#line 1 "library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) \
  for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T floor(T a, T b) {
  return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
  return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sm = 0;
  for (auto &&a: A) sm += a;
  return sm;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) \
  sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    (check(x) ? ok : ng) = x;
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    (check(x) ? ok : ng) = x;
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}
#endif
#line 1 "library/other/io.hpp"
#define FASTIO
#include <unistd.h>

// https://judge.yosupo.jp/submission/21623
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;

struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;

inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin);
  pil = 0;
  if (pir < SZ) ibuf[pir++] = '\n';
}

inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}

void rd(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
}

void rd(string &x) {
  x.clear();
  char c;
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
  do {
    x += c;
    if (pil == pir) load();
    c = ibuf[pil++];
  } while (!isspace(c));
}

template <typename T>
void rd_real(T &x) {
  string s;
  rd(s);
  x = stod(s);
}

template <typename T>
void rd_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  x = 0;
  while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (minus) x = -x;
  }
}

void rd(int &x) { rd_integer(x); }
void rd(ll &x) { rd_integer(x); }
void rd(i128 &x) { rd_integer(x); }
void rd(u32 &x) { rd_integer(x); }
void rd(u64 &x) { rd_integer(x); }
void rd(u128 &x) { rd_integer(x); }
void rd(double &x) { rd_real(x); }
void rd(long double &x) { rd_real(x); }
void rd(f128 &x) { rd_real(x); }

template <class T, class U>
void rd(pair<T, U> &p) {
  return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T>
void rd_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd(x);
    rd_tuple<N + 1>(t);
  }
}
template <class... T>
void rd(tuple<T...> &tpl) {
  rd_tuple(tpl);
}

template <size_t N = 0, typename T>
void rd(array<T, N> &x) {
  for (auto &d: x) rd(d);
}
template <class T>
void rd(vc<T> &x) {
  for (auto &d: x) rd(d);
}

void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd(h), read(t...);
}

void wt(const char c) {
  if (por == SZ) flush();
  obuf[por++] = c;
}
void wt(const string s) {
  for (char c: s) wt(c);
}
void wt(const char *s) {
  size_t len = strlen(s);
  for (size_t i = 0; i < len; i++) wt(s[i]);
}

template <typename T>
void wt_integer(T x) {
  if (por > SZ - 100) flush();
  if (x < 0) { obuf[por++] = '-', x = -x; }
  int outi;
  for (outi = 96; x >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[x % 10000], 4);
    x /= 10000;
  }
  if (x >= 1000) {
    memcpy(obuf + por, pre.num[x], 4);
    por += 4;
  } else if (x >= 100) {
    memcpy(obuf + por, pre.num[x] + 1, 3);
    por += 3;
  } else if (x >= 10) {
    int q = (x * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (x - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = x | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

template <typename T>
void wt_real(T x) {
  ostringstream oss;
  oss << fixed << setprecision(15) << double(x);
  string s = oss.str();
  wt(s);
}

void wt(int x) { wt_integer(x); }
void wt(ll x) { wt_integer(x); }
void wt(i128 x) { wt_integer(x); }
void wt(u32 x) { wt_integer(x); }
void wt(u64 x) { wt_integer(x); }
void wt(u128 x) { wt_integer(x); }
void wt(double x) { wt_real(x); }
void wt(long double x) { wt_real(x); }
void wt(f128 x) { wt_real(x); }

template <class T, class U>
void wt(const pair<T, U> val) {
  wt(val.first);
  wt(' ');
  wt(val.second);
}
template <size_t N = 0, typename T>
void wt_tuple(const T t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt(' '); }
    const auto x = std::get<N>(t);
    wt(x);
    wt_tuple<N + 1>(t);
  }
}
template <class... T>
void wt(tuple<T...> tpl) {
  wt_tuple(tpl);
}
template <class T, size_t S>
void wt(const array<T, S> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}
template <class T>
void wt(const vector<T> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}

void print() { wt('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt(head);
  if (sizeof...(Tail)) wt(' ');
  print(forward<Tail>(tail)...);
}

