/**
 *  date : 2021-11-24 22:03:31
 */

#define NDEBUG
using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// utility
namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;

template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;

template <typename T, typename U>
struct P : pair<T, U> {
  template <typename... Args>
  P(Args... args) : pair<T, U>(args...) {}

  using pair<T, U>::first;
  using pair<T, U>::second;

  T &x() { return first; }
  const T &x() const { return first; }
  U &y() { return second; }
  const U &y() const { return second; }

  P &operator+=(const P &r) {
    first += r.first;
    second += r.second;
    return *this;
  }
  P &operator-=(const P &r) {
    first -= r.first;
    second -= r.second;
    return *this;
  }
  P &operator*=(const P &r) {
    first *= r.first;
    second *= r.second;
    return *this;
  }
  P operator+(const P &r) const { return P(*this) += r; }
  P operator-(const P &r) const { return P(*this) -= r; }
  P operator*(const P &r) const { return P(*this) *= r; }
};

using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;

constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;

template <typename T>
int sz(const T &t) {
  return t.size();
}

template <typename T, typename U>
inline bool amin(T &x, U y) {
  return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
  return (x < y) ? (x = y, true) : false;
}

template <typename T>
inline T Max(const vector<T> &v) {
  return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
  return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
  return accumulate(begin(v), end(v), 0LL);
}

template <typename T>
int lb(const vector<T> &v, const T &a) {
  return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
  return upper_bound(begin(v), end(v), a) - begin(v);
}

constexpr long long TEN(int n) {
  long long ret = 1, x = 10;
  for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
  return ret;
}

template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
  return make_pair(t, u);
}

template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
  vector<T> ret(v.size() + 1);
  if (rev) {
    for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
  } else {
    for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
  }
  return ret;
};

template <typename T>
vector<T> mkuni(const vector<T> &v) {
  vector<T> ret(v);
  sort(ret.begin(), ret.end());
  ret.erase(unique(ret.begin(), ret.end()), ret.end());
  return ret;
}

template <typename F>
vector<int> mkord(int N, F f) {
  vector<int> ord(N);
  iota(begin(ord), end(ord), 0);
  sort(begin(ord), end(ord), f);
  return ord;
}

template <typename T>
vector<int> mkinv(vector<T> &v) {
  int max_val = *max_element(begin(v), end(v));
  vector<int> inv(max_val + 1, -1);
  for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
  return inv;
}

}  // namespace Nyaan

// bit operation
namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
  return _mm_popcnt_u64(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
  return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
  if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
}  // namespace Nyaan

// inout
namespace Nyaan {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
  os << p.first << " " << p.second;
  return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
  is >> p.first >> p.second;
  return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
  int s = (int)v.size();
  for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
  return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
  for (auto &x : v) is >> x;
  return is;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &... u) {
  cin >> t;
  in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &... u) {
  cout << t;
  if (sizeof...(u)) cout << sep;
  out(u...);
}

void outr() {}
template <typename T, class... U, char sep = ' '>
void outr(const T &t, const U &... u) {
  cout << t;
  outr(u...);
}

struct IoSetupNya {
  IoSetupNya() {
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(15);
    cerr << fixed << setprecision(7);
  }
} iosetupnya;

}  // namespace Nyaan

// debug
namespace DebugImpl {

template <typename U, typename = void>
struct is_specialize : false_type {};
template <typename U>
struct is_specialize<
    U, typename conditional<false, typename U::iterator, void>::type>
    : true_type {};
template <typename U>
struct is_specialize<
    U, typename conditional<false, decltype(U::first), void>::type>
    : true_type {};
template <typename U>
struct is_specialize<U, enable_if_t<is_integral<U>::value, void>> : true_type {
};

void dump(const char& t) { cerr << t; }

void dump(const string& t) { cerr << t; }

void dump(const bool& t) { cerr << (t ? "true" : "false"); }

template <typename U,
          enable_if_t<!is_specialize<U>::value, nullptr_t> = nullptr>
void dump(const U& t) {
  cerr << t;
}

