Problem p03717 (C++)

Buggy submission (status: Memory Limit Exceeded):
// todo 文字数を少なくする
// mintより高速に計算したい時用に modmul() modadd (正の値限定) を作る
// #pragma GCC optimize ("-O3")
#include <bits/stdc++.h>
using namespace std;
//@起動時
struct initon {
  initon() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout.setf(ios::fixed);
    cout.precision(16);
    srand((unsigned)clock() + (unsigned)time(NULL));
  };
} __initon;
// 衝突対策
#define ws ___ws

//@必須構造
struct T {
  int f, s, t;
  T() { f = -1, s = -1, t = -1; }
  T(int f, int s, int t) : f(f), s(s), t(t) {}
  bool operator<(const T &r) const {
    return f != r.f ? f < r.f : s != r.s ? s < r.s : t < r.t;
    // return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 大きい順
  }
  bool operator>(const T &r) const {
    return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t;
    // return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 小さい順
  }
  bool operator==(const T &r) const { return f == r.f && s == r.s && t == r.t; }
  bool operator!=(const T &r) const { return f != r.f || s != r.s || t != r.t; }
  int operator[](int i) {
    assert(i < 3);
    return i == 0 ? f : i == 1 ? s : t;
  }
};
struct F {
  int a, b, c, d;
  F() { a = -1, b = -1, c = -1, d = -1; }
  F(int a, int b, int c, int d) : a(a), b(b), c(c), d(d) {}
  bool operator<(const F &r) const {
    return a != r.a   ? a < r.a
           : b != r.b ? b < r.b
           : c != r.c ? c < r.c
                      : d < r.d;
    //        return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c >
    //        r.c : d > r.d;
  }
  bool operator>(const F &r) const {
    return a != r.a   ? a > r.a
           : b != r.b ? b > r.b
           : c != r.c ? c > r.c
                      : d > r.d;
    //        return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c <
    //        r.c : d < r.d;
  }
  bool operator==(const F &r) const {
    return a == r.a && b == r.b && c == r.c && d == r.d;
  }
  bool operator!=(const F &r) const {
    return a != r.a || b != r.b || c != r.c || d != r.d;
  }
  int operator[](int i) {
    assert(i < 4);
    return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;
  }
};

T mt(int a, int b, int c) { return T(a, b, c); }
//@マクロ省略系 型,構造

#define int long long
#define ll long long
#define double long double
#define ull unsigned long long
using dou = double;
using itn = int;
using str = string;
using bo = bool;
#define au auto
using P = pair<ll, ll>;
using pd = pair<dou, dou>;
#define fi first
#define se second
#define vec vector
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==

#define maxq 1
#define minq -1

#define ZERO(a) memset(a, 0, sizeof(a))
#define MINUS(a) memset(a, 0xff, sizeof(a))
#define MALLOC(type, len) (type *)malloc((len) * sizeof(type))

// マクロ省略系 コンテナ

using vi = vector<int>;
#define _overloadvvi(_1, _2, _3, _4, name, ...) name
#define vvi0() vec<vi>
#define vvi1(a) vec<vi> a
#define vvi2(a, b) vec<vi> a(b)
#define vvi3(a, b, c) vec<vi> a(b, vi(c))
#define vvi4(a, b, c, d) vec<vi> a(b, vi(c, d))
#define vvi(...)                                                               \
  _overloadvvi(__VA_ARGS__, vvi4, vvi3, vvi2, vvi1, vvi0)(__VA_ARGS__)

using vl = vector<ll>;
#define _overloadvvl(_1, _2, _3, _4, name, ...) name
#define vvl1(a) vec<vl> a
#define vvl2(a, b) vec<vl> a(b)
#define vvl3(a, b, c) vec<vl> a(b, vl(c))
#define vvl4(a, b, c, d) vec<vl> a(b, vl(c, d))
#define vvl(...) _overloadvvl(__VA_ARGS__, vvl4, vvl3, vvl2, vvl1)(__VA_ARGS__)

using vb = vector<bool>;
#define _overloadvvb(_1, _2, _3, _4, name, ...) name
#define vvb1(a) vec<vb> a
#define vvb2(a, b) vec<vb> a(b)
#define vvb3(a, b, c) vec<vb> a(b, vb(c))
#define vvb4(a, b, c, d) vec<vb> a(b, vb(c, d))
#define vvb(...) _overloadvvb(__VA_ARGS__, vvb4, vvb3, vvb2, vvb1)(__VA_ARGS__)

using vs = vector<string>;
#define _overloadvvs(_1, _2, _3, _4, name, ...) name
#define vvs1(a) vec<vs> a
#define vvs2(a, b) vec<vs> a(b)
#define vvs3(a, b, c) vec<vs> a(b, vs(c))
#define vvs4(a, b, c, d) vec<vs> a(b, vs(c, d))
#define vvs(...) _overloadvvs(__VA_ARGS__, vvs4, vvs3, vvs2, vvs1)(__VA_ARGS__)

using vd = vector<double>;
#define _overloadvvd(_1, _2, _3, _4, name, ...) name
#define vvd1(a) vec<vd> a
#define vvd2(a, b) vec<vd> a(b)
#define vvd3(a, b, c) vec<vd> a(b, vd(c))
#define vvd4(a, b, c, d) vec<vd> a(b, vd(c, d))
#define vvd(...) _overloadvvd(__VA_ARGS__, vvd4, vvd3, vvd2, vvd1)(__VA_ARGS__)

using vc = vector<char>;
#define _overloadvvc(_1, _2, _3, _4, name, ...) name
#define vvc1(a) vec<vc> a
#define vvc2(a, b) vec<vc> a(b)
#define vvc3(a, b, c) vec<vc> a(b, vc(c))
#define vvc4(a, b, c, d) vec<vc> a(b, vc(c, d))
#define vvc(...) _overloadvvc(__VA_ARGS__, vvc4, vvc3, vvc2, vvc1)(__VA_ARGS__)

using vp = vector<P>;
#define _overloadvvp(_1, _2, _3, _4, name, ...) name
#define vvp1(a) vec<vp> a
#define vvp2(a, b) vec<vp> a(b)
#define vvp3(a, b, c) vec<vp> a(b, vp(c))
#define vvp4(a, b, c, d) vec<vp> a(b, vp(c, d))
#define vvp(...) _overloadvvp(__VA_ARGS__, vvp4, vvp3, vvp2, vvp1)(__VA_ARGS__)

using vt = vector<T>;
#define _overloadvvt(_1, _2, _3, _4, name, ...) name
#define vvt1(a) vec<vt> a
#define vvt2(a, b) vec<vt> a(b)
#define vvt3(a, b, c) vec<vt> a(b, vt(c))
#define vvt4(a, b, c, d) vec<vt> a(b, vt(c, d))
#define vvt(...) _overloadvvt(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1)(__VA_ARGS__)

#define v3i(a, b, c, d) vector<vector<vi>> a(b, vector<vi>(c, vi(d)))
#define v3d(a, b, c, d) vector<vector<vd>> a(b, vector<vd>(c, vd(d)))
#define v3m(a, b, c, d) vector<vector<vm>> a(b, vector<vm>(c, vm(d)))

#define _vvi vector<vi>
#define _vvl vector<vl>
#define _vvb vector<vb>
#define _vvs vector<vs>
#define _vvd vector<vd>
#define _vvc vector<vc>
#define _vvp vector<vp>

template <typename T> vector<T> make_v(size_t a) { return vector<T>(a); }
template <typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {
  return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}
#define vni(name, ...) auto name = make_v<int>(__VA_ARGS__)
#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)
#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)
#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)
#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)
#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)

#define PQ priority_queue<ll, vector<ll>, greater<ll>>
#define tos to_string
using mapi = map<int, int>;
using mapp = map<P, int>;
using mapd = map<dou, int>;
using mapc = map<char, int>;
using maps = map<str, int>;
using seti = set<int>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<int>;
#define bset bitset
#define uset unordered_set
#define mset multiset
#define umap unordered_map
#define umapi unordered_map<int, int>
#define umapp unordered_map<P, int>
#define mmap multimap

// マクロ 繰り返し
#define _overloadrep(_1, _2, _3, _4, name, ...) name
#define _rep(i, n) for (int i = 0, _lim = n; i < _lim; i++)
#define repi(i, m, n) for (int i = m, _lim = n; i < _lim; i++)
#define repadd(i, m, n, ad) for (int i = m, _lim = n; i < _lim; i += ad)
#define rep(...) _overloadrep(__VA_ARGS__, repadd, repi, _rep, )(__VA_ARGS__)
#define _rer(i, n) for (int i = n; i >= 0; i--)
#define reri(i, m, n) for (int i = m, _lim = n; i >= _lim; i--)
#define rerdec(i, m, n, dec) for (int i = m, _lim = n; i >= _lim; i -= dec)
#define rer(...) _overloadrep(__VA_ARGS__, rerdec, reri, _rer, )(__VA_ARGS__)

#define reps2(i, j, n)                                                         \
  for (int i = 0, _lim = n; i < _lim; i++)                                     \
    for (int j = 0; j < _lim; j++)
#define reps3(i, j, k, n)                                                      \
  for (int i = 0, _lim = n; i < _lim; i++)                                     \
    for (int j = 0; j < _lim; j++)                                             \
      for (int k = 0; k < _lim; k++)
#define reps4(i, j, k, l, n)                                                   \
  for (int i = 0, _lim = n; i < _lim; i++)                                     \
    for (int j = 0; j < _lim; j++)                                             \
      for (int k = 0; k < _lim; k++)                                           \
        for (int l = 0; l < _lim; l++)

#define _overloadreps(_1, _2, _3, _4, _5, name, ...) name
#define reps(...)                                                              \
  _overloadreps(__VA_ARGS__, reps4, reps3, reps2, _rep, )(__VA_ARGS__)
#define fora(a, b) for (auto &&a : b)
#define forg(gi, ve)                                                           \
  for (int gi = 0, f, t, c;                                                    \
       gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true);     \
       gi++)
#define fort(gi, ve)                                                           \
  for (int gi = 0, f, t, c;                                                    \
       gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true);     \
       gi++)                                                                   \
    if (t != p)
// #define fort(gi, ve) for (int gi = 0, f, t, c;gi<ve.size()&& (gi+=
// (ve[gi].t==p))< ve.size() && (f = ve[gi].f,t=ve[gi].t, c = ve[gi].c,true);
// gi++)

// マクロ 定数
#define k3 1010
#define k4 10101
#define k5 101010
#define k6 1010101
#define k7 10101010
const int inf = (int)1e9 + 100;
const ll linf = (ll)1e18 + 100;
const char infc = '{';
const string infs = "{";
const double eps = 1e-9;
const double PI = 3.1415926535897932384626433832795029L;
ll ma = numeric_limits<ll>::min();
ll mi = numeric_limits<ll>::max();
const int y4[] = {-1, 1, 0, 0};
const int x4[] = {0, 0, -1, 1};
const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};
const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};

// マクロ省略形 関数等
#define arsz(a) (sizeof(a) / sizeof(a[0]))
#define sz(a) ((int)(a).size())
// #define rs resize
#define mp make_pair
#define pb push_back
#define pf push_front
#define eb emplace_back
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()

constexpr bool ev(int a) { return !(a & 1); }
constexpr bool od(int a) { return (a & 1); }
//@拡張系 こう出来るべきというもの

// 埋め込み 存在を意識せずに機能を増やされているもの
namespace std {
template <> class hash<std::pair<signed, signed>> {
public:
  size_t operator()(const std::pair<signed, signed> &x) const {
    return hash<ll>()(((ll)x.first << 32) + x.second);
  }
};
template <> class hash<std::pair<ll, ll>> {
public:
  // 大きいllが渡されると、<<32でオーバーフローするがとりあえず問題ないと判断
  size_t operator()(const std::pair<ll, ll> &x) const {
    return hash<ll>()(((ll)x.first << 32) + x.second);
  }
};
} // namespace std
istream &operator>>(istream &iss, P &a) {
  iss >> a.first >> a.second;
  return iss;
}
template <typename T> istream &operator>>(istream &iss, vector<T> &vec) {
  for (T &x : vec)
    iss >> x;
  return iss;
}
template <class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {
  os << p.fi << " " << p.se << endl;
  return os;
}
ostream &operator<<(ostream &os, T p) {
  os << p.f << " " << p.s << " " << p.t;
  return os;
}
ostream &operator<<(ostream &os, F p) {
  os << p.a << " " << p.b << " " << p.c << " " << p.d;
  return os;
}
template <typename T> ostream &operator<<(ostream &os, vector<T> &vec) {
  for (int i = 0; i < vec.size(); i++)
    os << vec[i] << (i + 1 == vec.size() ? "" : " ");
  return os;
}
template <typename T> ostream &operator<<(ostream &os, vector<vector<T>> &vec) {
  for (int i = 0; i < vec.size(); i++) {
    for (int j = 0; j < vec[0].size(); j++) {
      os << vec[i][j];
    }
    os << endl;
  }
  return os;
}
template <typename V, typename H> void resize(vector<V> &vec, const H head) {
  vec.resize(head);
}
template <typename V, typename H, typename... T>
void resize(vector<V> &vec, const H &head, const T... tail) {
  vec.resize(head);
  for (auto &v : vec)
    resize(v, tail...);
}
template <typename T, typename _Pr>
bool all_of(const vector<T> &vec, _Pr pred) {
  return std::all_of(vec.begin(), vec.end(), pred);
}
template <typename T, typename _Pr>
bool any_of(const vector<T> &vec, _Pr pred) {
  return std::any_of(vec.begin(), vec.end(), pred);
}
template <typename T, typename _Pr>
bool none_of(const vector<T> &vec, _Pr pred) {
  return std::none_of(vec.begin(), vec.end(), pred);
}
template <typename T, typename _Pr>
const typename vector<T>::const_iterator find_if(const vector<T> &vec,
                                                 _Pr pred) {
  return std::find_if(vec.begin(), vec.end(), pred);
}
template <typename T> bool contains(const vector<T> &vec, const T &val) {
  return std::find(vec.begin(), vec.end(), val) != vec.end();
}
template <typename T, typename _Pr>
bool contains_if(const vector<T> &vec, _Pr pred) {
  return std::find_if(vec.begin(), vec.end(), pred) != vec.end();
}
template <class T, class U> void replace(vector<T> &a, T key, U v) {
  replace(a.begin(), a.end(), key, v);
}
template <class T> void replace(str &a, T key, T v) {
  a.replace(all(a), key, v);
}
// keyと同じかどうか01で置き換える
template <class T, class U> void replace(T &a, U k) {
  rep(i, sz(a)) a[i] = a[i] == k;
}
template <class T, class U> void replace(T &a) { replace(a, '#'); }