// gcc expansion. called automaticall after main.
void __attribute__((destructor)) _d() { flush(); }
} // namespace fastio
using fastio::read;
using fastio::print;
using fastio::flush;

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __VA_ARGS__; \
  read(__VA_ARGS__)
#define U32(...)   \
  u32 __VA_ARGS__; \
  read(__VA_ARGS__)
#define U64(...)   \
  u64 __VA_ARGS__; \
  read(__VA_ARGS__)
#define STR(...)      \
  string __VA_ARGS__; \
  read(__VA_ARGS__)
#define CHAR(...)   \
  char __VA_ARGS__; \
  read(__VA_ARGS__)
#define DBL(...)      \
  double __VA_ARGS__; \
  read(__VA_ARGS__)

#define VEC(type, name, size) \
  vector<type> name(size);    \
  read(name)
#define VV(type, name, h, w)                     \
  vector<vector<type>> name(h, vector<type>(w)); \
  read(name)

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 3 "main.cpp"

#line 2 "library/graph/tree.hpp"

#line 2 "library/graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    if (len(used_e) != M) used_e.assign(M, 0);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 4 "library/graph/tree.hpp"

// HLD euler tour をとっていろいろ。
template <typename GT>
struct Tree {
  using Graph_type = GT;
  GT &G;
  using WT = typename GT::cost_type;
  int N;
  vector<int> LID, RID, head, V, parent, VtoE;
  vc<int> depth;
  vc<WT> depth_weighted;

  Tree(GT &G, int r = 0, bool hld = 1) : G(G) { build(r, hld); }

  void build(int r = 0, bool hld = 1) {
    if (r == -1) return; // build を遅延したいとき
    N = G.N;
    LID.assign(N, -1), RID.assign(N, -1), head.assign(N, r);
    V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1);
    depth.assign(N, -1), depth_weighted.assign(N, 0);
    assert(G.is_prepared());
    int t1 = 0;
    dfs_sz(r, -1, hld);
    dfs_hld(r, t1);
  }

  void dfs_sz(int v, int p, bool hld) {
    auto &sz = RID;
    parent[v] = p;
    depth[v] = (p == -1 ? 0 : depth[p] + 1);
    sz[v] = 1;
    int l = G.indptr[v], r = G.indptr[v + 1];
    auto &csr = G.csr_edges;
    // 使う辺があれば先頭にする
    for (int i = r - 2; i >= l; --i) {
      if (hld && depth[csr[i + 1].to] == -1) swap(csr[i], csr[i + 1]);
    }
    int hld_sz = 0;
    for (int i = l; i < r; ++i) {
      auto e = csr[i];
      if (depth[e.to] != -1) continue;
      depth_weighted[e.to] = depth_weighted[v] + e.cost;
      VtoE[e.to] = e.id;
      dfs_sz(e.to, v, hld);
      sz[v] += sz[e.to];
      if (hld && chmax(hld_sz, sz[e.to]) && l < i) { swap(csr[l], csr[i]); }
    }
  }

  void dfs_hld(int v, int &times) {
    LID[v] = times++;
    RID[v] += LID[v];
    V[LID[v]] = v;
    bool heavy = true;
    for (auto &&e: G[v]) {
      if (depth[e.to] <= depth[v]) continue;
      head[e.to] = (heavy ? head[v] : e.to);
      heavy = false;
      dfs_hld(e.to, times);
    }
  }

  vc<int> heavy_path_at(int v) {
    vc<int> P = {v};
    while (1) {
      int a = P.back();
      for (auto &&e: G[a]) {
        if (e.to != parent[a] && head[e.to] == v) {
          P.eb(e.to);
          break;
        }
      }
      if (P.back() == a) break;
    }
    return P;
  }

  int heavy_child(int v) {
    int k = LID[v] + 1;
    if (k == N) return -1;
    int w = V[k];
    return (parent[w] == v ? w : -1);
  }

  int e_to_v(int eid) {
    auto e = G.edges[eid];
    return (parent[e.frm] == e.to ? e.frm : e.to);
  }
  int v_to_e(int v) { return VtoE[v]; }

  int ELID(int v) { return 2 * LID[v] - depth[v]; }
  int ERID(int v) { return 2 * RID[v] - depth[v] - 1; }