template <typename T>
void dump(const T& t, enable_if_t<is_integral<T>::value>* = nullptr) {
  string res;
  if (t == Nyaan::inf) res = "inf";
  if constexpr (is_signed<T>::value) {
    if (t == -Nyaan::inf) res = "-inf";
  }
  if constexpr (sizeof(T) == 8) {
    if (t == Nyaan::infLL) res = "inf";
    if constexpr (is_signed<T>::value) {
      if (t == -Nyaan::infLL) res = "-inf";
    }
  }
  if (res.empty()) res = to_string(t);
  cerr << res;
}

template <typename T, typename U>
void dump(const pair<T, U>&);
template <typename T>
void dump(const pair<T*, int>&);

template <typename T>
void dump(const T& t,
          enable_if_t<!is_void<typename T::iterator>::value>* = nullptr) {
  cerr << "[ ";
  for (auto it = t.begin(); it != t.end();) {
    dump(*it);
    cerr << (++it == t.end() ? "" : ", ");
  }
  cerr << " ]";
}

template <typename T, typename U>
void dump(const pair<T, U>& t) {
  cerr << "( ";
  dump(t.first);
  cerr << ", ";
  dump(t.second);
  cerr << " )";
}

template <typename T>
void dump(const pair<T*, int>& t) {
  cerr << "[ ";
  for (int i = 0; i < t.second; i++) {
    dump(t.first[i]);
    cerr << (i == t.second - 1 ? "" : ", ");
  }
  cerr << " ]";
}

void trace() { cerr << endl; }
template <typename Head, typename... Tail>
void trace(Head&& head, Tail&&... tail) {
  cerr << " ";
  dump(head);
  if (sizeof...(tail) != 0) cerr << ",";
  trace(forward<Tail>(tail)...);
}

}  // namespace DebugImpl

#ifdef NyaanDebug
#define trc(...)                            \
  do {                                      \
    cerr << "## " << #__VA_ARGS__ << " = "; \
    DebugImpl::trace(__VA_ARGS__);          \
  } while (0)
#else
#define trc(...) (void(0))
#endif

// macro
#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...)   \
  int __VA_ARGS__; \
  in(__VA_ARGS__)
#define inl(...)         \
  long long __VA_ARGS__; \
  in(__VA_ARGS__)
#define ins(...)      \
  string __VA_ARGS__; \
  in(__VA_ARGS__)
#define in2(s, t)                           \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i]);                         \
  }
#define in3(s, t, u)                        \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i], u[i]);                   \
  }
#define in4(s, t, u, v)                     \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i], u[i], v[i]);             \
  }
#define die(...)             \
  do {                       \
    Nyaan::out(__VA_ARGS__); \
    return;                  \
  } while (0)

namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }

//

template <typename T>
struct edge {
  int src, to;
  T cost;

  edge(int _to, T _cost) : src(-1), to(_to), cost(_cost) {}
  edge(int _src, int _to, T _cost) : src(_src), to(_to), cost(_cost) {}

  edge &operator=(const int &x) {
    to = x;
    return *this;
  }

  operator int() const { return to; }
};
template <typename T>
using Edges = vector<edge<T>>;
template <typename T>
using WeightedGraph = vector<Edges<T>>;
using UnweightedGraph = vector<vector<int>>;

// Input of (Unweighted) Graph
UnweightedGraph graph(int N, int M = -1, bool is_directed = false,
                      bool is_1origin = true) {
  UnweightedGraph g(N);
  if (M == -1) M = N - 1;
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    if (is_1origin) x--, y--;
    g[x].push_back(y);
    if (!is_directed) g[y].push_back(x);
  }
  return g;
}

