#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;

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); }
// (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, typename U>
T ceil(T x, U y) {
  return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U>
T floor(T x, U y) {
  return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T, typename U>
pair<T, T> divmod(T x, U y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

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

#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) {
  assert(!que.empty());
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  assert(!que.empty());
  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;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, 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;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, 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"
// based on yosupo's fastio
#include <unistd.h>

namespace fastio {
#define FASTIO
// クラスが read(), print() を持っているかを判定するメタ関数
struct has_write_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.write(), std::true_type{});

  template <class T>
  static auto check(...) -> std::false_type;
};

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

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

  template <class T>
  static auto check(...) -> std::false_type;
};

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

struct Scanner {
  FILE *fp;
  char line[(1 << 15) + 1];
  size_t st = 0, ed = 0;
  void reread() {
    memmove(line, line + st, ed - st);
    ed -= st;
    st = 0;
    ed += fread(line + ed, 1, (1 << 15) - ed, fp);
    line[ed] = '\0';
  }
  bool succ() {
    while (true) {
      if (st == ed) {
        reread();
        if (st == ed) return false;
      }
      while (st != ed && isspace(line[st])) st++;
      if (st != ed) break;
    }
    if (ed - st <= 50) {
      bool sep = false;
      for (size_t i = st; i < ed; i++) {
        if (isspace(line[i])) {
          sep = true;
          break;
        }
      }
      if (!sep) reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    while (true) {
      size_t sz = 0;
      while (st + sz < ed && !isspace(line[st + sz])) sz++;
      ref.append(line + st, sz);
      st += sz;
      if (!sz || st != ed) break;
      reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    ref = T(0);
    while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); }
    if (neg) ref = -ref;
    return true;
  }
  template <typename T,
            typename enable_if<has_read<T>::value>::type * = nullptr>
  inline bool read_single(T &x) {
    x.read();
    return true;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }
  bool read_single(char &ref) {
    string s;
    if (!read_single(s) || s.size() != 1) return false;
    ref = s[0];
    return true;
  }
  template <class T>
  bool read_single(vector<T> &ref) {
    for (auto &d: ref) {
      if (!read_single(d)) return false;
    }
    return true;
  }
  template <class T, class U>
  bool read_single(pair<T, U> &p) {
    return (read_single(p.first) && read_single(p.second));
  }
  template <size_t N = 0, typename T>
  void read_single_tuple(T &t) {
    if constexpr (N < std::tuple_size<T>::value) {
      auto &x = std::get<N>(t);
      read_single(x);
      read_single_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool read_single(tuple<T...> &tpl) {
    read_single_tuple(tpl);
    return true;
  }
  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }
  Scanner(FILE *fp) : fp(fp) {}
};

struct Printer {
  Printer(FILE *_fp) : fp(_fp) {}
  ~Printer() { flush(); }

  static constexpr size_t SIZE = 1 << 15;
  FILE *fp;
  char line[SIZE], small[50];
  size_t pos = 0;
  void flush() {
    fwrite(line, 1, pos, fp);
    pos = 0;
  }
  void write(const char val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  void write(T val) {
    if (pos > (1 << 15) - 50) flush();
    if (val == 0) {
      write('0');
      return;
    }
    if (val < 0) {
      write('-');
      val = -val; // todo min
    }
    size_t len = 0;
    while (val) {
      small[len++] = char(0x30 | (val % 10));
      val /= 10;
    }
    for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; }
    pos += len;
  }
  void write(const string s) {
    for (char c: s) write(c);
  }
  void write(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write(s[i]);
  }
  void write(const double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  void write(const long double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  template <typename T,
            typename enable_if<has_write<T>::value>::type * = nullptr>
  inline void write(T x) {
    x.write();
  }
  template <class T>
  void write(const vector<T> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  template <class T, class U>
  void write(const pair<T, U> val) {
    write(val.first);
    write(' ');
    write(val.second);
  }
  template <size_t N = 0, typename T>
  void write_tuple(const T t) {
    if constexpr (N < std::tuple_size<T>::value) {
      if constexpr (N > 0) { write(' '); }
      const auto x = std::get<N>(t);
      write(x);
      write_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool write(tuple<T...> tpl) {
    write_tuple(tpl);
    return true;
  }
  template <class T, size_t S>
  void write(const array<T, S> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  void write(i128 val) {
    string s;
    bool negative = 0;
    if (val < 0) {
      negative = 1;
      val = -val;
    }
    while (val) {
      s += '0' + int(val % 10);
      val /= 10;
    }
    if (negative) s += "-";
    reverse(all(s));
    if (len(s) == 0) s = "0";
    write(s);
  }
};
Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);
void flush() { printer.flush(); }
void print() { printer.write('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  printer.write(head);
  if (sizeof...(Tail)) printer.write(' ');
  print(forward<Tail>(tail)...);
}