template <class T> bool includes(vector<T> &a, vector<T> &b) {
  vi c = a;
  vi d = b;
  sort(all(c));
  sort(all(d));
  return includes(all(c), all(d));
}
template <class T> bool is_permutation(vector<T> &a, vector<T> &b) {
  return is_permutation(all(a), all(b));
}
template <class T> bool next_permutation(vector<T> &a) {
  return next_permutation(all(a));
}

template <class T> T pop(set<T> &set) {
  T res = *set.begin();
  set.erase(set.find(res));
  return res;
}
template <class T> T pop(mset<T> &set) {
  T res = *set.begin();
  set.erase(set.find(res));
  return res;
}
template <class T> T popBack(set<T> &set) {
  T res = *set.rbegin();
  set.erase(set.find(res));
  return res;
}
template <class T> T popBack(mset<T> &set) {
  T res = *set.rbegin();
  set.erase(set.find(res));
  return res;
}
inline void sort(string &a) { sort(a.begin(), a.end()); }
template <class T> inline void sort(vector<T> &a) { sort(a.begin(), a.end()); }
template <class T> inline void sort(vector<T> &a, int len) {
  sort(a.begin(), a.begin() + len);
}
template <class T, class F> inline void sort(vector<T> &a, F f) {
  sort(a.begin(), a.end(), [&](T a, T b) { return f(a) < f(b); });
}
enum ___pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
inline void sort(vector<P> &a, ___pcomparator type) {
  switch (type) {
  case fisi:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });
    break;
  case fisd:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });
    break;
  case fdsi:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });
    break;
  case fdsd:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });
    break;
  case sifi:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });
    break;
  case sifd:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });
    break;
  case sdfi:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });
    break;
  case sdfd:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });
    break;
  }
};
inline void sort(vector<T> &a, ___pcomparator type) {
  switch (type) {
  case fisi:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });
    break;
  case fisd:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });
    break;
  case fdsi:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });
    break;
  case fdsd:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });
    break;
  case sifi:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });
    break;
  case sifd:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });
    break;
  case sdfi:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });
    break;
  case sdfd:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });
    break;
  }
};
template <class T> inline void rsort(vector<T> &a) {
  sort(a.begin(), a.end(), greater<T>());
};
template <class T> inline void rsort(vector<T> &a, int len) {
  sort(a.begin(), a.begin() + len, greater<T>());
};
template <class U, class F> inline void rsort(vector<U> &a, F f) {
  sort(a.begin(), a.end(), [&](U l, U r) { return f(l) > f(r); });
};
template <class U> inline void sortp(vector<U> &a, vector<U> &b) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  sort(c);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
    ;
  }
};
// F = T<T>
// 例えばreturn p.fi + p.se;
template <class U, class F> inline void sortp(vector<U> &a, vector<U> &b, F f) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  sort(c, f);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U, class F>
inline void sortp(vector<U> &a, vector<U> &b, char type) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  sort(c, type);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U> inline void rsortp(vector<U> &a, vector<U> &b) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  rsort(c);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U, class F>
inline void rsortp(vector<U> &a, vector<U> &b, F f) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  rsort(c, f);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c) {
  vt r;
  int n = sz(a);
  assert(n == sz(b));
  assert(n == sz(c));
  rep(i, n) r.eb(a[i], b[i], c[i]);
  sort(r);
  rep(i, n) {
    a[i] = r[i].f;
    b[i] = r[i].s;
    c[i] = r[i].t;
  }
};
template <class U, class F>
inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c, F f) {
  vt r;
  int n = sz(a);
  assert(n == sz(b));
  assert(n == sz(c));
  rep(i, n) r.eb(a[i], b[i], c[i]);
  sort(r, f);
  rep(i, n) {
    a[i] = r[i].f;
    b[i] = r[i].s;
    c[i] = r[i].t;
  }
};
template <class U, class F>
inline void rsortt(vector<U> &a, vector<U> &b, vector<U> &c, F f) {
  vt r;
  int n = sz(a);
  assert(n == sz(b));
  assert(n == sz(c));
  rep(i, n) r.eb(a[i], b[i], c[i]);
  rsort(r, f);
  rep(i, n) {
    a[i] = r[i].f;
    b[i] = r[i].s;
    c[i] = r[i].t;
  }
};
// indexをつけてsortする (second)
template <class T> auto sorti(vector<T> &a) {
  int n = sz(a);
  vp ret(n);
  rep(i, n) ret[i] = P(a[i], i);
  sort(ret);
  return ret;
};
template <class T> auto rsorti(vector<T> &a) {
  int n = sz(a);
  vp ret(n);
  rep(i, n) ret[i] = P(a[i], i);
  rsort(ret);
  return ret;
};
template <class T> inline void sort2(vector<vector<T>> &a) {
  for (int i = 0, n = a.size(); i < n; i++)
    sort(a[i]);
}
template <class T> inline void rsort2(vector<vector<T>> &a) {
  for (int i = 0, n = a.size(); i < n; i++)
    rsort(a[i]);
}
template <typename A, size_t N, typename T>
inline void fill(A (&a)[N], const T &v) {
  rep(i, N) a[i] = v;
}
template <typename A, size_t N, size_t O, typename T>
inline void fill(A (&a)[N][O], const T &v) {
  rep(i, N) rep(j, O) a[i][j] = v;
}
template <typename A, size_t N, size_t O, size_t P, typename T>
inline void fill(A (&a)[N][O][P], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) a[i][j][k] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, typename T>
inline void fill(A (&a)[N][O][P][Q], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) a[i][j][k][l] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          typename T>
inline void fill(A (&a)[N][O][P][Q][R], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) a[i][j][k][l][m] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S, typename T>
inline void fill(A (&a)[N][O][P][Q][R][S], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S)
      a[i][j][k][l][m][n] = v;
}

template <typename A, size_t N> A sum(A (&a)[N]) {
  A res = 0;
  rep(i, N) res += a[i];
  return res;
}
template <typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {
  A res = 0;
  rep(i, N) rep(j, O) res += a[i][j];
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A sum(A (&a)[N][O][P]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) res += a[i][j][k];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A sum(A (&a)[N][O][P][Q]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) res += a[i][j][k][l];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A sum(A (&a)[N][O][P][Q][R]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) res += a[i][j][k][l][m];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A sum(A (&a)[N][O][P][Q][R][S]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S) res +=
      a[i][j][k][l][m][n];
  return res;
}

template <typename V, typename T> void fill(V &xx, const T vall) { xx = vall; }
template <typename V, typename T> void fill(vector<V> &vecc, const T vall) {
  for (auto &&vx : vecc)
    fill(vx, vall);
}

//@汎用便利関数 入力
inline int in() {
  int ret;
  //    scanf("%lld", &ret);
  cin >> ret;
  return ret;
}
inline string sin() {
  string ret;
  cin >> ret;
  return ret;
}
template <class T> inline void in(T &head) { cin >> head; }
template <class T, class... U> inline void in(T &head, U &...tail) {
  cin >> head;
  in(tail...);
}

#define _overloaddin(_1, _2, _3, _4, name, ...) name
#define din1(a)                                                                \
  int a;                                                                       \
  cin >> a
#define din2(a, b)                                                             \
  int a, b;                                                                    \
  cin >> a >> b
#define din3(a, b, c)                                                          \
  int a, b, c;                                                                 \
  cin >> a >> b >> c
#define din4(a, b, c, d)                                                       \
  int a, b, c, d;                                                              \
  cin >> a >> b >> c >> d
#define din(...) _overloaddin(__VA_ARGS__, din4, din3, din2, din1)(__VA_ARGS__)

#define _overloaddind(_1, _2, _3, _4, name, ...) name
#define din1d(a)                                                               \
  din1(a);                                                                     \
  a--
#define din2d(a, b)                                                            \
  din2(a, b);                                                                  \
  a--, b--
#define din3d(a, b, c)                                                         \
  din3(a, b, c);                                                               \
  a--, b--, c--
#define din4d(a, b, c, d)                                                      \
  din4(a, b, c, d);                                                            \
  a--, b--, c--, d--
#define dind(...)                                                              \
  _overloaddind(__VA_ARGS__, din4d, din3d, din2d, din1d)(__VA_ARGS__)

#define _overloadout(_1, _2, _3, _4, name, ...) name
#define out1(a) cout << a << endl
#define out2(a, b) cout << a << " " << b << endl
#define out3(a, b, c) cout << a << " " << b << " " << c << endl
#define out4(a, b, c, d) cout << a << " " << b << " " << c << " " << d << endl
#define out(...) _overloadout(__VA_ARGS__, out4, out3, out2, out1)(__VA_ARGS__)

template <class T> void outl(vector<T> &a) { fora(v, a) cout << v << endl; }
#define na(a, n)                                                               \
  a.resize(n);                                                                 \
  rep(_i, n) cin >> a[_i];
#define nao(a, n)                                                              \
  a.resize(n + 1);                                                             \
  rep(_i, n) cin >> a[_i + 1];
#define nad(a, n)                                                              \
  a.resize(n);                                                                 \
  rep(_i, n) {                                                                 \
    cin >> a[_i];                                                              \
    a[_i]--;                                                                   \
  }
#define na2(a, b, n)                                                           \
  a.resize(n), b.resize(n);                                                    \
  rep(_i, n) cin >> a[_i] >> b[_i];
#define nao2(a, b, n)                                                          \
  a.resize(n + 1), b.resize(n + 1);                                            \
  rep(_i, n) cin >> a[_i + 1] >> b[_i + 1];
#define na2d(a, b, n)                                                          \
  a.resize(n), b.resize(n);                                                    \
  rep(_i, n) {                                                                 \
    cin >> a[_i] >> b[_i];                                                     \
    a[_i]--, b[_i]--;                                                          \
  }
#define na3(a, b, c, n)                                                        \
  a.resize(n), b.resize(n), c.resize(n);                                       \
  rep(_i, n) cin >> a[_i] >> b[_i] >> c[_i];
#define na3d(a, b, c, n)                                                       \
  a.resize(n), b.resize(n), c.resize(n);                                       \
  rep(_i, n) {                                                                 \
    cin >> a[_i] >> b[_i] >> c[_i];                                            \
    a[_i]--, b[_i]--, c[_i]--;                                                 \
  }
#define nt(a, h, w)                                                            \
  resize(a, h, w);                                                             \
  rep(_hi, h) rep(_wi, w) cin >> a[_hi][_wi];
#define ntd(a, h, w)                                                           \
  rs(a, h, w);                                                                 \
  rep(_hi, h) rep(_wi, w) cin >> a[_hi][_wi], a[_hi][_wi]--;
#define ntp(a, h, w)                                                           \
  fill(a, '#');                                                                \
  rep(_hi, 1, h + 1) rep(_wi, 1, w + 1) cin >> a[_hi][_wi];

// デバッグ
#define sp << " " <<

#define debugName(VariableName) #VariableName

#define _deb1(x) debugName(x) << " = " << x
#define _deb2(x, ...) _deb1(x) << ", " << _deb1(__VA_ARGS__)
#define _deb3(x, ...) _deb1(x) << ", " << _deb2(__VA_ARGS__)
#define _deb4(x, ...) _deb1(x) << ", " << _deb3(__VA_ARGS__)
#define _deb5(x, ...) _deb1(x) << ", " << _deb4(__VA_ARGS__)
#define _deb6(x, ...) _deb1(x) << ", " << _deb5(__VA_ARGS__)
#define _deb7(x, ...) _deb1(x) << ", " << _deb6(__VA_ARGS__)
#define _deb8(x, ...) _deb1(x) << ", " << _deb7(__VA_ARGS__)
#define _deb9(x, ...) _deb1(x) << ", " << _deb8(__VA_ARGS__)
#define _deb10(x, ...) _deb1(x) << ", " << _deb9(__VA_ARGS__)

#define _overloadebug(_1, _2, _3, _4, _5, _6, _7, _8, _9, _10, name, ...) name

#ifdef _DEBUG
#define deb(...)                                                               \
  cerr << _overloadebug(__VA_ARGS__, _deb10, _deb9, _deb8, _deb7, _deb6,       \
                        _deb5, _deb4, _deb3, _deb2, _deb1)(__VA_ARGS__)        \
       << endl
#else
#define deb(...) ;
#endif

#define debugline(x)                                                           \
  cerr << x << " "                                                             \
       << "(L:" << __LINE__ << ")" << '\n'

