ModInt inverse() const {
    int a = x, b = mod, u = 1, v = 0, t;
    while(b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ModInt(u);
  }

  ModInt pow(int64_t n) const {
    ModInt ret(1), mul(x);
    while(n > 0) {
      if(n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  pair<int,int>frac(){
    for(int j=1;j<=300;j++){
      for(int i=-300;i<=300;i++){
        if(ModInt(i)/j==*this){
          return make_pair(i,j);
        }
      }
    }
    return make_pair(-1,-1);
  }
  friend ostream &operator<<(ostream &os, const ModInt &p) {
    return os << p.x;
  }

  friend istream &operator>>(istream &is, ModInt &a) {
    int64_t t;
    is >> t;
    a = ModInt< mod >(t);
    return (is);
  }

  static int get_mod() { return mod; }
};

template< typename T >
struct Combination {
  vector< T > _fact, _rfact, _inv;

  Combination(ll sz) : _fact(sz + 1), _rfact(sz + 1), _inv(sz + 1) {
    _fact[0] = _rfact[sz] = _inv[0] = 1;
    for(ll i = 1; i <= sz; i++) _fact[i] = _fact[i - 1] * i;
    _rfact[sz] /= _fact[sz];
    for(ll i = sz - 1; i >= 0; i--) _rfact[i] = _rfact[i + 1] * (i + 1);
    for(ll i = 1; i <= sz; i++) _inv[i] = _rfact[i] * _fact[i - 1];
  }

  inline T fact(ll k) const { return _fact[k]; }

  inline T rfact(ll k) const { return _rfact[k]; }

  inline T inv(ll k) const { return _inv[k]; }

  T P(ll n, ll r) const {
    if(r < 0 || n < r) return 0;
    return fact(n) * rfact(n - r);
  }

  T C(ll p, ll q) const {
    if(q < 0 || p < q) return 0;
    return fact(p) * rfact(q) * rfact(p - q);
  }
  
  T RC(ll p, ll q) const {
    if(q < 0 || p < q) return 0;
    return rfact(p) * fact(q) * fact(p - q);
  }

  T H(ll n, ll r) const {
    if(n < 0 || r < 0) return (0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
  //+1がm個、-1がn個で prefix sumが常にk以上
  T catalan(ll m,ll n,ll k){
    if(n>m-k)return 0;
    else return C(n+m,m)-C(n+m,n+k-1);
  }
};
using modint = ModInt< MOD9 >;modint mpow(ll n, ll x){return modint(n).pow(x);}modint mpow(modint n, ll x){return n.pow(x);}
//using modint=ld;modint mpow(ll n, ll x){return pow(n,x);}modint mpow(modint n, ll x){return pow(n,x);}
using Comb=Combination<modint>;


void solve(){
	ll res=0,buf=0;
  bool judge = true;

  using SQ=SquareMatrix<modint,3>;
  int n,q;cin>>n>>q;
  vector<int>lch(n);
  Graph<int>g(n);
  struct A{
    SQ lx;
    //rx;
    int v; //一番上の頂点
    array<modint,3>val;
  };
  A ident;
  ident.lx=SQ(3);
  //ident.rx=SQ(3);
  ident.v=-1;
  ident.val[0]=1,ident.val[1]=1,ident.val[2]=1;
  
  auto get=[&](int v,int x,int y,int z){
    auto ret=ident;
    ret.v=v;
    ret.lx[1][0]=z;
    ret.lx[0][1]=-z;
    ret.lx[2][0]=-y;
    ret.lx[0][2]=y;
    ret.lx[1][2]=-x;
    ret.lx[2][1]=x;

    // ret.rx[1][0]=-z;
    // ret.rx[0][1]=z;
    // ret.rx[2][0]=y;
    // ret.rx[0][2]=-y;
    // ret.rx[1][2]=x;
    // ret.rx[2][1]=-x;
    
    ret.val[0]=x,ret.val[1]=y,ret.val[2]=z;
    //OUT(ret.x);
    return ret;
  };
  auto fh=[&](A h,A l){
    if(h.v==-1)return l;
    if(l.v==-1)return h;
    //OUT(h.val,l.val);
    //OUT(h.lx);
    h.val[0]=0,h.val[1]=0,h.val[2]=0;
    if(lch[l.v]){
      //OUT("l");
      rep(i,0,3)rep(j,0,3)h.val[j]+=l.val[i]*h.lx[i][j];
      h.lx=l.lx*h.lx;
      //h.lx=h.lx*l.lx;
    }
    else{
      //assert(false);
      //OUT("r");
      rep(i,0,3)rep(j,0,3)h.val[i]+=h.lx[i][j]*l.val[j];
      h.lx=h.lx*l.lx;
      //h.lx=l.lx*h.lx;
      //rep(i,0,3)rep(j,0,3)h.val[j]+=l.val[i]*h.lx[i][j];
    }
    //OUT(h.v);
    return h;
  };
  auto fl=[&](A l,A r){
    if(l.v==-1)return r;
    if(r.v==-1)return l;
    //OUT(l.v,r.v);
    assert(false);
    //二分木なので単位元以外ないはず
  };
  
  auto fhl=[&](A h,A l){
    if(h.v==-1)return l;
    if(l.v==-1)return h;
    //OUT(l.v,h.v);
    auto ret=get(h.v,l.val[0].x,l.val[1].x,l.val[2].x);
    return ret;
  };
  
  vector<int>ix(n,0),iy(n,0),iz(n,0);
  rep(i,0,n){
    char c;cin>>c;
    if(c=='x'){
      int l,r;cin>>l>>r;l--;r--;
      lch[l]=true;
      g[i].EB(l);
      g[i].EB(r);
      g[l].PB(i);
      g[r].PB(i);
    }
    else{
      cin>>ix[i]>>iy[i]>>iz[i];
    }
  }
  HLD hld(g,0);
  StaticToptree st(hld);
  DynamicReRooting seg(st,fh,fl,fhl,ident);
  rep(i,0,n){
    seg.set(i,get(i,ix[i],iy[i],iz[i]));
  }
  seg.build();
  // rep(i,0,seg.seg.size()){
  //   OUT(i,seg.seg[i].v);
  // }
  while(q--){
    int v,x,y,z;cin>>v>>x>>y>>z;v--;
    seg.update(v,get(v,x,y,z));
    //OUT(seg.seg[0].x,seg.seg[0].v);
    //auto ret=seg.reroot(0);
    auto ret=seg.seg[2*n-2];
    // rep(i,0,seg.seg.size()){
    //   if(q==2){
    //     OUT(i,st.l[i],st.r[i],seg.seg[i].lx,seg.seg[i].v,seg.seg[i].val);
    //     OUT("-----");
    //   }
    // }
    //OUT(ret.v,ret.val);
    debug(ret.val);
    // OUT(mat1.x,mat2.x,mat3.x);
    // OUT(mat2.x*mat3.x);
    // OUT(mat1.x*(mat2.x*mat3.x));
  }
}

int main(){
  cin.tie(nullptr);
  ios_base::sync_with_stdio(false);
  ll res=0,buf=0;
  bool judge = true;
  int T = 1;
  //cin>>T;
  while(T--){
    solve();
  }
  return 0;
}