  // 目標地点へ進む個数が k
  int LA(int v, int k) {
    assert(k <= depth[v]);
    while (1) {
      int u = head[v];
      if (LID[v] - k >= LID[u]) return V[LID[v] - k];
      k -= LID[v] - LID[u] + 1;
      v = parent[u];
    }
  }
  int la(int u, int v) { return LA(u, v); }

  int LCA(int u, int v) {
    for (;; v = parent[head[v]]) {
      if (LID[u] > LID[v]) swap(u, v);
      if (head[u] == head[v]) return u;
    }
  }
  // root を根とした場合の lca
  int LCA_root(int u, int v, int root) {
    return LCA(u, v) ^ LCA(u, root) ^ LCA(v, root);
  }
  int lca(int u, int v) { return LCA(u, v); }
  int lca_root(int u, int v, int root) { return LCA_root(u, v, root); }

  int subtree_size(int v, int root = -1) {
    if (root == -1) return RID[v] - LID[v];
    if (v == root) return N;
    int x = jump(v, root, 1);
    if (in_subtree(v, x)) return RID[v] - LID[v];
    return N - RID[x] + LID[x];
  }

  int dist(int a, int b) {
    int c = LCA(a, b);
    return depth[a] + depth[b] - 2 * depth[c];
  }

  WT dist_weighted(int a, int b) {
    int c = LCA(a, b);
    return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c];
  }

  // a is in b
  bool in_subtree(int a, int b) { return LID[b] <= LID[a] && LID[a] < RID[b]; }

  int jump(int a, int b, ll k) {
    if (k == 1) {
      if (a == b) return -1;
      return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]);
    }
    int c = LCA(a, b);
    int d_ac = depth[a] - depth[c];
    int d_bc = depth[b] - depth[c];
    if (k > d_ac + d_bc) return -1;
    if (k <= d_ac) return LA(a, k);
    return LA(b, d_ac + d_bc - k);
  }

  vc<int> collect_child(int v) {
    vc<int> res;
    for (auto &&e: G[v])
      if (e.to != parent[v]) res.eb(e.to);
    return res;
  }

  vc<int> collect_light(int v) {
    vc<int> res;
    bool skip = true;
    for (auto &&e: G[v])
      if (e.to != parent[v]) {
        if (!skip) res.eb(e.to);
        skip = false;
      }
    return res;
  }

  vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) {
    // [始点, 終点] の"閉"区間列。
    vc<pair<int, int>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        down.eb(LID[head[v]], LID[v]);
        v = parent[head[v]];
      } else {
        up.eb(LID[u], LID[head[u]]);
        u = parent[head[u]];
      }
    }
    if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]);
    elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge);
    reverse(all(down));
    up.insert(up.end(), all(down));
    return up;
  }

  vc<int> restore_path(int u, int v) {
    vc<int> P;
    for (auto &&[a, b]: get_path_decomposition(u, v, 0)) {
      if (a <= b) {
        FOR(i, a, b + 1) P.eb(V[i]);
      } else {
        FOR_R(i, b, a + 1) P.eb(V[i]);
      }
    }
    return P;
  }
};
#line 5 "main.cpp"

#line 2 "library/ds/segtree/segtree.hpp"

template <class Monoid>
struct SegTree {
  using MX = Monoid;
  using X = typename MX::value_type;
  using value_type = X;
  vc<X> dat;
  int n, log, size;

  SegTree() {}
  SegTree(int n) { build(n); }
  template <typename F>
  SegTree(int n, F f) {
    build(n, f);
  }
  SegTree(const vc<X>& v) { build(v); }

  void build(int m) {
    build(m, [](int i) -> X { return MX::unit(); });
  }
  void build(const vc<X>& v) {
    build(len(v), [&](int i) -> X { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m, log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    dat.assign(size << 1, MX::unit());
    FOR(i, n) dat[size + i] = f(i);
    FOR_R(i, 1, size) update(i);
  }