// よく使うクラス、構造体
class UnionFind {
public:
  vi par, rank, sizes;
  int n, trees;
  UnionFind(int n) : n(n), trees(n) {
    par.resize(n), rank.resize(n), sizes.resize(n);
    rep(i, n) par[i] = i, sizes[i] = 1;
  }
  int root(int x) {
    if (par[x] == x)
      return x;
    else
      return par[x] = root(par[x]);
  }
  int find(int x) { return root(x); }
  void unite(int x, int y) {
    x = root(x);
    y = root(y);
    if (x == y)
      return;
    if (rank[x] < rank[y])
      swap(x, y);
    trees--;
    par[y] = x;
    sizes[x] += sizes[y];
    if (rank[x] == rank[y])
      rank[x]++;
  }
  bool same(int x, int y) { return root(x) == root(y); }
  int size(int x) { return sizes[root(x)]; }
  // 順不同 umapなので
  vec<vi> sets() {
    vec<vi> res(trees);
    umap<int, vi> map;
    rep(i, n) map[root(i)].push_back(i);
    int i = 0;
    for (auto &&p : map) {
      int r = p.fi;
      res[i].push_back(r);
      for (auto &&v : p.se) {
        if (r == v)
          continue;
        res[i].push_back(v);
      }
      i++;
    }
    return res;
  }
};

using bint = __int128;
using u32 = unsigned int;
using u64 = unsigned long long;
using u128 = __uint128_t;
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;
}

__int128 toi128(string &s) {
  __int128 ret = 0;
  for (int i = 0; i < s.length(); i++)
    if ('0' <= s[i] && s[i] <= '9')
      ret = 10 * ret + s[i] - '0';
  return ret;
}

template <typename T> T minv(T a, T m);
template <typename T> T minv(T a);

template <typename T> class Modular {
public:
  using Type = typename decay<decltype(T::value)>::type;

  constexpr Modular() : value() {}
  template <typename U> Modular(const U &x) { value = normalize(x); }

  template <typename U> static Type normalize(const U &x) {
    Type v;
    if (-mod() <= x && x < mod())
      v = static_cast<Type>(x);
    else
      v = static_cast<Type>(x % mod());
    if (v < 0)
      v += mod();
    return v;
  }

  const Type &operator()() const { return value; }
  template <typename U> explicit operator U() const {
    return static_cast<U>(value);
  }
  constexpr static Type mod() { return T::value; }

  Modular &operator+=(const Modular &other) {
    if ((value += other.value) >= mod())
      value -= mod();
    return *this;
  }
  Modular &operator-=(const Modular &other) {
    if ((value -= other.value) < 0)
      value += mod();
    return *this;
  }
  template <typename U> Modular &operator+=(const U &other) {
    return *this += Modular(other);
  }
  template <typename U> Modular &operator-=(const U &other) {
    return *this -= Modular(other);
  }
  Modular &operator++() { return *this += 1; }
  Modular &operator--() { return *this -= 1; }
  Modular operator++(signed) {
    Modular result(*this);
    *this += 1;
    return result;
  }
  Modular operator--(signed) {
    Modular result(*this);
    *this -= 1;
    return result;
  }
  Modular operator-() const { return Modular(-value); }

  template <typename U = T>
  typename enable_if<is_same<typename Modular<U>::Type, signed>::value,
                     Modular>::type &
  operator*=(const Modular &rhs) {
#ifdef _WIN32
    uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);
    uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x),
             d, m;
    asm("divl %4; \n\t" : "=a"(d), "=d"(m) : "d"(xh), "a"(xl), "r"(mod()));
    value = m;
#else
    value = normalize(static_cast<int64_t>(value) *
                      static_cast<int64_t>(rhs.value));
#endif
    return *this;
  }
  template <typename U = T>
  typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value,
                     Modular>::type &
  operator*=(const Modular &rhs) {
    int64_t q =
        static_cast<int64_t>(static_cast<double>(value) * rhs.value / mod());
    value = normalize(value * rhs.value - q * mod());
    return *this;
  }
  template <typename U = T>
  typename enable_if<!is_integral<typename Modular<U>::Type>::value,
                     Modular>::type &
  operator*=(const Modular &rhs) {
    value = normalize(value * rhs.value);
    return *this;
  }
  Modular &operator/=(const Modular &other) {
    return *this *= Modular(minv(other.value));
  }
  template <typename U>
  friend bool operator==(const Modular<U> &lhs, const Modular<U> &rhs);
  template <typename U>
  friend bool operator<(const Modular<U> &lhs, const Modular<U> &rhs);
  template <typename U>
  friend std::istream &operator>>(std::istream &stream, Modular<U> &number);
  operator int() { return normalize(value); }

private:
  Type value;
};

template <typename T>
bool operator==(const Modular<T> &lhs, const Modular<T> &rhs) {
  return lhs.value == rhs.value;
}
template <typename T, typename U>
bool operator==(const Modular<T> &lhs, U rhs) {
  return lhs == Modular<T>(rhs);
}
template <typename T, typename U>
bool operator==(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) == rhs;
}

template <typename T>
bool operator!=(const Modular<T> &lhs, const Modular<T> &rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U>
bool operator!=(const Modular<T> &lhs, U rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U>
bool operator!=(U lhs, const Modular<T> &rhs) {
  return !(lhs == rhs);
}

template <typename T>
bool operator<(const Modular<T> &lhs, const Modular<T> &rhs) {
  return lhs.value < rhs.value;
}

template <typename T>
Modular<T> operator+(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) += rhs;
}
template <typename T, typename U>
Modular<T> operator+(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) += rhs;
}
template <typename T, typename U>
Modular<T> operator+(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) += rhs;
}

template <typename T>
Modular<T> operator-(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) -= rhs;
}
template <typename T, typename U>
Modular<T> operator-(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) -= rhs;
}
template <typename T, typename U>
Modular<T> operator-(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) -= rhs;
}

template <typename T>
Modular<T> operator*(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) *= rhs;
}
template <typename T, typename U>
Modular<T> operator*(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) *= rhs;
}
template <typename T, typename U>
Modular<T> operator*(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) *= rhs;
}

template <typename T>
Modular<T> operator/(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) /= rhs;
}
template <typename T, typename U>
Modular<T> operator/(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) /= rhs;
}
template <typename T, typename U>
Modular<T> operator/(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) /= rhs;
}

constexpr signed MOD = 1e9 + 7; // MOD
using mint = Modular<std::integral_constant<decay<decltype(MOD)>::type, MOD>>;

constexpr int mint_len = 1400001;
vi fac, finv, inv, p2;
void setmod() {
  fac.resize(mint_len);
  finv.resize(mint_len);
  inv.resize(mint_len);
  p2.resize(mint_len);
  inv[1] = fac[0] = finv[0] = p2[0] = 1;
  for (int i = 2; i < mint_len; ++i)
    inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
  for (int i = 1; i < mint_len; ++i)
    fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD,
    p2[i] = p2[i - 1] * 2 % MOD;
}
#define smod setmod
inline mint com(int n, int r) {
  if (r < 0 || r > n)
    return 0;
  return mint(finv[r] * fac[n] % MOD * finv[n - r]);
}
inline mint pom(int n, int r) {
  //    if (!sz(fac)) com(0, -1);
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - 1]);
}
inline mint npr(int n, int r) {
  //    if (!sz(fac)) com(0, -1);
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - r]);
}
inline int nprin(int n, int r) {
  //    if (!sz(fac)) com(0, -1);
  if (r < 0 || r > n)
    return 0;
  return fac[n] * finv[n - r] % MOD;
}

inline int icom(int n, int r) {
  const int NUM_ = 1400001;
  static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];
  if (fac[0] == 0) {
    inv[1] = fac[0] = finv[0] = 1;
    for (int i = 2; i <= NUM_; ++i)
      inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
    for (int i = 1; i <= NUM_; ++i)
      fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
  }
  if (r < 0 || r > n)
    return 0;
  return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;
}
#define ncr com
#define ncri icom
inline mint nhr(int n, int r) { return com(n + r - 1, r); }
inline mint hom(int n, int r) { return com(n + r - 1, r); }
inline int nhri(int n, int r) { return icom(n + r - 1, r); }

template <typename T> inline T minv(T a, T m) {
  T u = 0, v = 1;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}
template <typename T> inline T minv(T a) {
  if (a < mint_len)
    return inv[a];
  T u = 0, v = 1;
  T m = MOD;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}

template <typename T, typename U>
inline Modular<T> mpow(const Modular<T> &a, const U &b) {
  assert(b >= 0);
  int x = a(), res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= MOD;
    (x *= x) %= MOD;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U, typename V>
inline mint mpow(const T a, const U b, const V m = MOD) {
  assert(b >= 0);
  int x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= m;
    (x *= x) %= m;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U> inline mint mpow(const T a, const U b) {
  assert(b >= 0);
  int x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= MOD;
    (x *= x) %= MOD;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U, typename V>
inline int mpowi(const T &a, const U &b, const V &m = MOD) {
  assert(b >= 0);
  int x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= m;
    (x *= x) %= m;
    p >>= 1;
  }
  return res;
}
template <typename T> inline string to_string(const Modular<T> &number) {
  return to_string(number());
}

template <typename T>
std::ostream &operator<<(std::ostream &stream, const Modular<T> &number) {
  return stream << number();
}

template <typename T>
std::istream &operator>>(std::istream &stream, Modular<T> &number) {
  typename common_type<typename Modular<T>::Type, int64_t>::type x;
  stream >> x;
  number.value = Modular<T>::normalize(x);
  return stream;
}

using PM = pair<mint, mint>;
using vm = vector<mint>;
using mapm = map<int, mint>;
using umapm = umap<int, mint>;
#define _overloadvvm(_1, _2, _3, _4, name, ...) name
#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)

#define vvm1(a) vec<vm> a
#define vvm2(a, b) vec<vm> a(b)
#define vvm3(a, b, c) vec<vm> a(b, vm(c))
#define vvm4(a, b, c, d) vec<vm> a(b, vm(c, d))
#define vvm(...) _overloadvvm(__VA_ARGS__, vvm4, vvm3, vvm2, vvm1)(__VA_ARGS__)

// エラー
void ole() {
#ifdef _DEBUG
  debugline("ole");
  exit(0);
#endif
  string a = "a";
  rep(i, 30) a += a;
  rep(i, 1 << 17) cout << a << endl;
  cout << "OLE 出力長制限超過" << endl;
  exit(0);
}
void tle() {
  while (inf)
    cout << inf << endl;
}

// 便利関数

// テスト用
char ranc() { return (char)('a' + rand() % 26); }
int rand(int min, int max) {
  assert(min <= max);
  if (min >= 0 && max >= 0) {
    return rand() % (max + 1 - min) + min;
  } else if (max < 0) {
    return -rand(-max, -min);
  } else {
    //+
    if (rand() % 2) {
      return rand(0, max);
      //-
    } else {
      return -rand(0, -min);
    }
  }
}
vi ranv(int n, int min, int max) {
  vi v(n);
  rep(i, n) v[i] = rand(min, max);
  return v;
}
// 単調増加
vi ranvi(int n, int min, int max) {
  vi v(n);
  bool bad = 1;
  while (bad) {
    bad = 0;
    v.resize(n);
    rep(i, n) {
      if (i && min > max - v[i - 1]) {
        bad = 1;
        break;
      }
      if (i)
        v[i] = v[i - 1] + rand(min, max - v[i - 1]);
      else
        v[i] = rand(min, max);
    }
  }
  return v;
}
void ranvlr(int n, int min, int max, vi &l, vi &r) {
  l.resize(n);
  r.resize(n);
  rep(i, n) {
    l[i] = rand(min, max);
    r[i] = l[i] + rand(0, max - l[i]);
  }
}
// 便利 汎用
vp run_length(vi &a) {
  vp ret;
  ret.eb(a[0], 1);
  rep(i, 1, sz(a)) {
    if (ret.back().fi == a[i]) {
      ret.back().se++;
    } else {
      ret.eb(a[i], 1);
    }
  }
  return ret;
}
vector<pair<char, int>> run_length(string &a) {
  vector<pair<char, int>> ret;
  ret.eb(a[0], 1);
  rep(i, 1, sz(a)) {
    if (ret.back().fi == a[i]) {
      ret.back().se++;
    } else {
      ret.eb(a[i], 1);
    }
  }
  return ret;
}
template <class F> inline int mgr(int ok, int ng, F f) {
#define _mgrbody                                                               \
  int mid = (ok + ng) / 2;                                                     \
  if (f(mid))                                                                  \
    ok = mid;                                                                  \
  else                                                                         \
    ng = mid;
  if (ok < ng)
    while (ng - ok > 1) {
      _mgrbody
    }
  else
    while (ok - ng > 1) {
      _mgrbody
    }
  return ok;
}

template <class F> inline int mgr(int ok, int ng, int second, F f) {
#define _mgrbody2                                                              \
  int mid = (ok + ng) / 2;                                                     \
  if (f(mid, second))                                                          \
    ok = mid;                                                                  \
  else                                                                         \
    ng = mid;
  if (ok < ng)
    while (ng - ok > 1) {
      _mgrbody2
    }
  else
    while (ok - ng > 1) {
      _mgrbody2
    }
  return ok;
}
// strを整数として比較
string smax(str &a, str b) {
  if (sz(a) < sz(b)) {
    return b;
  } else if (sz(a) > sz(b)) {
    return a;
  } else {
    rep(i, sz(a)) {
      if (a[i] < b[i]) {
        return b;
      } else if (a[i] > b[i])
        return a;
    }
  }
  return a;
}
// strを整数として比較
string smin(str &a, str b) {
  if (sz(a) < sz(b)) {
    return a;
  } else if (sz(a) > sz(b)) {
    return b;
  } else {
    rep(i, sz(a)) {
      if (a[i] < b[i]) {
        return a;
      } else if (a[i] > b[i])
        return b;
    }
  }
  return a;
}
template <typename V, typename T> int find(vector<V> &a, const T key) {
  rep(i, sz(a)) if (a[i] == key) return i;
  return -1;
}
template <typename V, typename T> P find(vector<vector<V>> &a, const T key) {
  rep(i, sz(a)) rep(j, sz(a[0])) if (a[i][j] == key) return mp(i, j);
  return mp(-1, -1);
}
template <typename V, typename U>
T find(vector<vector<vector<V>>> &a, const U key) {
  rep(i, sz(a)) rep(j, sz(a[0]))
      rep(k, sz(a[0][0])) if (a[i][j][k] == key) return mt(i, j, k);
  return mt(-1, -1, -1);
}