// Input of Weighted Graph
template <typename T>
WeightedGraph<T> wgraph(int N, int M = -1, bool is_directed = false,
                        bool is_1origin = true) {
  WeightedGraph<T> g(N);
  if (M == -1) M = N - 1;
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    T c;
    cin >> c;
    if (is_1origin) x--, y--;
    g[x].emplace_back(x, y, c);
    if (!is_directed) g[y].emplace_back(y, x, c);
  }
  return g;
}

// Input of Edges
template <typename T>
Edges<T> esgraph(int N, int M, int is_weighted = true, bool is_1origin = true) {
  Edges<T> es;
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    T c;
    if (is_weighted)
      cin >> c;
    else
      c = 1;
    if (is_1origin) x--, y--;
    es.emplace_back(x, y, c);
  }
  return es;
}

// Input of Adjacency Matrix
template <typename T>
vector<vector<T>> adjgraph(int N, int M, T INF, int is_weighted = true,
                           bool is_directed = false, bool is_1origin = true) {
  vector<vector<T>> d(N, vector<T>(N, INF));
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    T c;
    if (is_weighted)
      cin >> c;
    else
      c = 1;
    if (is_1origin) x--, y--;
    d[x][y] = c;
    if (!is_directed) d[y][x] = c;
  }
  return d;
}
//


namespace HashMapImpl {
using u32 = uint32_t;
using u64 = uint64_t;

template <typename Key, typename Data>
struct HashMapBase;

template <typename Key, typename Data>
struct itrB
    : iterator<bidirectional_iterator_tag, Data, ptrdiff_t, Data*, Data&> {
  using base =
      iterator<bidirectional_iterator_tag, Data, ptrdiff_t, Data*, Data&>;
  using ptr = typename base::pointer;
  using ref = typename base::reference;

  u32 i;
  HashMapBase<Key, Data>* p;

  explicit constexpr itrB() : i(0), p(nullptr) {}
  explicit constexpr itrB(u32 _i, HashMapBase<Key, Data>* _p) : i(_i), p(_p) {}
  explicit constexpr itrB(u32 _i, const HashMapBase<Key, Data>* _p)
      : i(_i), p(const_cast<HashMapBase<Key, Data>*>(_p)) {}
  friend void swap(itrB& l, itrB& r) { swap(l.i, r.i), swap(l.p, r.p); }
  friend bool operator==(const itrB& l, const itrB& r) { return l.i == r.i; }
  friend bool operator!=(const itrB& l, const itrB& r) { return l.i != r.i; }
  const ref operator*() const {
    return const_cast<const HashMapBase<Key, Data>*>(p)->data[i];
  }
  ref operator*() { return p->data[i]; }
  ptr operator->() const { return &(p->data[i]); }

  itrB& operator++() {
    assert(i != p->cap && "itr::operator++()");
    do {
      i++;
      if (i == p->cap) break;
      if (p->flag[i] == true && p->dflag[i] == false) break;
    } while (true);
    return (*this);
  }
  itrB operator++(int) {
    itrB it(*this);
    ++(*this);
    return it;
  }
  itrB& operator--() {
    do {
      i--;
      if (p->flag[i] == true && p->dflag[i] == false) break;
      assert(i != 0 && "itr::operator--()");
    } while (true);
    return (*this);
  }
  itrB operator--(int) {
    itrB it(*this);
    --(*this);
    return it;
  }
};

template <typename Key, typename Data>
struct HashMapBase {
  using u32 = uint32_t;
  using u64 = uint64_t;
  using iterator = itrB<Key, Data>;
  using itr = iterator;

 protected:
  template <typename K>
  inline u64 randomized(const K& key) const {
    return u64(key) ^ r;
  }