void read() {}
template <class Head, class... Tail>
void read(Head &head, Tail &... tail) {
  scanner.read(head);
  read(tail...);
}
} // namespace fastio
using fastio::print;
using fastio::flush;
using fastio::read;

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __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/ds/unionfind/unionfind.hpp"

struct UnionFind {
  int n, n_comp;
  vc<int> dat; // par or (-size)
  UnionFind(int n = 0) { build(n); }

  void build(int m) {
    n = m, n_comp = m;
    dat.assign(n, -1);
  }

  void reset() { build(n); }

  int operator[](int x) {
    while (dat[x] >= 0) {
      int pp = dat[dat[x]];
      if (pp < 0) { return dat[x]; }
      x = dat[x] = pp;
    }
    return x;
  }

  ll size(int x) {
    assert(dat[x] < 0);
    return -dat[x];
  }

  bool merge(int x, int y) {
    x = (*this)[x], y = (*this)[y];
    if (x == y) return false;
    if (-dat[x] < -dat[y]) swap(x, y);
    dat[x] += dat[y], dat[y] = x, n_comp--;
    return true;
  }
};
#line 2 "library/graph/ds/link_cut.hpp"

template <typename Node, int NODES>
struct LinkCutTree_base {
  int n;
  Node *nodes;

  LinkCutTree_base(int n = 0) : n(n) {
    nodes = new Node[NODES];
    FOR(i, n) nodes[i] = Node(i);
  }

  Node *operator[](int v) { return &nodes[v]; }

  // パスを表す splay tree の根になっているかどうか
  bool is_root(Node *c) { return state(c) == 0; }
  bool is_root(int c) { return state(&nodes[c]) == 0; }

  Node *get_root(Node *c) {
    expose(c);
    while (c->l) {
      c->push();
      c = c->l;
    }
    splay(c);
    return c;
  }

  int get_root(int c) { return get_root(&nodes[c])->idx; }

  // c の親を p にする。内部で evert するのでいつ呼んでも大丈夫。
  virtual void link(Node *c, Node *p) {
    evert(c);
    expose(p);
    c->p = p;
    p->r = c;
    p->update();
  }

  // c の親を p にする。内部で evert するのでいつ呼んでも大丈夫。
  virtual void link(int c, int p) { return link(&nodes[c], &nodes[p]); }

  void cut(Node *a, Node *b) {
    evert(a);
    expose(b);
    assert(!b->p);
    assert((b->l) == a);
    b->l->p = nullptr;
    b->l = nullptr;
    b->update();
  }

  void cut(int a, int b) { return cut(&nodes[a], &nodes[b]); }

  void evert(Node *c) {
    expose(c);
    c->reverse();
    c->push();
  }

  void evert(int c) { evert(&nodes[c]); }

  Node *lca(Node *u, Node *v) {
    assert(get_root(u) == get_root(v));
    expose(u);
    return expose(v);
  }

  int lca(int u, int v) { return lca(&nodes[u], &nodes[v])->idx; }

  // c と根までが繋がれている状態に変更して、根を return する
  virtual Node *expose(Node *c) {
    Node *now = c;
    Node *rp = nullptr; // 今まで作ったパス
    while (now) {
      splay(now);
      now->r = rp; // 子方向の変更
      now->update();
      rp = now;
      now = now->p;
    }
    splay(c);
    return rp;
  }

  int expose(int c) {
    Node *x = expose(&nodes[c]);
    if (!x) return -1;
    return x->idx;
  }