template <typename V, typename T> int count(V &a, const T k) { return a == k; }
template <typename V, typename T> int count(vector<V> &a, const T k) {
  int ret = 0;
  fora(v, a) ret += count(v, k);
  return ret;
}
template <typename T> int count(char &a, const T k) { return a == k; }
template <typename T> int count(str &a, const T k) {
  int ret = 0;
  rep(i, sz(a)) ret += count(a[i], k);
  return ret;
}
template <typename V> int count_odd(V &a) { return a % 2; }
template <typename V> int count_odd(vector<V> &a) {
  int ret = 0;
  fora(v, a) ret += count_odd(v);
  return ret;
}
template <typename V> int count_even(V &a) { return a % 2 == 0; }
template <typename V> int count_even(vector<V> &a) {
  int ret = 0;
  fora(v, a) ret += count_even(v);
  return ret;
}
// algorythm
void iota(vector<int> &ve, int s, int n) {
  ve.resize(n);
  iota(all(ve), s);
}
vi iota(int s, int n) {
  vi ve(n);
  iota(all(ve), s);
  return ve;
}

inline ll rev(ll a) {
  ll res = 0;
  while (a) {
    res *= 10;
    res += a % 10;
    a /= 10;
  }
  return res;
}
template <class T> inline void rev(vector<T> &a) { reverse(all(a)); }
void inline rev(string &a) { reverse(all(a)); }

int get(int a, int keta) { return (a / (int)pow(10, keta)) % 10; }
int keta(int v) {
  int cou = 0;
  while (v) {
    cou++, v %= 10;
  }
  return cou;
}
int dsum(int v) {
  int ret = 0;
  for (; v; v /= 10)
    ret += v % 10;
  return ret;
}
inline int sumd(int v) { return dsum(v); }
// 変換系
template <class T, class U> inline vector<T> keys(vector<pair<T, U>> a) {
  vector<T> res;
  for (auto &&k : a)
    res.pb(k.fi);
  return res;
}
template <class T, class U> inline vector<U> keys(map<T, U> a) {
  vector<U> res;
  for (auto &&k : a)
    res.pb(k.fi);
  return res;
}

template <class T, class U> inline vector<U> keys(umap<T, U> a) {
  vector<U> res;
  for (auto &&k : a)
    res.pb(k.fi);
  return res;
}

template <class T, class U> vector<U> inline values(vector<pair<T, U>> a) {
  vector<U> res;
  for (auto &&k : a)
    res.pb(k.se);
  return res;
}

template <class T, class U> inline vector<T> values(map<T, U> a) {
  vector<T> res;
  for (auto &&k : a)
    res.pb(k.se);
  return res;
}

template <class T, class U> inline vector<T> values(umap<T, U> a) {
  vector<T> res;
  for (auto &&k : a)
    res.pb(k.se);
  return res;
}

template <class T, class U> inline bool chma(T &a, const U &b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}
template <class U> inline bool chma(const U &b) { return chma(ma, b); }
template <class T, class U> inline bool chmi(T &a, const U &b) {
  if (b < a) {
    a = b;
    return true;
  }
  return false;
}
template <class U> inline bool chmi(const U &b) { return chmi(mi, b); }
template <class T> inline T min(T a, signed b) { return a < b ? a : b; }
template <class T> inline T max(T a, signed b) { return a < b ? b : a; }
template <class T> inline T min(T a, T b, T c) {
  return a >= b ? b >= c ? c : b : a >= c ? c : a;
}
template <class T> inline T max(T a, T b, T c) {
  return a <= b ? b <= c ? c : b : a <= c ? c : a;
}
template <class T> inline T min(vector<T> a) { return *min_element(all(a)); }
template <class T> inline T mini(vector<T> a) {
  return min_element(all(a)) - a.begin();
}
template <class T> inline T min(vector<T> a, int n) {
  return *min_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> inline T min(vector<T> a, int s, int n) {
  return *min_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <class T> inline T max(vector<T> a) { return *max_element(all(a)); }
template <class T> inline T maxi(vector<T> a) {
  return max_element(all(a)) - a.begin();
}
template <class T> inline T max(vector<T> a, int n) {
  return *max_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> inline T max(vector<T> a, int s, int n) {
  return *max_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <typename A, size_t N> inline A max(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = max(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> inline A max(A (&a)[N][O]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P>
inline A max(A (&a)[N][O][P]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
inline A max(A (&a)[N][O][P][Q], const T &v) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
inline A max(A (&a)[N][O][P][Q][R]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
inline A max(A (&a)[N][O][P][Q][R][S]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N> inline A min(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = min(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> inline A min(A (&a)[N][O]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P>
inline A min(A (&a)[N][O][P]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
inline A min(A (&a)[N][O][P][Q], const T &v) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
inline A min(A (&a)[N][O][P][Q][R]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
inline A min(A (&a)[N][O][P][Q][R][S]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}

template <class T> inline T sum(T &v) { return v; }
template <class T> inline auto sum(vector<T> &v) {
  auto ret = sum(v[0]);
  rep(i, 1, sz(v)) ret += sum(v[i]);
  return ret;
}
template <class T, class U, class... V>
inline auto sum(vector<T> &v, U head, V... tail) {
  auto ret = sum(v[0], tail...);
  rep(i, 1, min(sz(v), head)) ret += sum(v[i], tail...);
  return ret;
}

inline P sump(vp &v, int len = -1) {
  if (len == -1)
    len = v.size();
  P res = {0, 0};
  chmi(len, v.size());
  rep(i, len) {
    res.fi += v[i].fi;
    res.se += v[i].se;
  }
  return res;
}

/// 要素が0の時、返り値は0か1か
template <class T> inline T mul(vector<T> &v, int len = -1) {
  if (len == -1)
    len = v.size();
  T res = 1;
  chmi(len, v.size());
  rep(i, len) res *= v[i];
  return res;
}
inline void clear(PQ &q) {
  while (q.size())
    q.pop();
}
template <class T> inline void clear(queue<T> &q) {
  while (q.size())
    q.pop();
}
template <class T> inline T *negarr(int size) {
  T *body = (T *)malloc((size * 2 + 1) * sizeof(T));
  return body + size;
}
template <class T> inline T *negarr2(int h, int w) {
  double **dummy1 = new double *[2 * h + 1];
  double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];
  dummy1[0] = dummy2 + w;
  for (int i = 1; i <= 2 * h + 1; i++) {
    dummy1[i] = dummy1[i - 1] + 2 * w + 1;
  }
  double **a = dummy1 + h;
  return a;
}
// imoは0-indexed
// ruiは1-indexed
template <class T> inline vector<T> imo(vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}
// 漸化的に最小を持つ
template <class T> inline vector<T> imi(vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) chmi(ret[i + 1], ret[i]);
  return ret;
}
template <class T> struct ruiC {
  const vector<T> rui;
  ruiC(vector<T> &ru) : rui(ru) {}
  inline T operator()(int l, int r) {
    assert(l <= r);
    return rui[r] - rui[l];
  }
  inline T operator[](int i) { return rui[i]; }
};

template <class T> struct rruic {
  const T *rrui;
  rruic(T *ru) : rrui(ru) {}
  // n-1から-1へ
  inline T operator()(int l, int r) {
    assert(l >= r);
    return rrui[r] - rrui[l];
  }
  inline T operator[](int i) { return rrui[i]; }
};
template <class T> inline vector<T> ruiv(vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + a[i];
  return ret;
}
template <class T> inline ruiC<T> ruic(vector<T> &a) {
  vector<T> ret = ruiv(a);
  return ruiC<T>(ret);
}
// kと同じものの数
template <class T, class U> inline vi ruiv(T &a, U k) {
  vi ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + (a[i] == k);
  return ret;
}
template <class T, class U> inline ruiC<int> ruic(T &a, U k) {
  vi ret = ruiv(a, k);
  return ruiC<int>(ret);
}

// xor
template <class T> inline vector<T> ruix(vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] ^ a[i];
  return ret;
}
template <class T> inline vector<T> ruim(vector<T> &a) {
  vector<T> res(a.size() + 1, 1);
  rep(i, a.size()) res[i + 1] = res[i] * a[i];
  return res;
}
// 漸化的に最小を1indexで持つ
template <class T> inline vector<T> ruimi(vector<T> &a) {
  int n = sz(a);
  vector<T> ret(n + 1);
  rep(i, 1, n) {
    ret[i] = a[i - 1];
    chmi(ret[i + 1], ret[i]);
  }
  return ret;
}
// template<class T> T *rrui(vector<T> &a) {
// 右から左にかけての半開区間 (-1 n-1]
template <class T> rruic<T> rrui(vector<T> &a) {
  int len = a.size();
  T *body = (T *)malloc((len + 1) * sizeof(T));
  T *res = body + 1;
  rer(i, len - 1) res[i - 1] = res[i] + a[i];
  return rruic<T>(res);
}
// 掛け算
template <class T> T *rruim(vector<T> &a) {
  int len = a.size();
  T *body = (T *)malloc((len + 1) * sizeof(T));
  T *res = body + 1;
  res[len - 1] = 1;
  rer(i, len - 1) res[i - 1] = res[i] * a[i];
  return res;
}
template <class T, class U> void inc(T &a, U v = 1) { a += v; }
template <class T, class U> void inc(vector<T> &a, U v = 1) {
  for (auto &u : a)
    inc(u, v);
}
template <class T> void inc(vector<T> &a) {
  for (auto &u : a)
    inc(u, 1);
}
template <class T, class U> void plus(T &a, U v = 1) { a += v; }
template <class T, class U> void plus(vector<T> &a, U v = 1) {
  for (auto &u : a)
    inc(u, v);
}
template <class T> void plus(vector<T> &a) {
  for (auto &u : a)
    inc(u, 1);
}
template <class T, class U> void dec(T &a, U v = 1) { a -= v; }
template <class T, class U> void dec(vector<T> &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class T> void dec(vector<T> &a) {
  for (auto &u : a)
    dec(u, 1);
}
template <class T, class U> void minu(T &a, U v = 1) { a -= v; }
template <class T, class U> void minu(vector<T> &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class T> void minu(vector<T> &a) {
  for (auto &u : a)
    dec(u, 1);
}
template <class U> void minu(string &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
void minu(string &a) {
  for (auto &u : a)
    dec(u, 1);
}
inline bool inside(int h, int w, int H, int W) {
  return h >= 0 && w >= 0 && h < H && w < W;
}
inline bool inside(int l, int v, int r) { return l <= v && v < r; }
#define ins inside
ll u(ll a) { return a < 0 ? 0 : a; }
template <class T> vector<T> u(const vector<T> &a) {
  vector<T> ret = a;
  fora(v, ret) v = u(v);
  return ret;
}
#define MIN(a) numeric_limits<a>::min()
#define MAX(a) numeric_limits<a>::max()
ll goldd(ll left, ll right, function<ll(ll)> calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl < fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  ll minScore = MAX(ll);
  ll resIndex = left;
  for (ll i = left; i < right + 1; i++) {
    ll score = calc(i);
    if (minScore > score) {
      minScore = score;
      resIndex = i;
    }
  }
  return resIndex;
}
ll goldt(ll left, ll right, function<ll(ll)> calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl > fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  if (left > right) {
    ll l = left;
    left = right;
    right = l;
  }
  ll maxScore = MIN(ll);
  ll resIndex = left;
  for (ll i = left; i < right + 1; i++) {
    ll score = calc(i);
    if (maxScore < score) {
      maxScore = score;
      resIndex = i;
    }
  }
  return resIndex;
}
template <class T> T min(vector<vector<T>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, *min_element(all(a[i])));
  return res;
}
template <class T> T max(vector<vector<T>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, *max_element(all(a[i])));
  return res;
}

constexpr bool bget(ll m, int keta) {
  assert(keta < 64);
  return (m >> keta) & 1;
}
int bget(ll m, int keta, int sinsuu) {
  m /= (ll)pow(sinsuu, keta);
  return m % sinsuu;
}
inline ll bit(int n) { return (1LL << (n)); }
inline ll bit(int n, int sinsuu) { return (ll)pow(sinsuu, n); }
// int bcou(ll m) { return __builtin_popcount(m & 0xFFFFFFFF) +
// __builtin_popcount(m >> 32); }
#define bcou __builtin_popcountll
// 最下位ビット
int lbit(int n) { return n & -n; }
// 最上位ビット
int hbit(int n) {
  n |= (n >> 1);
  n |= (n >> 2);
  n |= (n >> 4);
  n |= (n >> 8);
  n |= (n >> 16);
  n |= (n >> 32);
  return n - (n >> 1);
}
// 初期化は0を渡す
ll nextComb(ll &mask, int n, int r) {
  if (!mask)
    return mask = (1LL << r) - 1;
  ll x = mask & -mask; // 最下位の1
  ll y = mask + x;     // 連続した下の1を繰り上がらせる
  ll res = ((mask & ~y) / x >> 1) | y;
  if (bget(res, n))
    return mask = 0;
  else
    return mask = res;
}