  X get(int i) { return dat[size + i]; }
  vc<X> get_all() { return {dat.begin() + size, dat.begin() + size + n}; }

  void update(int i) { dat[i] = Monoid::op(dat[2 * i], dat[2 * i + 1]); }
  void set(int i, const X& x) {
    assert(i < n);
    dat[i += size] = x;
    while (i >>= 1) update(i);
  }

  void multiply(int i, const X& x) {
    assert(i < n);
    i += size;
    dat[i] = Monoid::op(dat[i], x);
    while (i >>= 1) update(i);
  }

  X prod(int L, int R) {
    assert(0 <= L && L <= R && R <= n);
    X vl = Monoid::unit(), vr = Monoid::unit();
    L += size, R += size;
    while (L < R) {
      if (L & 1) vl = Monoid::op(vl, dat[L++]);
      if (R & 1) vr = Monoid::op(dat[--R], vr);
      L >>= 1, R >>= 1;
    }
    return Monoid::op(vl, vr);
  }

  X prod_all() { return dat[1]; }

  template <class F>
  int max_right(F check, int L) {
    assert(0 <= L && L <= n && check(Monoid::unit()));
    if (L == n) return n;
    L += size;
    X sm = Monoid::unit();
    do {
      while (L % 2 == 0) L >>= 1;
      if (!check(Monoid::op(sm, dat[L]))) {
        while (L < size) {
          L = 2 * L;
          if (check(Monoid::op(sm, dat[L]))) { sm = Monoid::op(sm, dat[L++]); }
        }
        return L - size;
      }
      sm = Monoid::op(sm, dat[L++]);
    } while ((L & -L) != L);
    return n;
  }

  template <class F>
  int min_left(F check, int R) {
    assert(0 <= R && R <= n && check(Monoid::unit()));
    if (R == 0) return 0;
    R += size;
    X sm = Monoid::unit();
    do {
      --R;
      while (R > 1 && (R % 2)) R >>= 1;
      if (!check(Monoid::op(dat[R], sm))) {
        while (R < size) {
          R = 2 * R + 1;
          if (check(Monoid::op(dat[R], sm))) { sm = Monoid::op(dat[R--], sm); }
        }
        return R + 1 - size;
      }
      sm = Monoid::op(dat[R], sm);
    } while ((R & -R) != R);
    return 0;
  }

  // prod_{l<=i<r} A[i xor x]
  X xor_prod(int l, int r, int xor_val) {
    static_assert(Monoid::commute);
    X x = Monoid::unit();
    for (int k = 0; k < log + 1; ++k) {
      if (l >= r) break;
      if (l & 1) { x = Monoid::op(x, dat[(size >> k) + ((l++) ^ xor_val)]); }
      if (r & 1) { x = Monoid::op(x, dat[(size >> k) + ((--r) ^ xor_val)]); }
      l /= 2, r /= 2, xor_val /= 2;
    }
    return x;
  }
};
#line 2 "library/mod/modint_common.hpp"

struct has_mod_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};

template <typename mint>
mint inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {0, 1};
  assert(0 <= n);
  if (n >= mod) n %= mod;
  while (len(dat) <= n) {
    int k = len(dat);
    int q = (mod + k - 1) / k;
    dat.eb(dat[k * q - mod] * mint::raw(q));
  }
  return dat[n];
}

template <typename mint>
mint fact(int n) {
  static const int mod = mint::get_mod();
  assert(0 <= n && n < mod);
  static vector<mint> dat = {1, 1};
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat)));
  return dat[n];
}

template <typename mint>
mint fact_inv(int n) {
  static vector<mint> dat = {1, 1};
  if (n < 0) return mint(0);
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
  return dat[n];
}

template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
  return (mint(1) * ... * fact_inv<mint>(xs));
}