  template <typename K,
            enable_if_t<is_same<K, Key>::value, nullptr_t> = nullptr,
            enable_if_t<is_integral<K>::value, nullptr_t> = nullptr>
  inline u32 inner_hash(const K& key) const {
    return (randomized(key) * 11995408973635179863ULL) >> shift;
  }
  template <
      typename K, enable_if_t<is_same<K, Key>::value, nullptr_t> = nullptr,
      enable_if_t<is_integral<decltype(K::first)>::value, nullptr_t> = nullptr,
      enable_if_t<is_integral<decltype(K::second)>::value, nullptr_t> = nullptr>
  inline u32 inner_hash(const K& key) const {
    u64 a = randomized(key.first), b = randomized(key.second);
    a *= 11995408973635179863ULL;
    b *= 10150724397891781847ULL;
    return (a + b) >> shift;
  }
  template <typename K,
            enable_if_t<is_same<K, Key>::value, nullptr_t> = nullptr,
            enable_if_t<is_integral<typename K::value_type>::value, nullptr_t> =
                nullptr>
  inline u32 inner_hash(const K& key) const {
    static constexpr u64 mod = (1LL << 61) - 1;
    static constexpr u64 base = 950699498548472943ULL;
    u64 res = 0;
    for (auto& elem : key) {
      __uint128_t x = __uint128_t(res) * base + (randomized(elem) & mod);
      res = (x & mod) + (x >> 61);
    }
    __uint128_t x = __uint128_t(res) * base;
    res = (x & mod) + (x >> 61);
    if (res >= mod) res -= mod;
    return res >> (shift - 3);
  }

  template <typename D = Data,
            enable_if_t<is_same<D, Key>::value, nullptr_t> = nullptr>
  inline u32 hash(const D& dat) const {
    return inner_hash(dat);
  }
  template <
      typename D = Data,
      enable_if_t<is_same<decltype(D::first), Key>::value, nullptr_t> = nullptr>
  inline u32 hash(const D& dat) const {
    return inner_hash(dat.first);
  }

  template <typename D = Data,
            enable_if_t<is_same<D, Key>::value, nullptr_t> = nullptr>
  inline Key dtok(const D& dat) const {
    return dat;
  }
  template <
      typename D = Data,
      enable_if_t<is_same<decltype(D::first), Key>::value, nullptr_t> = nullptr>
  inline Key dtok(const D& dat) const {
    return dat.first;
  }

  void reallocate(u32 ncap) {
    vector<Data> ndata(ncap);
    vector<bool> nf(ncap);
    shift = 64 - __lg(ncap);
    for (u32 i = 0; i < cap; i++) {
      if (flag[i] == true && dflag[i] == false) {
        u32 h = hash(data[i]);
        while (nf[h]) h = (h + 1) & (ncap - 1);
        ndata[h] = move(data[i]);
        nf[h] = true;
      }
    }
    data.swap(ndata);
    flag.swap(nf);
    cap = ncap;
    dflag.resize(cap);
    fill(std::begin(dflag), std::end(dflag), false);
  }

  inline bool extend_rate(u32 x) const { return x * 2 >= cap; }

  inline bool shrink_rate(u32 x) const {
    return HASHMAP_DEFAULT_SIZE < cap && x * 10 <= cap;
  }

  inline void extend() { reallocate(cap << 1); }

  inline void shrink() { reallocate(cap >> 1); }

 public:
  u32 cap, s;
  vector<Data> data;
  vector<bool> flag, dflag;
  u32 shift;
  static u64 r;
  static constexpr uint32_t HASHMAP_DEFAULT_SIZE = 4;

  explicit HashMapBase()
      : cap(HASHMAP_DEFAULT_SIZE),
        s(0),
        data(cap),
        flag(cap),
        dflag(cap),
        shift(64 - __lg(cap)) {}

  itr begin() const {
    u32 h = 0;
    while (h != cap) {
      if (flag[h] == true && dflag[h] == false) break;
      h++;
    }
    return itr(h, this);
  }
  itr end() const { return itr(this->cap, this); }

  friend itr begin(const HashMapBase& h) { return h.begin(); }
  friend itr end(const HashMapBase& h) { return h.end(); }

  itr find(const Key& key) const {
    u32 h = inner_hash(key);
    while (true) {
      if (flag[h] == false) return this->end();
      if (dtok(data[h]) == key) {
        if (dflag[h] == true) return this->end();
        return itr(h, this);
      }
      h = (h + 1) & (cap - 1);
    }
  }