// n桁以下でビットがr個立っているもののvectorを返す
vl bitCombList(int n, int r) {
  vl res;
  int m = 0;
  while (nextComb(m, n, r)) {
    res.pb(m);
  }
  return res;
}
// 大文字小文字を区別する
int altoiaZ(char c) {
  if ('A' <= c && c <= 'Z')
    return c - 'A';
  return c - 'a' + 26;
}
char itoalaZ(int i) {
  if (i < 26)
    return 'A' + i;
  return 'a' + i - 26;
}
// aもAも0を返す 基本小文字
int altoi(char c) {
  if ('A' <= c && c <= 'Z')
    return c - 'A';
  return c - 'a';
}
char itoal(int i) { return 'a' + i; }
int ctoi(char c) { return c - '0'; }
char itoc(int i) { return i + '0'; }
int vtoi(vi &v) {
  int res = 0;
  if (sz(v) > 18) {
    debugline("vtoi");
    deb(sz(v));
    ole();
  }
  rep(i, sz(v)) {
    res *= 10;
    res += v[i];
  }
  return res;
}
vi itov(int i) {
  vi res;
  while (i) {
    res.pb(i % 10);
    i /= 10;
  }
  rev(res);
  return res;
}
vi stov(string &a) {
  int n = sz(a);
  vi ret(n);
  rep(i, n) { ret[i] = a[i] - '0'; }
  return ret;
}
vector<vector<int>> ctoi(vector<vector<char>> s, char c) {
  int n = sz(s), m = sz(s[0]);
  vector<vector<int>> res(n, vector<int>(m));
  rep(i, n) rep(j, m) res[i][j] = s[i][j] == c;
  return res;
}
#define unique(v) v.erase(unique(v.begin(), v.end()), v.end());

void compress(vi &a) {
  vi b;
  int len = a.size();
  for (int i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (int i = 0; i < len; ++i) {
    a[i] = lower_bound(all(b), a[i]) - b.begin();
  }
}

void compress(int a[], int len) {
  vi b;
  for (int i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (int i = 0; i < len; ++i) {
    a[i] = lower_bound(all(b), a[i]) - b.begin();
  }
}
// 要素が見つからなかったときに困る
#define binarySearch(a, v) (binary_search(all(a), v))
#define lowerIndex(a, v) (lower_bound(all(a), v) - a.begin())
#define lowerBound(a, v) (*lower_bound(all(a), v))
#define upperIndex(a, v) (upper_bound(all(a), v) - a.begin())
#define upperBound(a, v) (*upper_bound(all(a), v))
#define ans(a)                                                                 \
  cout << a << endl;                                                           \
  continue;
#define poll(a)                                                                \
  q.front();                                                                   \
  q.pop()
#define dpoll(a)                                                               \
  q.front();                                                                   \
  q.pop_front()
#define pollLast(a)                                                            \
  q.back();                                                                    \
  q.pop_back()
#define pollBack(a)                                                            \
  q.back();                                                                    \
  q.pop_back()
template <class T> inline void fin(T s) { cout << s << endl, exit(0); }

// 便利 数学 math
int mod(int a, int m) { return (a % m + m) % m; }
int pow(int a) { return a * a; };
ll fact(int v) { return v <= 1 ? 1 : v * fact(v - 1); }
ll comi(int n, int r) {
  assert(n < 100);
  static vvi(pas, 100, 100);
  if (pas[0][0])
    return pas[n][r];
  pas[0][0] = 1;
  rep(i, 1, 100) {
    pas[i][0] = 1;
    rep(j, 1, i + 1) pas[i][j] = pas[i - 1][j - 1] + pas[i - 1][j];
  }
  return pas[n][r];
}
double comd(int n, int r) {
  assert(n < 2020);
  static vvd(comb, 2020, 2020);
  if (comb[0][0] == 0) {
    comb[0][0] = 1;
    rep(i, 2000) {
      comb[i + 1][0] = 1;
      rep(j, 1, i + 2) { comb[i + 1][j] = comb[i][j] + comb[i][j - 1]; }
    }
  }
  return comb[n][r];
}
int gcd(int a, int b) {
  while (b)
    a %= b, swap(a, b);
  return abs(a);
}
int gcd(vi b) {
  ll res = b[0];
  rep(i, 1, sz(b)) res = gcd(b[i], res);
  return res;
}
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
ll ceil(ll a, ll b) {
  if (b == 0) {
    debugline("ceil");
    deb(a, b);
    ole();
    return -1;
  } else if (a < 0) {
    return 0;
  } else {
    return (a + b - 1) / b;
  }
}
// v * v >= aとなる最小のvを返す
ll sqrt(ll a) {
  if (a < 0) {
    debugline("sqrt");
    deb(a);
    ole();
  }
  ll res = (ll)std::sqrt(a);
  while (res * res < a)
    res++;
  return res;
}
double log(double e, double x) { return log(x) / log(e); }
ll sig(ll t) { return (1 + t) * t / 2; }
ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; }

namespace pnumber_tools {
#define Eratosthenes30_2                                                       \
  class Eratosthenes {                                                         \
  public:                                                                      \
    virtual void initialize() {}                                               \
    virtual void generate(const int64_t x) = 0;                                \
    virtual void generate(const int64_t from, const int64_t to) = 0;           \
    virtual int64_t count() = 0;                                               \
    virtual bool prime(int x) = 0;                                             \
    virtual int32_t version() = 0;                                             \
    inline uint64_t PopCnt(uint64_t);                                          \
    inline int BitToIndex(uint8_t);                                            \
  };                                                                           \
  uint64_t Eratosthenes::PopCnt(uint64_t bits) {                               \
    bits = (bits & 0x5555555555555555ULL) +                                    \
           ((bits >> 1) & 0x5555555555555555ULL);                              \
    bits = (bits & 0x3333333333333333ULL) +                                    \
           ((bits >> 2) & 0x3333333333333333ULL);                              \
    bits = (bits & 0x0f0f0f0f0f0f0f0fULL) +                                    \
           ((bits >> 4) & 0x0f0f0f0f0f0f0f0fULL);                              \
    return (bits * 0x0101010101010101ULL) >> (64 - 8);                         \
  }                                                                            \
  int Eratosthenes::BitToIndex(uint8_t b) {                                    \
    switch (b) {                                                               \
    case 1 << 0:                                                               \
      return 0;                                                                \
    case 1 << 1:                                                               \
      return 1;                                                                \
    case 1 << 2:                                                               \
      return 2;                                                                \
    case 1 << 3:                                                               \
      return 3;                                                                \
    case 1 << 4:                                                               \
      return 4;                                                                \
    case 1 << 5:                                                               \
      return 5;                                                                \
    case 1 << 6:                                                               \
      return 6;                                                                \
    case 1 << 7:                                                               \
      return 7;                                                                \
    }                                                                          \
    return -1;                                                                 \
  }                                                                            \
  namespace {                                                                  \
  const uint64_t kSegmentSize = 1000000;                                       \
  const int32_t kMod30[] = {1, 7, 11, 13, 17, 19, 23, 29};                     \
  const int C1[] = {6, 4, 2, 4, 2, 4, 6, 2};                                   \
  const int C0[][8] = {                                                        \
      {0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 0, 1, 1, 1, 1},                      \
      {2, 2, 0, 2, 0, 2, 2, 1}, {3, 1, 1, 2, 1, 1, 3, 1},                      \
      {3, 3, 1, 2, 1, 3, 3, 1}, {4, 2, 2, 2, 2, 2, 4, 1},                      \
      {5, 3, 1, 4, 1, 3, 5, 1}, {6, 4, 2, 4, 2, 4, 6, 1},                      \
  };                                                                           \
  const uint8_t kMask[][8] = {                                                 \
      {0xfe, 0xfd, 0xfb, 0xf7, 0xef, 0xdf, 0xbf, 0x7f},                        \
      {0xfd, 0xdf, 0xef, 0xfe, 0x7f, 0xf7, 0xfb, 0xbf},                        \
      {0xfb, 0xef, 0xfe, 0xbf, 0xfd, 0x7f, 0xf7, 0xdf},                        \
      {0xf7, 0xfe, 0xbf, 0xdf, 0xfb, 0xfd, 0x7f, 0xef},                        \
      {0xef, 0x7f, 0xfd, 0xfb, 0xdf, 0xbf, 0xfe, 0xf7},                        \
      {0xdf, 0xf7, 0x7f, 0xfd, 0xbf, 0xfe, 0xef, 0xfb},                        \
      {0xbf, 0xfb, 0xf7, 0x7f, 0xfe, 0xef, 0xdf, 0xfd},                        \
      {0x7f, 0xbf, 0xdf, 0xef, 0xf7, 0xfb, 0xfd, 0xfe},                        \
  };                                                                           \
  const int64_t kSievedPrimes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23};            \
  const int64_t kInitialSize = 7 * 11 * 13 * 17 * 19 * 23;                     \
  class Eratosthenes2 final : public Eratosthenes {                            \
  public:                                                                      \
    void generate(const int64_t x) override;                                   \
    void generate(const int64_t, const int64_t) override { assert(false); }    \
    bool prime(int x) override;                                                \
    int64_t count() override;                                                  \
    int32_t version() override { return 2; }                                   \
    std::vector<uint8_t> flags_;                                               \
  };                                                                           \
  void Eratosthenes2::generate(const int64_t x) {                              \
    flags_.clear();                                                            \
    if (x > 10000000000)                                                       \
      return;                                                                  \
    int64_t size = x / 30 + (x % 30 != 0);                                     \
    flags_.resize(size, 0xff);                                                 \
    if (int r = x % 30) {                                                      \
      if (r <= 1)                                                              \
        flags_[size - 1] = 0x0;                                                \
      else if (r <= 7)                                                         \
        flags_[size - 1] = 0x1;                                                \
      else if (r <= 11)                                                        \
        flags_[size - 1] = 0x3;                                                \
      else if (r <= 13)                                                        \
        flags_[size - 1] = 0x7;                                                \
      else if (r <= 17)                                                        \
        flags_[size - 1] = 0xf;                                                \
      else if (r <= 19)                                                        \
        flags_[size - 1] = 0x1f;                                               \
      else if (r <= 23)                                                        \
        flags_[size - 1] = 0x3f;                                               \
      else if (r <= 29)                                                        \
        flags_[size - 1] = 0x7f;                                               \
    }                                                                          \
    flags_[0] = 0xfe;                                                          \
    const uint64_t sqrt_x = std::ceil(std::sqrt(x) + 0.1);                     \
    const uint64_t sqrt_xi = sqrt_x / 30 + 1;                                  \
    for (uint64_t i = 0; i < sqrt_xi; ++i) {                                   \
      for (uint8_t flags = flags_[i]; flags; flags &= flags - 1) {             \
        uint8_t lsb = flags & (-flags);                                        \
        int ibit = BitToIndex(lsb);                                            \
        const int32_t m = kMod30[ibit];                                        \
        int32_t pm = 30 * i + 2 * m;                                           \
        for (uint64_t j = i * pm + (m * m) / 30, k = ibit; j < flags_.size();  \
             j += i * C1[k] + C0[ibit][k], k = (k + 1) & 7) {                  \
          flags_[j] &= kMask[ibit][k];                                         \
        }                                                                      \
      }                                                                        \
    }                                                                          \
  }                                                                            \
  int64_t Eratosthenes2::count() {                                             \
    if (flags_.empty())                                                        \
      return -1;                                                               \
    int64_t ret = 3;                                                           \
    for (uint8_t f : flags_) {                                                 \
      ret += PopCnt(f);                                                        \
    }                                                                          \
    return ret;                                                                \
  }                                                                            \
  inline bool Eratosthenes2::prime(int x) {                                    \
    static unordered_map<int, int> imap = {                                    \
        {1, 0}, {7, 1}, {11, 2}, {13, 3}, {17, 4}, {19, 5}, {23, 6}, {29, 7}}; \
    if (x == 2 || x == 3 || x == 5)                                            \
      return true;                                                             \
    else if (!(x & 1) || x % 3 == 0 || x % 5 == 0)                             \
      return false;                                                            \
    return (flags_[x / 30] >> imap[x % 30]) & 1;                               \
  }                                                                            \
  }
Eratosthenes30_2 Eratosthenes2 era;
int prime_len = 3e6;
vector<int> primes;
void setprime(int size = 3e6) {
  prime_len = size;
  era.generate(size);
  primes = {2, 3, 5};
  int nums[129] = {};
  nums[1] = 1;
  nums[2] = 7;
  nums[4] = 11;
  nums[8] = 13;
  nums[16] = 17;
  nums[32] = 19;
  nums[64] = 23;
  nums[128] = 29;
  // p = 30i + j
  int j = 0;
  int sflim = era.flags_.size();
  int i30 = 0;
  for (int i = 0; i < sflim; i++, i30 += 30) {
    uint8_t m = era.flags_[i];
    while (m) {
      j = m & -m;
      primes.push_back(i30 + nums[j]);
      m ^= j;
    }
  }
}
using u32 = unsigned int; // longlongになっていると思うが放置していいと思う
using u64 = unsigned long long;
using u128 = __uint128_t;