template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
  return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}

template <typename mint>
mint C_dense(int n, int k) {
  static vvc<mint> C;
  static int H = 0, W = 0;
  auto calc = [&](int i, int j) -> mint {
    if (i == 0) return (j == 0 ? mint(1) : mint(0));
    return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
  };
  if (W <= k) {
    FOR(i, H) {
      C[i].resize(k + 1);
      FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
    }
    W = k + 1;
  }
  if (H <= n) {
    C.resize(n + 1);
    FOR(i, H, n + 1) {
      C[i].resize(W);
      FOR(j, W) { C[i][j] = calc(i, j); }
    }
    H = n + 1;
  }
  return C[n][k];
}

template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
  assert(n >= 0);
  if (k < 0 || n < k) return 0;
  if constexpr (dense) return C_dense<mint>(n, k);
  if constexpr (!large) return multinomial<mint>(n, k, n - k);
  k = min(k, n - k);
  mint x(1);
  FOR(i, k) x *= mint(n - i);
  return x * fact_inv<mint>(k);
}

template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
  assert(n >= 0);
  assert(0 <= k && k <= n);
  if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
  return mint(1) / C<mint, 1>(n, k);
}

// [x^d](1-x)^{-n}
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
  assert(n >= 0);
  if (d < 0) return mint(0);
  if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
  return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "library/mod/modint.hpp"

template <int mod>
struct modint {
  static constexpr u32 umod = u32(mod);
  static_assert(umod < u32(1) << 31);
  u32 val;

  static modint raw(u32 v) {
    modint x;
    x.val = v;
    return x;
  }
  constexpr modint() : val(0) {}
  constexpr modint(u32 x) : val(x % umod) {}
  constexpr modint(u64 x) : val(x % umod) {}
  constexpr modint(u128 x) : val(x % umod) {}
  constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
  bool operator<(const modint &other) const { return val < other.val; }
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += umod - p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = u64(val) * p.val % umod;
    return *this;
  }
  modint &operator/=(const modint &p) {
    *this *= p.inverse();
    return *this;
  }
  modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
  modint operator+(const modint &p) const { return modint(*this) += p; }
  modint operator-(const modint &p) const { return modint(*this) -= p; }
  modint operator*(const modint &p) const { return modint(*this) *= p; }
  modint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const modint &p) const { return val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }
  modint inverse() const {
    int a = val, 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(ll n) const {
    assert(n >= 0);
    modint ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  static constexpr int get_mod() { return mod; }
  // (n, r), r は 1 の 2^n 乗根
  static constexpr pair<int, int> ntt_info() {
    if (mod == 120586241) return {20, 74066978};
    if (mod == 167772161) return {25, 17};
    if (mod == 469762049) return {26, 30};
    if (mod == 754974721) return {24, 362};
    if (mod == 880803841) return {23, 211};
    if (mod == 943718401) return {22, 663003469};
    if (mod == 998244353) return {23, 31};
    if (mod == 1045430273) return {20, 363};
    if (mod == 1051721729) return {20, 330};
    if (mod == 1053818881) return {20, 2789};
    return {-1, -1};
  }
  static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};

#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
  fastio::rd(x.val);
  x.val %= mod;
  // assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
  fastio::wt(x.val);
}
#endif

using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 3 "library/linalg/matrix_mul.hpp"

template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1,
                     int N2 = -1, int N3 = -1) {
  if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); }
  vv(u32, b, N3, N2);
  FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j].val;
  vv(T, C, N1, N3);

  if ((T::get_mod() < (1 << 30)) && N2 <= 16) {
    FOR(i, N1) FOR(j, N3) {
      u64 sm = 0;
      FOR(m, N2) sm += u64(A[i][m].val) * b[j][m];
      C[i][j] = sm;
    }
  } else {
    FOR(i, N1) FOR(j, N3) {
      u128 sm = 0;
      FOR(m, N2) sm += u64(A[i][m].val) * b[j][m];
      C[i][j] = T::raw(sm % (T::get_mod()));
    }
  }
  return C;
}

template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1,
                     int N2 = -1, int N3 = -1) {
  if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); }
  vv(T, b, N2, N3);
  FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j];
  vv(T, C, N1, N3);
  FOR(n, N1) FOR(m, N2) FOR(k, N3) C[n][k] += A[n][m] * b[k][m];
  return C;
}