  bool contain(const Key& key) const { return find(key) != this->end(); }

  itr insert(const Data& d) {
    u32 h = hash(d);
    while (true) {
      if (flag[h] == false) {
        if (extend_rate(s + 1)) {
          extend();
          h = hash(d);
          continue;
        }
        data[h] = d;
        flag[h] = true;
        ++s;
        return itr(h, this);
      }
      if (dtok(data[h]) == dtok(d)) {
        if (dflag[h] == true) {
          data[h] = d;
          dflag[h] = false;
          ++s;
        }
        return itr(h, this);
      }
      h = (h + 1) & (cap - 1);
    }
  }

  // tips for speed up :
  // if return value is unnecessary, make argument_2 false.
  itr erase(itr it, bool get_next = true) {
    if (it == this->end()) return this->end();
    s--;
    if (shrink_rate(s)) {
      Data d = data[it.i];
      shrink();
      it = find(dtok(d));
    }
    int ni = (it.i + 1) & (cap - 1);
    if (this->flag[ni]) {
      this->dflag[it.i] = true;
    } else {
      this->flag[it.i] = false;
    }
    if (get_next) ++it;
    return it;
  }

  itr erase(const Key& key) { return erase(find(key)); }

  bool empty() const { return s == 0; }

  int size() const { return s; }

  void clear() {
    fill(std::begin(flag), std::end(flag), false);
    fill(std::begin(dflag), std::end(dflag), false);
    s = 0;
  }

  void reserve(int n) {
    if (n <= 0) return;
    n = 1 << min(23, __lg(n) + 2);
    if (cap < u32(n)) reallocate(n);
  }
};

template <typename Key, typename Data>
uint64_t HashMapBase<Key, Data>::r =
    chrono::duration_cast<chrono::nanoseconds>(
        chrono::high_resolution_clock::now().time_since_epoch())
        .count();

}  // namespace HashMapImpl

/**
 * @brief Hash Map(base)　(ハッシュマップ・基底クラス)
 */

template <typename Key, typename Val>
struct HashMap : HashMapImpl::HashMapBase<Key, pair<Key, Val>> {
  using base = typename HashMapImpl::HashMapBase<Key, pair<Key, Val>>;
  using HashMapImpl::HashMapBase<Key, pair<Key, Val>>::HashMapBase;
  using Data = pair<Key, Val>;

  Val& operator[](const Key& k) {
    typename base::u32 h = base::inner_hash(k);
    while (true) {
      if (base::flag[h] == false) {
        if (base::extend_rate(base::s + 1)) {
          base::extend();
          h = base::hash(k);
          continue;
        }
        base::data[h].first = k;
        base::data[h].second = Val();
        base::flag[h] = true;
        ++base::s;
        return base::data[h].second;
      }
      if (base::data[h].first == k) {
        if (base::dflag[h] == true) base::data[h].second = Val();
        return base::data[h].second;
      }
      h = (h + 1) & (base::cap - 1);
    }
  }

  typename base::itr emplace(const Key& key, const Val& val) {
    return base::insert(Data(key, val));
  }
};

/* 
 * @brief ハッシュマップ(連想配列)
 * @docs docs/hashmap/hashmap.md
**/

//


template <typename mint>
std::pair<int, mint> GaussElimination(vector<vector<mint>> &a,
                                      int pivot_end = -1,
                                      bool diagonalize = false) {
  int H = a.size(), W = a[0].size();
  int rank = 0, je = pivot_end;
  if (je == -1) je = W;
  mint det = 1;
  for (int j = 0; j < je; j++) {
    int idx = -1;
    for (int i = rank; i < H; i++) {
      if (a[i][j] != mint(0)) {
        idx = i;
        break;
      }
    }
    if (idx == -1) {
      det = 0;
      continue;
    }
    if (rank != idx) {
      det = -det;
      swap(a[rank], a[idx]);
    }
    det *= a[rank][j];
    if (diagonalize && a[rank][j] != mint(1)) {
      mint coeff = a[rank][j].inverse();
      for (int k = j; k < W; k++) a[rank][k] *= coeff;
    }
    int is = diagonalize ? 0 : rank + 1;
    for (int i = is; i < H; i++) {
      if (i == rank) continue;
      if (a[i][j] != mint(0)) {
        mint coeff = a[i][j] / a[rank][j];
        for (int k = j; k < W; k++) a[i][k] -= a[rank][k] * coeff;
      }
    }
    rank++;
  }
  return make_pair(rank, det);
}