// O(k log(n)^3)
// 32bit -> 270回程度
// 64bit -> 1260回程度
// 64bit以下で精度100%
template <class BinOp>
bool miller_rabin(const uint64_t &n, const uint64_t *witness, BinOp modmul) {
  if (n == 2)
    return true;
  if (n < 2 || n % 2 == 0)
    return false;
  const int64_t m = n - 1, d = m / (m & -m);
  auto modpow = [&](int64_t a, int64_t b) {
    int64_t res = 1;
    for (; b; b /= 2) {
      if (b & 1)
        res = modmul(res, a);
      a = modmul(a, a);
    }
    return res;
  };
  auto suspect = [&](uint64_t a, uint64_t t) {
    a = modpow(a, t);
    while (t != n - 1 && a != 1 && a != n - 1) {
      a = modmul(a, a);
      t = modmul(t, 2);
    }
    return a == n - 1 || t % 2 == 1;
  };
  for (const uint64_t *w = witness; *w; w++) {
    if (*w % n != 0 && !suspect(*w, d))
      return false;
  }
  return true;
}

// miller_rabin & エラトステネス
bool prime(const int &n) {
  assert(n < 9223372036854775808ULL);
  if (!sz(primes))
    setprime();
  // エラトステネスの篩 偶数省略ver
  //[i] = i*2+1
  if (n < prime_len)
    return era.prime(n);
  if (n < 4294967296) {
    rep(i, 30) if (!(n % primes[i])) return false;
    constexpr uint64_t witness[] = {2, 7, 61, 0};
    auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t { return a * b % n; };
    return miller_rabin(n, witness, modmul);
  } else {
    rep(i, 50) if (!(n % primes[i])) return false;
    constexpr uint64_t witness[] = {2,      325,     9375,       28178,
                                    450775, 9780504, 1795265022, 0};
    auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t {
      return (uint64_t)((__uint128_t)a * b % n);
    };
    return miller_rabin(n, witness, modmul);
  }
}
vi divisors(int v) {
  vi res;
  double lim = std::sqrt(v);
  for (int i = 1; i <= lim; ++i) {
    if (v % i == 0) {
      res.pb(i);
      if (i != v / i)
        res.pb(v / i);
    }
  }
  return res;
}
vp factorization(int v) {
  assert(v);
  int tv = v;
  vp res;
  if (!sz(primes))
    setprime();
  for (auto &&p : primes) {
    if (v % p == 0)
      res.eb(p, 0);
    while (v % p == 0) {
      v /= p;
      res.back().second++;
    }
    if (v == 1 || p * p > tv)
      break;
  }
  if (v > 1)
    res.eb(v, 1);
  return res;
}

// 注意 gcdを取る間隔 i%10 が場合によってはtleを引き起こす
// うまくいかなかったら%1で動かすように変更した
vp rho(int v) {
  assert(v);
  if (v == 1)
    return vp();
  vi ret;
  if (!sz(primes))
    setprime();
  int lim = 300; // 64bit
  if (v < 4294967296)
    lim = 80; // 32bit
  else if (v < 281474976710656LL)
    lim = 120; // 48bit
  for (int i = 0; i < lim; i += 5) {
    // v<=1の判定回数を減らせる分高速化できる事を期待している
    while (v % primes[i] == 0)
      ret.pb(primes[i]), v /= primes[i];
    while (v % primes[i + 1] == 0)
      ret.pb(primes[i + 1]), v /= primes[i + 1];
    while (v % primes[i + 2] == 0)
      ret.pb(primes[i + 2]), v /= primes[i + 2];
    while (v % primes[i + 3] == 0)
      ret.pb(primes[i + 3]), v /= primes[i + 3];
    while (v % primes[i + 4] == 0)
      ret.pb(primes[i + 4]), v /= primes[i + 4];
    if (v <= 1) {
      sort(ret); // 試し割りだけで終わった時
      return run_length(ret);
    }
  }
  int lower_prime_bound = primes[lim - 1] * primes[lim - 1];

  // 試し割りするか
  mt19937 mr;
  mr.seed(clock());
  int c;
  auto fac = [&](int v, const int loop = 10) {
    //        if (v == 4)return 2ll; //試し割りで消えた
    c = mr() % v;
    int x = mr() % v;
    int y = x;
    int d = 1;
    int lim = std::sqrt(v) + 2;
    // brentによる改良 差分を一操作で求められる
    for (int i = 1, mul = 1, next_2pow = 2; d == 1 && i <= lim; i++) {
      if (i == next_2pow) {
        y = x;
        next_2pow <<= 1;
      }
      x = ((u128)x * x + c) % v;
      mul = ((u128)mul * abs(x - y)) %
            v; // ユークリッドの互除法と同じ原理で因数は保たれる
      if (i % loop == 0) { // 10とかなんとか iによっては死ぬ
        d = gcd(mul, v);
        mul = 1;
      }
    }
    if (d == v)
      return -1ll;
    return d;
  };

  queue<int> q;
  q.push(v);
  while (!q.empty()) {
    int v = q.front();
    q.pop();
    if (prime(v)) {
      ret.pb(v);
      continue;
    }
    int d = -1;
    // 基本loop回数10でやり、駄目なら1で
    for (int i = 0; d == -1 && i < 5; i++)
      d = fac(v, 10);
    if (d == -1)
      while (d == -1)
        d = fac(v, 1);
    int d2 = v / d;
    rep(_, 2) {
      if (d <= lower_prime_bound)
        ret.pb(d);
      else
        q.push(d);
      d = d2;
    }
  }
  sort(ret);
  return run_length(ret);
}
} // namespace pnumber_tools
using namespace pnumber_tools;
// 幾何 Pをcomplexとして扱う
template <class T, class U> bool eq(T a, U b) { return fabs(a - b) < eps; }
dou atan2(pd a) { return atan2(a.se, a.fi); }
dou angle(pd f, pd t) { return atan2(t.se - f.se, t.fi - f.fi); }
dou distance(pd a, pd b) { return hypot(a.fi - b.fi, a.se - b.se); }
// bを中心とするabcのtheta aからcにかけて時計回り
dou angle(pd a, pd b, pd c) {
  dou ax = a.fi - b.fi;
  dou ay = a.se - b.se;
  dou cx = c.fi - b.fi;
  dou cy = c.se - b.se;
  double ret = atan2(cy, cx) - atan2(ay, ax);
  if (ret < 0)
    ret += 2 * PI;
  return ret;
}

dou dot(pd a, pd b) { return a.fi * b.fi + a.se + b.se; }
dou cro(pd a, pd b) { return a.fi * b.se - a.se + b.fi; }

template <class T> struct edge {
  int f, t;
  T c;
  //    int id;
  //    int type;
  edge(int f, int t, T c = 1) : f(f), t(t), c(c) {}
  bool operator<(const edge &b) const { return c < b.c; }
  bool operator>(const edge &b) const { return c > b.c; }
};
template <class T> ostream &operator<<(ostream &os, edge<T> &e) {
  os << e.f << " " << e.t << " " << e.c;
  return os;
}
template <typename T> class graph {
protected:
  vector<bool> _used;

public:
  vector<vector<edge<T>>> g;
  vector<edge<T>> edges;
  int n;
  graph(int n) : n(n) { g.resize(n), _used.resize(n); }
  void clear() { g.clear(), edges.clear(); }
  void resize(int n) {
    this->n = n;
    g.resize(n);
    _used.resize(n);
  }
  int size() { return g.size(); }
  vector<edge<T>> &operator[](int i) { return g[i]; }
  virtual void add(int f, int t, T c) = 0;
};
template <typename T = ll> class digraph : public graph<T> {
public:
  using graph<T>::g;
  using graph<T>::n;
  using graph<T>::edges;
  using graph<T>::_used;
  int eid = 0;

  digraph(int n) : graph<T>(n) {}
  void add(int f, int t, T c = 1) {
    g[f].emplace_back(f, t, c);
    //        edges.emplace_back(f, t, c);
  }
  //    bool used(edge<T> &e) { return _used[e.id]; }
  //    bool used(int id) { return _used[id]; }
  //    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }
  //    void del(int id) { _used[id] = _used[id ^ 1] = 1; }
  void set_edges() {
    if (sz(edges))
      return;
    rep(i, n) fora(e, g[i]) edges.pb(e);
  }
};
template <class T = int> class undigraph : public graph<T> {
public:
  using graph<T>::g;
  using graph<T>::n;
  using graph<T>::edges;
  using graph<T>::_used;
  int eid = 0;
  undigraph(int n) : graph<T>(n) {}
  void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
    if (!(0 <= f && f < n && 0 <= t && t < n)) {
      debugline("undigraph add");
      deb(f, t, c, id, ty);
      ole();
    }
    if (id == -1)
      id = eid, eid += 2;
    g[f].emplace_back(f, t, c, id, ty);
    g[t].emplace_back(t, f, c, id + 1, ty);
    edges.emplace_back(f, t, c, id, ty);
    edges.emplace_back(t, f, c, id + 1, ty);
  }
  void add(edge<T> &e) {
    int f = e.f, t = e.t, ty = e.type;
    T c = e.c;
    add(f, t, c, ty);
  }
  bool used(edge<T> &e) { return _used[e.id]; }
  bool used(int id) { return _used[id]; }
  void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }
  void del(int id) { _used[id] = _used[id ^ 1] = 1; }
  void set_edges() {
    if (sz(edges))
      return;
    rep(i, n) fora(e, g[i]) edges.pb(e);
  }
};
template <typename T> struct radixheap {
  vector<pair<u64, T>> v[65];
  u64 size, last;
  radixheap() : size(0), last(0) {}
  bool empty() const { return size == 0; }
  inline int getbit(int a) { return a ? 64 - __builtin_clzll(a) : 0; }
  void push(u64 key, const T &value) {
    ++size;
    v[getbit(key ^ last)].emplace_back(key, value);
  }
  pair<u64, T> pop() {
    if (v[0].empty()) {
      int idx = 1;
      while (v[idx].empty())
        ++idx;
      last = min_element(begin(v[idx]), end(v[idx]))->first;
      for (auto &p : v[idx])
        v[getbit(p.first ^ last)].emplace_back(p);
      v[idx].clear();
    }
    --size;
    auto ret = v[0].back();
    v[0].pop_back();
    return ret;
  }
};
vi dijkstra2(const graph<int> &g, int s, int cant_arrive = -1) {
  if (!(0 <= s && s < g.n)) {
    debugline("dijkstra");
    deb(s, g.n);
    ole();
  }
  int initValue = MAX(int);
  vi dis(g.n, initValue);
  radixheap<int> q;
  dis[s] = 0;
  q.push(0, s);
  while (!q.empty()) {
    int nowc, i;
    tie(nowc, i) = q.pop();
    if (dis[i] != nowc)
      continue;
    for (auto &&e : g.g[i]) {
      int to = e.t;
      int c = nowc + e.c;
      if (dis[to] > c) {
        dis[to] = c;
        q.push(dis[to], to);
      }
    }
  }
  // 基本、たどり着かないなら-1
  if (cant_arrive == -1)
    for (auto &&d : dis)
      if (d == initValue)
        d = -1;
  return dis;
}

template <class T>
vector<T> dijkstra(const graph<T> &g, int s, int cant_arrive = -1) {
  if (!(0 <= s && s < g.n)) {
    debugline("dijkstra");
    deb(s, g.n);
    ole();
  }
  T initValue = MAX(T);
  vector<T> dis(g.n, initValue);
  priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>> q;
  dis[s] = 0;
  q.emplace(0, s);
  while (q.size()) {
    T nowc = q.top().fi;
    int i = q.top().se;
    q.pop();
    if (dis[i] != nowc)
      continue;
    for (auto &&e : g.g[i]) {
      int to = e.t;
      T c = nowc + e.c;
      if (dis[to] > c) {
        dis[to] = c;
        q.emplace(dis[to], to);
      }
    }
  }
  // 基本、たどり着かないなら-1
  if (cant_arrive == -1)
    for (auto &&d : dis)
      if (d == initValue)
        d = -1;
  return dis;
}
template <class T>
vector<vector<T>> warshall(const graph<T> &g, int cant_arrive = -1) {
  int n = g.n;
  vector<vector<T>> dis(n, vector<T>(n, linf));
  rep(i, n) fora(e, g.g[i]) chmi(dis[e.f][e.t], e.c);
  rep(i, n) dis[i][i] = 0;
  rep(k, n) rep(i, n) rep(j, n) chmi(dis[i][j], dis[i][k] + dis[k][j]);
  // 基本、たどり着かないなら-1
  if (cant_arrive == -1)
    rep(i, n) rep(j, n) if (dis[i][j] == linf) dis[i][j] = -1;
  return dis;
}

template <class T = int> class tree : public undigraph<T> {
public:
  using undigraph<T>::g;
  using undigraph<T>::n;
  using undigraph<T>::edges;
  using undigraph<T>::_used;
  vi disv;

private:
  bool never = 1;
  int root = -1;
  vi dep;            // 到着した順の深さ
  vi vis;            // 到着した順
  vi first;          // 初めてiに到着した時
  vector<int> bestt; // タイミングを持つ
  int sn = 0;        // seg木の底辺の長さ euler tourの長さ
  void built() {
    never = 0;
    n = g.size();
    // euler tour
    disv.resize(n);
    first.resize(n);
    dep.resize(n * 2);
    vis.resize(n * 2);
    function<void(int, int, int, int &)> dfs = [&](int i, int p, int d,
                                                   int &k) {
      first[i] = k;
      vis[k] = i;
      dep[k++] = d;
      forg(gi, g[i]) {
        if (t == p)
          continue;
        disv[t] = disv[i] + c;
        dfs(t, i, d + 1, k);
        vis[k] = i;
        dep[k++] = d;
      }
    };
    int k = 0;
    dfs(root, -1, 0, k);
    // rmq
    sn = 1;
    while (sn < n * 2)
      sn <<= 1;

    bestt.resize((sn << 1) - 1);
    rep(i, k) bestt[i + sn - 1] = i; // 時間を持つ
    rer(i, sn - 2) {
      int lt = (i << 1) + 1, rt = (i << 1) + 2;
      if (dep[bestt[lt]] < dep[bestt[rt]]) {
        bestt[i] = bestt[lt];
      } else {
        bestt[i] = bestt[rt];
      }
    }
  }
  // 最小の深さを持つタイミングが知りたい
  inline int lca_rmq(int l, int r) {
    int midv = inf;
    int midi = -1;
    l += sn - 1;
    r += sn - 1;
    while (l < r) {
      // 右なら、次に右上へ移動してしまうので
      if (!(l & 1)) {
        if (chmi(midv, dep[bestt[l]])) {
          midi = bestt[l];
        }
      }
      // 右なら、左を取らないと機会を逃す
      if (!(r & 1)) {
        if (chmi(midv, dep[bestt[r - 1]])) {
          midi = bestt[r - 1];
        }
      }
      l >>= 1;
      r = (r - 1) >> 1;
    }
    return vis[midi];
  }

public:
  tree(int n, int root = 0) : undigraph<T>(n), root(root) {}
  bool leaf(int v) { return sz(g[v]) == 1 && v != root; }
  int lca(int u, int v) {
    if (never)
      built();
    if (first[u] > first[v])
      swap(u, v);
    return lca_rmq(first[u], first[v] + 1);
  }
  int dis(int u, int v) {
    if (never)
      built();
    if (first[u] > first[v])
      swap(u, v);
    int p = lca_rmq(first[u], first[v] + 1);
    return disv[u] + disv[v] - disv[p] * 2;
  }
};
// 辺によりメモリを大量消費
//  よってedgesを消している
// 頂点10^6でメモリを190MB(制限の8割)使う