  Node *get_parent(Node *x) {
    expose(x);
    if (!x->l) return nullptr;
    x = x->l;
    while (x->r) x = x->r;
    return x;
  }

  int get_parent(int x) {
    Node *p = get_parent((*this)[x]);
    return (p ? p->idx : -1);
  }

  void debug() {
    FOR(i, n) { nodes[i].debug(); }
  }

  virtual void rotate(Node *n) {
    // n を根に近づける
    Node *pp, *p, *c;
    p = n->p;
    pp = p->p;

    if (p->l == n) {
      c = n->r;
      n->r = p;
      p->l = c;
    } else {
      c = n->l;
      n->l = p;
      p->r = c;
    }

    if (pp && pp->l == p) pp->l = n;
    if (pp && pp->r == p) pp->r = n;
    n->p = pp;
    p->p = n;
    if (c) c->p = p;
  }

  inline int state(Node *n) {
    if (!n->p) return 0;
    if (n->p->l == n) return 1;
    if (n->p->r == n) return -1;
    return 0;
  }

  void splay(Node *c) {
    c->push();
    while (!is_root(c)) {
      Node *p = c->p;
      Node *pp = (p ? p->p : nullptr);
      if (state(p) == 0) {
        p->push(), c->push();
        rotate(c);
        if (p) p->update();
      }
      elif (state(c) == state(p)) {
        pp->push(), p->push(), c->push();
        rotate(p);
        rotate(c);
        if (pp) pp->update();
        if (p) p->update();
      }
      else {
        pp->push(), p->push(), c->push();
        rotate(c);
        rotate(c);
        if (p) p->update();
        if (pp) pp->update();
      }
    }
    c->update();
  }
};

struct LCT_Node_base {
  LCT_Node_base *l, *r, *p;
  int idx;
  bool rev;
  LCT_Node_base(int i = 0) : l(nullptr), r(nullptr), p(nullptr), idx(i) {}

  void update() {}

  void push() {
    if (rev) {
      if (l) l->reverse();
      if (r) r->reverse();
      rev = 0;
    }
  }

  void reverse() {
    rev ^= 1;
    swap(l, r);
  }
};

template <int NODES>
using LinkCutTree = LinkCutTree_base<LCT_Node_base, NODES>;
#line 2 "library/graph/ds/link_cut_path.hpp"

template <typename Node, int NODES>
struct LinkCutTree_Path_base : public LinkCutTree_base<Node, NODES> {
  using X = typename Node::X;

  LinkCutTree_Path_base(int n) : LinkCutTree_base<Node, NODES>(n) {}

  LinkCutTree_Path_base(vc<X> dat) : LinkCutTree_base<Node, NODES>(len(dat)) {
    FOR(v, len(dat)) {
      Node *c = (*this)[v];
      set(c, dat[v]);
    }
  }

  template <typename F>
  LinkCutTree_Path_base(int n, F f) : LinkCutTree_base<Node, NODES>(n) {
    FOR(v, n) {
      X x = f(v);
      Node *c = (*this)[v];
      set(c, x);
    }
  }

  void set(Node *c, X x) {
    this->evert(c);
    c->x = x;
    c->update();
  }

  void set(int c, X x) { set((*this)[c], x); }

  void multiply(Node *c, X x) { set(c, Node::Mono::op(c->x, x)); }

  void multiply(int c, X x) { multiply((*this)[c], x); }

  X prod_path(Node *a, Node *b) {
    this->evert(a);
    this->expose(b);
    return b->prod;
  }

  X prod_path(int a, int b) { return prod_path((*this)[a], (*this)[b]); }
};

template <typename Monoid>
struct LCT_Node_Monoid {
  using Mono = Monoid;
  using X = typename Monoid::value_type;
  LCT_Node_Monoid *l, *r, *p;
  int idx;
  X x, prod, rev_prod;
  bool rev;
  LCT_Node_Monoid(int i = 0)
      : l(nullptr),
        r(nullptr),
        p(nullptr),
        idx(i),
        x(Monoid::unit()),
        prod(Monoid::unit()),
        rev_prod(Monoid::unit()) {}

  void update() {
    prod = rev_prod = x;
    if (l) {
      prod = Monoid::op(l->prod, prod);
      rev_prod = Monoid::op(rev_prod, l->rev_prod);
    }
    if (r) {
      prod = Monoid::op(prod, r->prod);
      rev_prod = Monoid::op(r->rev_prod, rev_prod);
    }
  }