template <typename mint>
vector<vector<mint>> LinearEquation(vector<vector<mint>> a, vector<mint> b) {
  int H = a.size(), W = a[0].size();
  for (int i = 0; i < H; i++) a[i].push_back(b[i]);
  auto p = GaussElimination(a, W, true);
  int rank = p.first;

  for (int i = rank; i < H; ++i) {
    if (a[i][W] != 0) return vector<vector<mint>>{};
  }

  vector<vector<mint>> res(1, vector<mint>(W));
  vector<int> pivot(W, -1);
  for (int i = 0, j = 0; i < rank; ++i) {
    while (a[i][j] == 0) ++j;
    res[0][j] = a[i][W], pivot[j] = i;
  }
  for (int j = 0; j < W; ++j) {
    if (pivot[j] == -1) {
      vector<mint> x(W);
      x[j] = 1;
      for (int k = 0; k < j; ++k) {
        if (pivot[k] != -1) x[k] = -a[pivot[k]][j];
      }
      res.push_back(x);
    }
  }
  return res;
}

//

template <typename mint>
struct FormalPowerSeries : vector<mint> {
  using vector<mint>::vector;
  using FPS = FormalPowerSeries;

  FPS &operator+=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
    return *this;
  }

  FPS &operator+=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }

  FPS &operator-=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
    return *this;
  }

  FPS &operator-=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }

  FPS &operator*=(const mint &v) {
    for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
    return *this;
  }

  FPS &operator/=(const FPS &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    if ((int)r.size() <= 64) {
      FPS f(*this), g(r);
      g.shrink();
      mint coeff = g.back().inverse();
      for (auto &x : g) x *= coeff;
      int deg = (int)f.size() - (int)g.size() + 1;
      int gs = g.size();
      FPS quo(deg);
      for (int i = deg - 1; i >= 0; i--) {
        quo[i] = f[i + gs - 1];
        for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
      }
      *this = quo * coeff;
      this->resize(n, mint(0));
      return *this;
    }
    return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
  }

  FPS &operator%=(const FPS &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }

  FPS operator+(const FPS &r) const { return FPS(*this) += r; }
  FPS operator+(const mint &v) const { return FPS(*this) += v; }
  FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
  FPS operator-(const mint &v) const { return FPS(*this) -= v; }
  FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
  FPS operator*(const mint &v) const { return FPS(*this) *= v; }
  FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
  FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
  FPS operator-() const {
    FPS ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }

  void shrink() {
    while (this->size() && this->back() == mint(0)) this->pop_back();
  }

  FPS rev() const {
    FPS ret(*this);
    reverse(begin(ret), end(ret));
    return ret;
  }

  FPS dot(FPS r) const {
    FPS ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }

  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }

  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }

  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret.insert(ret.begin(), sz, mint(0));
    return ret;
  }

  FPS diff() const {
    const int n = (int)this->size();
    FPS ret(max(0, n - 1));
    mint one(1), coeff(1);
    for (int i = 1; i < n; i++) {
      ret[i - 1] = (*this)[i] * coeff;
      coeff += one;
    }
    return ret;
  }

  FPS integral() const {
    const int n = (int)this->size();
    FPS ret(n + 1);
    ret[0] = mint(0);
    if (n > 0) ret[1] = mint(1);
    auto mod = mint::get_mod();
    for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
    for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
    return ret;
  }

  mint eval(mint x) const {
    mint r = 0, w = 1;
    for (auto &v : *this) r += w * v, w *= x;
    return r;
  }