// 軽量化のため、辺を持たないbig gridクラスがあってもいいかもしれない
//
template <class T = int> class grid_k6 : public undigraph<T> {
public:
  using undigraph<T>::g;
  using undigraph<T>::n;
  using undigraph<T>::edges;
  using undigraph<T>::_used;
  int H, W;
  int eid = 0;
  void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
    if (!(0 <= f && f < n && 0 <= t && t < n)) {
      debugline("grid_k6 add");
      deb(f, t, c, id, ty);
      ole();
    }
    g[f].emplace_back(f, t, c, eid++, ty);
    g[t].emplace_back(t, f, c, eid++, ty);
  }
  int getid(int h, int w) {
    if (!inside(h, w, H, W))
      return -1;
    return W * h + w;
  }
  P get2(int id) { return mp(id / W, id % W); }
  P operator()(int id) { return get2(id); }
  int operator()(int h, int w) { return getid(h, w); }
  grid_k6(int H, int W) : H(H), W(W), undigraph<T>(H * W) {
    rep(h, H) {
      rep(w, W) {
        int f = getid(h, w);
        if (w + 1 < W)
          add(f, getid(h, w + 1));
        if (h + 1 < H)
          add(f, getid(h + 1, w));
      }
    }
  }
  grid_k6(_vvc ba, char wall = '#')
      : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])) {
    rep(h, H) {
      rep(w, W) {
        if (ba[h][w] == wall)
          con;
        int f = getid(h, w);
        if (w + 1 < W && ba[h][w + 1] != wall) {
          add(f, getid(h, w + 1));
        }
        if (h + 1 < H && ba[h + 1][w] != wall) {
          add(f, getid(h + 1, w));
        }
      }
    }
  }
  void add(int fh, int fw, int th, int tw) {
    add(getid(fh, fw), getid(th, tw));
  }
  void set_edges() { rep(i, n) fora(e, g[i]) edges.pb(e); }
};
// 左上から右下に移動できる
template <class T = int> class digrid_k6 : public digraph<T> {
public:
  using digraph<T>::g;
  using digraph<T>::n;
  using digraph<T>::edges;
  using digraph<T>::_used;
  int H, W;
  int eid = 0;
  void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
    if (!(0 <= f && f < n && 0 <= t && t < n)) {
      debugline("digrid_k6 add");
      deb(f, t, c, id, ty);
      ole();
    }
    g[f].emplace_back(f, t, c, eid++, ty);
  }
  int getid(int h, int w) {
    if (!inside(h, w, H, W))
      return -1;
    return W * h + w;
  }
  P get2(int id) { return mp(id / W, id % W); }
  P operator()(int id) { return get2(id); }
  int operator()(int h, int w) { return getid(h, w); }
  digrid_k6(int H, int W) : H(H), W(W), digraph<T>(H * W) {
    rep(h, H) {
      rep(w, W) {
        int f = getid(h, w);
        if (w + 1 < W)
          add(f, getid(h, w + 1));
        if (h + 1 < H)
          add(f, getid(h + 1, w));
      }
    }
  }
  digrid_k6(_vvc ba, char wall = '#')
      : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])) {
    rep(h, H) {
      rep(w, W) {
        if (ba[h][w] == wall)
          con;
        int f = getid(h, w);
        if (w + 1 < W && ba[h][w + 1] != wall) {
          add(f, getid(h, w + 1));
        }
        if (h + 1 < H && ba[h + 1][w] != wall) {
          add(f, getid(h + 1, w));
        }
      }
    }
  }
  void add(int fh, int fw, int th, int tw) {
    add(getid(fh, fw), getid(th, tw));
  }
  void set_edges() { rep(i, n) fora(e, g[i]) edges.pb(e); }
};
template <class T> bool nibu(const graph<T> &g) {
  int size = 0;
  rep(i, g.n) size += sz(g.g[i]);
  if (size == 0)
    return true;

  UnionFind uf(g.n * 2);
  rep(i, g.n) fora(e, g.g[i]) uf.unite(e.f, e.t + g.n),
      uf.unite(e.f + g.n, e.t);
  rep(i, g.n) if (uf.same(i, i + g.n)) return 0;
  return 1;
}
// 二部グラフを色分けした際の頂点数を返す
template <class T> vp nibug(graph<T> &g) {
  vp cg;
  if (!nibu(g)) {
    debugline("nibu");
    ole();
  }
  int _n = g.size();
  vb _was(_n);
  queue<P> q;
  rep(i, _n) {
    if (_was[i])
      continue;
    q.push(mp(i, 1));
    _was[i] = 1;
    int red = 0;
    int coun = 0;
    while (q.size()) {
      int now = q.front().fi;
      int col = q.front().se;
      red += col;
      coun++;
      q.pop();
      forg(gi, g[now]) {
        if (_was[t])
          continue;
        q.push(mp(t, col ^ 1));
        _was[t] = 1;
      }
    }
    cg.push_back(mp(red, coun - red));
  }
  return cg;
}
template <class T> ostream &operator<<(ostream &os, graph<T> &g) {
  rep(i, sz(g)) {
    forg(gi, g[i]) { os << f << " " << t << " " << c << endl; }
  }
  return os;
}

// 機能拡張
vp vtop(vi &a, vi &b) {
  vp res(sz(a));
  rep(i, sz(a)) res[i] = mp(a[i], b[i]);
  return res;
}
void ptov(vp &p, vi &a, vi &b) {
  a.resize(sz(p));
  b.resize(sz(p));
  rep(i, sz(p)) a[i] = p[i].fi, b[i] = p[i].se;
}
template <typename _CharT, typename _Traits, typename _Alloc>
basic_string<_CharT, _Traits, _Alloc>
operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {
  basic_string<_CharT, _Traits, _Alloc> __str(__lhs);
  __str.append(to_string(__rv));
  return __str;
}
template <typename _CharT, typename _Traits, typename _Alloc>
void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {
  __lhs += to_string(__rv);
}
template <typename _CharT, typename _Traits, typename _Alloc>
basic_string<_CharT, _Traits, _Alloc>
operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs,
          const signed __rv) {
  basic_string<_CharT, _Traits, _Alloc> __str(__lhs);
  __str.append(to_string(__rv));
  return __str;
}
template <typename _CharT, typename _Traits, typename _Alloc>
void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs,
                const signed __rv) {
  __lhs += to_string(__rv);
}
template <class T, class U> void operator+=(queue<T> &a, U v) { a.push(v); }
template <class T, class U> void operator+=(deque<T> &a, U v) {
  a.push_back(v);
}
template <class T, class U>
priority_queue<T, vector<T>, greater<T>> &
operator+=(priority_queue<T, vector<T>, greater<T>> &a, U v) {
  a.push(v);
  return a;
}
template <class T, class U>
priority_queue<T> &operator+=(priority_queue<T> &a, U v) {
  a.push(v);
  return a;
}
template <class T, class U> set<T> &operator+=(set<T> &a, U v) {
  a.insert(v);
  return a;
}
template <class T, class U>
set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {
  a.insert(v);
  return a;
}
template <class T, class U> vector<T> &operator+=(vector<T> &a, U v) {
  a.pb(v);
  return a;
}
template <class T, class U> vector<T> operator+(const vector<T> &a, U v) {
  vector<T> ret = a;
  ret += v;
  return ret;
}
template <class T, class U> vector<T> operator+(U v, const vector<T> &a) {
  vector<T> ret = a;
  ret.insert(ret.begin(), v);
  return ret;
}
template <class T> vector<T> operator+(vector<T> &a, vector<T> &b) {
  vector<T> ret;
  ret = a;
  fora(v, b) ret += v;
  return ret;
}
template <class T> vector<T> &operator+=(vector<T> &a, vector<T> &b) {
  fora(v, b) a += v;
  return a;
}
template <class T, class U>
vector<T> &operator+=(vector<T> &a, initializer_list<U> v) {
  for (auto &&va : v)
    a.pb(va);
  return a;
}
template <class T> vector<T> &operator-=(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-=");
    deb(a);
    deb(b);
    ole();
  }
  rep(i, sz(a)) a[i] -= b[i];
  return a;
}
template <class T> vector<T> operator-(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-");
    deb(a);
    deb(b);
    ole();
  }
  vector<T> res(sz(a));
  rep(i, sz(a)) res[i] = a[i] - b[i];
  return res;
}

template <typename T> void remove(vector<T> &v, unsigned int i) {
  v.erase(v.begin() + i);
}
template <typename T>
void remove(vector<T> &v, unsigned int s, unsigned int e) {
  v.erase(v.begin() + s, v.begin() + e);
}
template <typename T>
void removen(vector<T> &v, unsigned int s, unsigned int n) {
  v.erase(v.begin() + s, v.begin() + s + n);
}
template <typename T> void erase(vector<T> &v, unsigned int i) {
  v.erase(v.begin() + i);
}
template <typename T> void erase(vector<T> &v, unsigned int s, unsigned int e) {
  v.erase(v.begin() + s, v.begin() + e);
}
template <typename T>
void erasen(vector<T> &v, unsigned int s, unsigned int n) {
  v.erase(v.begin() + s, v.begin() + s + n);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned int i, U t) {
  v.insert(v.begin() + i, t);
}
template <typename T, typename U> void push_front(vector<T> &v, U t) {
  v.insert(v.begin(), t);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned int i, vector<T> list) {
  for (auto &&va : list)
    v.insert(v.begin() + i++, va);
}
template <typename T, typename U>
void insert(vector<T> &v, initializer_list<U> list) {
  for (auto &&va : list)
    v.pb(va);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned int i, initializer_list<U> list) {
  for (auto &&va : list)
    v.insert(v.begin() + i++, va);
}
template <typename T> void insert(set<T> &v, vector<T> list) {
  for (auto &&va : list)
    v.insert(va);
}
template <typename T> void insert(set<T> &v, initializer_list<T> list) {
  for (auto &&va : list)
    v.insert(va);
}

// 閉路がなければtrue
bool topo(vi &res, digraph<int> &g) {
  int n = g.g.size();
  vi nyu(n);
  rep(i, n) for (auto &&e : g[i]) nyu[e.t]++;
  queue<int> st;
  rep(i, n) if (nyu[i] == 0) st.push(i);
  while (st.size()) {
    int v = st.front();
    st.pop();
    res.pb(v);
    fora(e, g[v]) if (--nyu[e.t] == 0) st.push(e.t);
  }
  return res.size() == n;
}
// 辞書順最小トポロジカルソート
bool topos(vi &res, digraph<int> &g) {
  int n = g.g.size();
  vi nyu(n);
  rep(i, n) for (auto &&e : g[i]) nyu[e.t]++;
  // 小さい順
  priority_queue<int, vector<int>, greater<int>> q;
  rep(i, n) if (nyu[i] == 0) q.push(i);
  while (q.size()) {
    int i = q.top();
    q.pop();
    res.pb(i);
    fora(e, g[i]) if (--nyu[e.t] == 0) q.push(e.t);
  }
  return res.size() == n;
}
vector<string> split(const string a, const char deli) {
  string b = a + deli;
  int l = 0, r = 0, n = b.size();
  vector<string> res;
  rep(i, n) {
    if (b[i] == deli) {
      r = i;
      if (l < r)
        res.push_back(b.substr(l, r - l));
      l = i + 1;
    }
  }
  return res;
}
vector<string> split(const string a, const string deli) {
  string b = a + deli;
  int l = 0, r = 0, n = b.size(), dn = deli.size();
  vector<string> res;
  rep(i, n) {
    if (i + dn <= n && b.substr(i, i + dn) == deli) {
      r = i;
      if (l < r)
        res.push_back(b.substr(l, r - l));
      i += dn - 1;
      l = i + 1;
    }
  }
  return res;
}
void yn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
}
void Yn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
}
void YN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
}
void fyn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
  exit(0);
}
void fYn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
  exit(0);
}
void fYN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
  exit(0);
}
void Possible(bool a) {
  if (a)
    cout << "Possible" << endl;
  else
    cout << "Impossible" << endl;
  exit(0);
}

int n, m, k, d, H, W, x, y, z, q;
int cou;
vi a, b, c;
// vvi (s, 0, 0);
vvc(ba, 0, 0);
vp p;
str s;
digraph<> rg(303);
int dp[303][303][303];
void solve() {
  setmod();
  cin >> n >> m;
  rep(i, m) {
    int f, t, c;
    cin >> f >> t >> c;
    rg.add(t, f, c);
  }
  mint res = 0;
  dp[0][0][0] = 1;

  rep(r, n + 1) {
    rep(g, n + 1) {
      rep(b, n + 1) {

        int i = max(r, g, b);
        vi p = {r, g, b};
        sort(p);
        int &now = dp[r][g][b];
        forg(gi, rg[i]) {
          if (c == 1) {
            // p1 dame p2 ok
            if (!(p[1] < t && t <= p[2])) {
              now = 0;
              brk;
            }
          } else if (c == 2) {
            // p0 dame p1 ok
            if (!(p[0] < t && t <= p[1])) {
              now = 0;
              brk;
            }
          } else {
            // p0 ok
            if (!(t <= p[0])) {
              now = 0;
              brk;
            }
          }
        }
        if (now == 0)
          con;
        if (i == n)
          res += now;
        dp[i + 1][g][b] += now;
        if (dp[i + 1][g][b] >= MOD)
          dp[i + 1][g][b] -= MOD;
        dp[r][i + 1][b] += now;
        if (dp[r][i + 1][b] >= MOD)
          dp[r][i + 1][b] -= MOD;
        dp[r][g][i + 1] += now;
        if (dp[r][g][i + 1] >= MOD)
          dp[r][g][i + 1] -= MOD;
      }
    }
  }
  cout << res << endl;
}
int my(int n, vi &a) { return 0; }
int sister(int n, vi &a) {
  int ret = 0;

  return ret;
}

signed main() {
  solve();