  void push() {
    if (rev) {
      if (l) l->reverse();
      if (r) r->reverse();
      rev = 0;
    }
  }

  void reverse() {
    rev ^= 1;
    swap(l, r);
    swap(prod, rev_prod);
  }

  void debug() {
    int li = (l ? l->idx : -1);
    int ri = (r ? r->idx : -1);
    int pi = (p ? p->idx : -1);
    print("idx", idx, "l", li, "r", ri, "p", pi, "x", x, "prod", prod,
          "rev_prod", rev_prod);
  }
};

template <typename Monoid>
struct LCT_Node_CommutativeMonoid {
  using Mono = Monoid;
  using X = typename Mono::value_type;
  LCT_Node_CommutativeMonoid *l, *r, *p;
  int idx;
  X x, prod;
  bool rev;
  LCT_Node_CommutativeMonoid(int i = 0)
      : l(nullptr),
        r(nullptr),
        p(nullptr),
        idx(i),
        x(Mono::unit()),
        prod(Mono::unit()) {}

  void update() {
    prod = x;
    if (l) { prod = Mono::op(l->prod, prod); }
    if (r) { prod = Mono::op(prod, r->prod); }
  }

  void push() {
    if (rev) {
      if (l) l->reverse();
      if (r) r->reverse();
      rev = 0;
    }
  }

  void reverse() {
    rev ^= 1;
    swap(l, r);
  }

  void debug() {
    int li = (l ? l->idx : -1);
    int ri = (r ? r->idx : -1);
    int pi = (p ? p->idx : -1);
    print("idx", idx, "l", li, "r", ri, "p", pi, "x", x, "prod", prod);
  }
};

template <typename Monoid, int NODES>
using LinkCutTree_Path = LinkCutTree_Path_base<LCT_Node_Monoid<Monoid>, NODES>;

template <typename Monoid, int NODES>
using LinkCutTree_Path_Commutative
    = LinkCutTree_Path_base<LCT_Node_CommutativeMonoid<Monoid>, NODES>;
#line 6 "main.cpp"

#line 2 "library/alg/monoid/min_idx.hpp"

template <typename T, bool tie_is_left = true>
struct Monoid_Min_Idx {
  using value_type = pair<T, int>;
  using X = value_type;
  static constexpr bool is_small(const X& x, const X& y) {
    if (x.fi < y.fi) return true;
    if (x.fi > y.fi) return false;
    return (tie_is_left ? (x.se < y.se) : (x.se >= y.se));
  }
  static X op(X x, X y) { return (is_small(x, y) ? x : y); }
  static constexpr X unit() { return {infty<T>, -1}; }
  static constexpr bool commute = true;
};
#line 8 "main.cpp"

void solve() {
  LL(N, M);
  vc<int> A(M), B(M), C(M);
  FOR(i, M) read(A[i]), read(B[i]), read(C[i]);
  for (auto&& x: A) --x;
  for (auto&& x: B) --x;
  FOR(i, N - 1) C[i] = 2 * C[i] - 1;
  FOR(i, N - 1, M) C[i] = 2 * C[i];

  vc<int> use(M);
  auto I = argsort(C);
  UnionFind uf(N);
  FOR(i, M) {
    int idx = I[i];
    if (uf.merge(A[idx], B[idx])) use[idx] = 1;
  }

  using Mono = Monoid_Min_Idx<int>;
  LinkCutTree_Path_Commutative<Mono, 5000> LCT(N + N - 1);
  FOR(e, N - 1) {
    LCT.link(N + e, A[e]);
    LCT.link(N + e, B[e]);
    if (!use[e]) { LCT.set(N + e, {0, e}); }
  }

  vc<pi> ANS;

  FOR(e, N - 1, M) {
    if (!use[e]) continue;
    auto [mi, idx] = LCT.prod_path(A[e], B[e]);
    assert(mi == 0);
    ANS.eb(idx, e);
    LCT.cut(A[idx], N + idx);
    LCT.cut(B[idx], N + idx);
    LCT.link(N + e, A[e]);
    LCT.link(N + e, B[e]);
  }

  print(len(ANS));
  for (auto&& [a, b]: ANS) print(1 + a, 1 + b);
}

signed main() {
  int T = 1;
  // INT(T);
  FOR(T) solve();
  return 0;
}