  FPS log(int deg = -1) const {
    assert((*this)[0] == mint(1));
    if (deg == -1) deg = (int)this->size();
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }

  FPS pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    for (int i = 0; i < n; i++) {
      if ((*this)[i] != mint(0)) {
        if (i * k > deg) return FPS(deg, mint(0));
        mint rev = mint(1) / (*this)[i];
        FPS ret =
            (((*this * rev) >> i).log(deg) * k).exp(deg) * ((*this)[i].pow(k));
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
    }
    return FPS(deg, mint(0));
  }

  static void *ntt_ptr;
  static void set_fft();
  FPS &operator*=(const FPS &r);
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_pr();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;

/**
 * @brief 多項式/形式的冪級数ライブラリ
 * @docs docs/fps/formal-power-series.md
 */




template <uint32_t mod>
struct LazyMontgomeryModInt {
  using mint = LazyMontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 ret = mod;
    for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
    return ret;
  }

  static constexpr u32 r = get_r();
  static constexpr u32 n2 = -u64(mod) % mod;
  static_assert(r * mod == 1, "invalid, r * mod != 1");
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");

  u32 a;

  constexpr LazyMontgomeryModInt() : a(0) {}
  constexpr LazyMontgomeryModInt(const int64_t &b)
      : a(reduce(u64(b % mod + mod) * n2)){};

  static constexpr u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
  }

  constexpr mint &operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

  constexpr mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
  constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
  constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
  constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
  constexpr bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr mint operator-() const { return mint() - mint(*this); }

  constexpr mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  
  constexpr mint inverse() const { return pow(mod - 2); }

  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }

  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = LazyMontgomeryModInt<mod>(t);
    return (is);
  }
  
  constexpr u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }

  static constexpr u32 get_mod() { return mod; }
};

template <typename T>
struct Binomial {
  vector<T> f, g, h;
  Binomial(int MAX = 0) : f(1, T(1)), g(1, T(1)), h(1, T(1)) {
    while (MAX >= (int)f.size()) extend();
  }

  void extend() {
    int n = f.size();
    int m = n * 2;
    f.resize(m);
    g.resize(m);
    h.resize(m);
    for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
    g[m - 1] = f[m - 1].inverse();
    h[m - 1] = g[m - 1] * f[m - 2];
    for (int i = m - 2; i >= n; i--) {
      g[i] = g[i + 1] * T(i + 1);
      h[i] = g[i] * f[i - 1];
    }
  }

  T fac(int i) {
    if (i < 0) return T(0);
    while (i >= (int)f.size()) extend();
    return f[i];
  }

  T finv(int i) {
    if (i < 0) return T(0);
    while (i >= (int)g.size()) extend();
    return g[i];
  }

  T inv(int i) {
    if (i < 0) return -inv(-i);
    while (i >= (int)h.size()) extend();
    return h[i];
  }

  T C(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r) * finv(r);
  }

  inline T operator()(int n, int r) { return C(n, r); }

  template <typename I>
  T multinomial(const vector<I>& r) {
    static_assert(is_integral<I>::value == true);
    int n = 0;
    for (auto& x : r) {
      if(x < 0) return T(0);
      n += x;
    }
    T res = fac(n);
    for (auto& x : r) res *= finv(x);
    return res;
  }

  template <typename I>
  T operator()(const vector<I>& r) {
    return multinomial(r);
  }

  T C_naive(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    T ret = T(1);
    r = min(r, n - r);
    for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
    return ret;
  }

  T P(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r);
  }

  T H(int n, int r) {
    if (n < 0 || r < 0) return T(0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};