// square-matrix defined as array
template <class T, int N>
array<array<T, N>, N> matrix_mul(const array<array<T, N>, N>& A,
                                 const array<array<T, N>, N>& B) {
  array<array<T, N>, N> C{};

  if ((T::get_mod() < (1 << 30)) && N <= 16) {
    FOR(i, N) FOR(k, N) {
      u64 sm = 0;
      FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val);
      C[i][k] = sm;
    }
  } else {
    FOR(i, N) FOR(k, N) {
      u128 sm = 0;
      FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val);
      C[i][k] = sm;
    }
  }
  return C;
}
#line 9 "main.cpp"

using mint = modint998;
using MAT = array<array<mint, 3>, 3>;
using VECT = array<mint, 3>;

struct Mono {
  using value_type = MAT;
  using X = value_type;
  static X op(X x, X y) { return matrix_mul<mint, 3>(x, y); }
  static constexpr X unit() {
    return {mint(1), mint(0), mint(0), mint(0), mint(1),
            mint(0), mint(0), mint(0), mint(1)};
  }
  static constexpr bool commute = false;
};

MAT cross(VECT X, bool right) {
  mint a = X[0], b = X[1], c = X[2];
  if (!right) a = -a, b = -b, c = -c;
  MAT A;
  A[0] = {mint(0), c, -b};
  A[1] = {-c, mint(0), a};
  A[2] = {b, -a, mint(0)};
  return A;
}

void solve() {
  LL(N, Q);
  vc<int> lch(N), rch(N);
  vc<VECT> A(N);

  Graph<int, 1> G(N);
  FOR(v, N) {
    CHAR(c);
    if (c == 'x') {
      LL(l, r);
      --l, --r;
      lch[v] = l, rch[v] = r;
      G.add(v, l), G.add(v, r);
    } else {
      mint a, b, c;
      read(a, b, c);
      A[v] = {a, b, c};
    }
  }
  G.build();

  // A: leaf に持たせているベクトル
  // seg: light child が持つ線形変換

  Tree<decltype(G)> tree(G);
  SegTree<Mono> seg(N);
  vc<pair<int, int>> range(N, {-1, -1});

  VECT ANS;

  FOR_R(i, N) {
    int v = tree.V[i];
    if (tree.head[v] == v) {
      auto path = tree.heavy_path_at(v);
      int a = tree.LID[path[0]];
      int b = tree.LID[path.back()];
      assert(a <= b);
      range[v] = {a, b};
      MAT F = seg.prod(a, b);
      VECT X = A[path.back()];
      VECT Y = {0, 0, 0};
      FOR(i, 3) FOR(j, 3) { Y[i] += F[i][j] * X[j]; }
      if (v == 0) {
        ANS = Y;
      } else {
        int p = tree.parent[v];
        int idx = tree.LID[p];
        seg.set(idx, cross(Y, rch[p] == v));
      }
      // print("v", v, ",", a, b, ",", Y);
    }
  }
  // print(ANS);

  FOR(Q) {
    LL(v);
    --v;
    read(A[v]);
    while (1) {
      v = tree.head[v];
      auto [a, b] = range[v];
      MAT F = seg.prod(a, b);
      VECT X = A[tree.V[b]];
      VECT Y = {0, 0, 0};
      FOR(i, 3) FOR(j, 3) { Y[i] += F[i][j] * X[j]; }
      if (v == 0) {
        print(Y);
        break;
      } else {
        int p = tree.parent[v];
        int idx = tree.LID[p];
        seg.set(idx, cross(Y, rch[p] == v));
        v = p;
      }
    }
  }
}

signed main() {
  solve();
  return 0;
}