//
using namespace Nyaan;
using mint = LazyMontgomeryModInt<998244353>;
// using mint = LazyMontgomeryModInt<1000000007>;
using vm = vector<mint>;
using vvm = vector<vm>;
using fps = FormalPowerSeries<mint>;
Binomial<mint> C;

using namespace Nyaan;

void Nyaan::solve() {
  inl(N);
  map<int, int> ws;
  Edges<ll> es;
  rep(i, N - 1) {
    inl(u, v, w);
    --u, --v;
    es.emplace_back(u, v, w);
    ws[w]++;
  }
  int X = 0, Y = 0;
  mint a = 0, b = 0;
  tie(a, X) = *begin(ws);
  if (sz(ws) == 2) tie(b, Y) = *next(begin(ws));
  if (X > Y) swap(X, Y), swap(a, b);

  vector<HashMap<int, mint>> A((X + 1) * (Y + 1));
  // for(auto&h:A)h.reserve(Y);
  vector<mint> B((X + 1) * (Y + 1));

  auto id = [&](int i, int j) { return i * (Y + 1) + j; };

  rep(i, X + 1) rep(j, Y + 1) {
    if (i == X and j == Y) continue;
    mint p = N * (N - 1) / 2 - (X + Y - 1);
    mint q = N - i - j;
    if (i != 0) A[id(i, j)][id(i - 0, j)] += a * i * q;
    if (i != 0) A[id(i, j)][id(i - 1, j)] += a * i * (p - q);
    if (j != 0) A[id(i, j)][id(i, j - 0)] += b * j * q;
    if (j != 0) A[id(i, j)][id(i, j - 1)] += b * j * (p - q);
    if (i != X) A[id(i, j)][id(i + 0, j)] += a * (X - i) * (p - q + 1);
    if (i != X) A[id(i, j)][id(i + 1, j)] += a * (X - i) * (q - 1);
    if (j != Y) A[id(i, j)][id(i, j + 0)] += b * (Y - j) * (p - q + 1);
    if (j != Y) A[id(i, j)][id(i, j + 1)] += b * (Y - j) * (q - 1);
    mint all = (a * X + b * Y) * p;
    A[id(i, j)][id(i, j)] -= all;
    B[id(i, j)] = -all / N;
  }

  vector<fps> f((X + 1) * (Y + 1), fps(Y + 2));
  rep(j, Y + 1) f[j][j] = 1;

  rep1(i, X) rep(j, Y + 1) {
    int imj = id(i - 1, j);
    int ij = id(i, j);

    auto& dst = f[ij];
    auto& eq = A[imj];

    each2(k, v, eq) {
      if (k == ij) continue;
      dst += f[k] * v;
    }
    dst[Y + 1] -= B[imj];
    dst *= -eq[ij].inverse();
    trc(dst, eq[ij]);
  }
  trc("f end");

  vector mat(Y, vector<mint>(Y + 1));
  vector<mint> vec(Y);

  {
    int i = X;
    rep(j, Y) {
      int ij = id(i, j);
      auto&eq = A[ij];

      fps sm(Y + 2);
      each2(k, v, eq) sm += f[k] * v;
      sm[Y + 1] -= B[ij];
      trc(sm);
      rep(k, Y + 1) mat[j][k] = sm[k];
      vec[j] = -sm[Y + 1];
    }
  }

  trc(mat);
  trc(vec);
  vm xs((X + 1) * (Y + 1));

  {
    auto xs2 = LinearEquation(mat, vec)[0];
    rep(i, (X + 1) * (Y + 1)) {
      xs[i] = f[i][Y + 1];
      rep(j, Y + 1) xs[i] += f[i][j] * xs2[j];
    }
  }

  vi cx(N), cy(N);
  each(e, es) {
    (a == e.cost ? cx : cy)[e.src]++;
    (a == e.cost ? cx : cy)[e.to]++;
  }
  trc(cx, cy);
  mint ans = 0;
  rep(i, N) ans += xs[id(cx[i], cy[i])];
  ans -= xs[id(1, 0)] * X + xs[id(0, 1)] * Y + xs[id(X, Y)];
  out(ans);
}