#define arg n, a
#ifdef _DEBUG
  bool bad = 0;
  for (int i = 0, ok = 1; i < k5 && ok; i++) {
    int n = rand(1, 8);
    vi a = ranv(n, 1, 10);
    int myres = my(arg);
    int res = sister(arg);
    ok = myres == res;
    if (!ok) {
      out(arg);
      cerr << "正解 : " << res << endl;
      cerr << "自分  : " << myres << endl;
      bad = 1;
      break;
    }
  }
  if (!bad) {
    //        cout << "完璧 : solveを書き直そう" << endl;
    //        cout << "     : そして、solve()を呼び出すのだ" << endl;
    //        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;
  }
#endif
  return 0;
};


Fixed submission:
// todo 文字数を少なくする
// mintより高速に計算したい時用に modmul() modadd (正の値限定) を作る
// #pragma GCC optimize ("-O3")
#include <bits/stdc++.h>
using namespace std;
//@起動時
struct initon {
  initon() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout.setf(ios::fixed);
    cout.precision(16);
    srand((unsigned)clock() + (unsigned)time(NULL));
  };
} __initon;
// 衝突対策
#define ws ___ws

//@必須構造
struct T {
  int f, s, t;
  T() { f = -1, s = -1, t = -1; }
  T(int f, int s, int t) : f(f), s(s), t(t) {}
  bool operator<(const T &r) const {
    return f != r.f ? f < r.f : s != r.s ? s < r.s : t < r.t;
    // return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 大きい順
  }
  bool operator>(const T &r) const {
    return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t;
    // return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 小さい順
  }
  bool operator==(const T &r) const { return f == r.f && s == r.s && t == r.t; }
  bool operator!=(const T &r) const { return f != r.f || s != r.s || t != r.t; }
  int operator[](int i) {
    assert(i < 3);
    return i == 0 ? f : i == 1 ? s : t;
  }
};
struct F {
  int a, b, c, d;
  F() { a = -1, b = -1, c = -1, d = -1; }
  F(int a, int b, int c, int d) : a(a), b(b), c(c), d(d) {}
  bool operator<(const F &r) const {
    return a != r.a   ? a < r.a
           : b != r.b ? b < r.b
           : c != r.c ? c < r.c
                      : d < r.d;
    //        return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c >
    //        r.c : d > r.d;
  }
  bool operator>(const F &r) const {
    return a != r.a   ? a > r.a
           : b != r.b ? b > r.b
           : c != r.c ? c > r.c
                      : d > r.d;
    //        return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c <
    //        r.c : d < r.d;
  }
  bool operator==(const F &r) const {
    return a == r.a && b == r.b && c == r.c && d == r.d;
  }
  bool operator!=(const F &r) const {
    return a != r.a || b != r.b || c != r.c || d != r.d;
  }
  int operator[](int i) {
    assert(i < 4);
    return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;
  }
};

T mt(int a, int b, int c) { return T(a, b, c); }
//@マクロ省略系 型,構造

#define int long long
#define ll long long
#define double long double
#define ull unsigned long long
using dou = double;
using itn = int;
using str = string;
using bo = bool;
#define au auto
using P = pair<ll, ll>;
using pd = pair<dou, dou>;
#define fi first
#define se second
#define vec vector
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==

#define maxq 1
#define minq -1

#define ZERO(a) memset(a, 0, sizeof(a))
#define MINUS(a) memset(a, 0xff, sizeof(a))
#define MALLOC(type, len) (type *)malloc((len) * sizeof(type))

// マクロ省略系 コンテナ

using vi = vector<int>;
#define _overloadvvi(_1, _2, _3, _4, name, ...) name
#define vvi0() vec<vi>
#define vvi1(a) vec<vi> a
#define vvi2(a, b) vec<vi> a(b)
#define vvi3(a, b, c) vec<vi> a(b, vi(c))
#define vvi4(a, b, c, d) vec<vi> a(b, vi(c, d))
#define vvi(...)                                                               \
  _overloadvvi(__VA_ARGS__, vvi4, vvi3, vvi2, vvi1, vvi0)(__VA_ARGS__)

using vl = vector<ll>;
#define _overloadvvl(_1, _2, _3, _4, name, ...) name
#define vvl1(a) vec<vl> a
#define vvl2(a, b) vec<vl> a(b)
#define vvl3(a, b, c) vec<vl> a(b, vl(c))
#define vvl4(a, b, c, d) vec<vl> a(b, vl(c, d))
#define vvl(...) _overloadvvl(__VA_ARGS__, vvl4, vvl3, vvl2, vvl1)(__VA_ARGS__)

using vb = vector<bool>;
#define _overloadvvb(_1, _2, _3, _4, name, ...) name
#define vvb1(a) vec<vb> a
#define vvb2(a, b) vec<vb> a(b)
#define vvb3(a, b, c) vec<vb> a(b, vb(c))
#define vvb4(a, b, c, d) vec<vb> a(b, vb(c, d))
#define vvb(...) _overloadvvb(__VA_ARGS__, vvb4, vvb3, vvb2, vvb1)(__VA_ARGS__)

using vs = vector<string>;
#define _overloadvvs(_1, _2, _3, _4, name, ...) name
#define vvs1(a) vec<vs> a
#define vvs2(a, b) vec<vs> a(b)
#define vvs3(a, b, c) vec<vs> a(b, vs(c))
#define vvs4(a, b, c, d) vec<vs> a(b, vs(c, d))
#define vvs(...) _overloadvvs(__VA_ARGS__, vvs4, vvs3, vvs2, vvs1)(__VA_ARGS__)

using vd = vector<double>;
#define _overloadvvd(_1, _2, _3, _4, name, ...) name
#define vvd1(a) vec<vd> a
#define vvd2(a, b) vec<vd> a(b)
#define vvd3(a, b, c) vec<vd> a(b, vd(c))
#define vvd4(a, b, c, d) vec<vd> a(b, vd(c, d))
#define vvd(...) _overloadvvd(__VA_ARGS__, vvd4, vvd3, vvd2, vvd1)(__VA_ARGS__)

using vc = vector<char>;
#define _overloadvvc(_1, _2, _3, _4, name, ...) name
#define vvc1(a) vec<vc> a
#define vvc2(a, b) vec<vc> a(b)
#define vvc3(a, b, c) vec<vc> a(b, vc(c))
#define vvc4(a, b, c, d) vec<vc> a(b, vc(c, d))
#define vvc(...) _overloadvvc(__VA_ARGS__, vvc4, vvc3, vvc2, vvc1)(__VA_ARGS__)

using vp = vector<P>;
#define _overloadvvp(_1, _2, _3, _4, name, ...) name
#define vvp1(a) vec<vp> a
#define vvp2(a, b) vec<vp> a(b)
#define vvp3(a, b, c) vec<vp> a(b, vp(c))
#define vvp4(a, b, c, d) vec<vp> a(b, vp(c, d))
#define vvp(...) _overloadvvp(__VA_ARGS__, vvp4, vvp3, vvp2, vvp1)(__VA_ARGS__)

using vt = vector<T>;
#define _overloadvvt(_1, _2, _3, _4, name, ...) name
#define vvt1(a) vec<vt> a
#define vvt2(a, b) vec<vt> a(b)
#define vvt3(a, b, c) vec<vt> a(b, vt(c))
#define vvt4(a, b, c, d) vec<vt> a(b, vt(c, d))
#define vvt(...) _overloadvvt(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1)(__VA_ARGS__)

#define v3i(a, b, c, d) vector<vector<vi>> a(b, vector<vi>(c, vi(d)))
#define v3d(a, b, c, d) vector<vector<vd>> a(b, vector<vd>(c, vd(d)))
#define v3m(a, b, c, d) vector<vector<vm>> a(b, vector<vm>(c, vm(d)))

#define _vvi vector<vi>
#define _vvl vector<vl>
#define _vvb vector<vb>
#define _vvs vector<vs>
#define _vvd vector<vd>
#define _vvc vector<vc>
#define _vvp vector<vp>

template <typename T> vector<T> make_v(size_t a) { return vector<T>(a); }
template <typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {
  return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}
#define vni(name, ...) auto name = make_v<int>(__VA_ARGS__)
#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)
#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)
#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)
#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)
#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)

#define PQ priority_queue<ll, vector<ll>, greater<ll>>
#define tos to_string
using mapi = map<int, int>;
using mapp = map<P, int>;
using mapd = map<dou, int>;
using mapc = map<char, int>;
using maps = map<str, int>;
using seti = set<int>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<int>;
#define bset bitset
#define uset unordered_set
#define mset multiset
#define umap unordered_map
#define umapi unordered_map<int, int>
#define umapp unordered_map<P, int>
#define mmap multimap

// マクロ 繰り返し
#define _overloadrep(_1, _2, _3, _4, name, ...) name
#define _rep(i, n) for (int i = 0, _lim = n; i < _lim; i++)
#define repi(i, m, n) for (int i = m, _lim = n; i < _lim; i++)
#define repadd(i, m, n, ad) for (int i = m, _lim = n; i < _lim; i += ad)
#define rep(...) _overloadrep(__VA_ARGS__, repadd, repi, _rep, )(__VA_ARGS__)
#define _rer(i, n) for (int i = n; i >= 0; i--)
#define reri(i, m, n) for (int i = m, _lim = n; i >= _lim; i--)
#define rerdec(i, m, n, dec) for (int i = m, _lim = n; i >= _lim; i -= dec)
#define rer(...) _overloadrep(__VA_ARGS__, rerdec, reri, _rer, )(__VA_ARGS__)

#define reps2(i, j, n)                                                         \
  for (int i = 0, _lim = n; i < _lim; i++)                                     \
    for (int j = 0; j < _lim; j++)
#define reps3(i, j, k, n)                                                      \
  for (int i = 0, _lim = n; i < _lim; i++)                                     \
    for (int j = 0; j < _lim; j++)                                             \
      for (int k = 0; k < _lim; k++)
#define reps4(i, j, k, l, n)                                                   \
  for (int i = 0, _lim = n; i < _lim; i++)                                     \
    for (int j = 0; j < _lim; j++)                                             \
      for (int k = 0; k < _lim; k++)                                           \
        for (int l = 0; l < _lim; l++)

#define _overloadreps(_1, _2, _3, _4, _5, name, ...) name
#define reps(...)                                                              \
  _overloadreps(__VA_ARGS__, reps4, reps3, reps2, _rep, )(__VA_ARGS__)
#define fora(a, b) for (auto &&a : b)
#define forg(gi, ve)                                                           \
  for (int gi = 0, f, t, c;                                                    \
       gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true);     \
       gi++)
#define fort(gi, ve)                                                           \
  for (int gi = 0, f, t, c;                                                    \
       gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true);     \
       gi++)                                                                   \
    if (t != p)
// #define fort(gi, ve) for (int gi = 0, f, t, c;gi<ve.size()&& (gi+=
// (ve[gi].t==p))< ve.size() && (f = ve[gi].f,t=ve[gi].t, c = ve[gi].c,true);
// gi++)

// マクロ 定数
#define k3 1010
#define k4 10101
#define k5 101010
#define k6 1010101
#define k7 10101010
const int inf = (int)1e9 + 100;
const ll linf = (ll)1e18 + 100;
const char infc = '{';
const string infs = "{";
const double eps = 1e-9;
const double PI = 3.1415926535897932384626433832795029L;
ll ma = numeric_limits<ll>::min();
ll mi = numeric_limits<ll>::max();
const int y4[] = {-1, 1, 0, 0};
const int x4[] = {0, 0, -1, 1};
const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};
const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};

// マクロ省略形 関数等
#define arsz(a) (sizeof(a) / sizeof(a[0]))
#define sz(a) ((int)(a).size())
// #define rs resize
#define mp make_pair
#define pb push_back
#define pf push_front
#define eb emplace_back
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()

constexpr bool ev(int a) { return !(a & 1); }
constexpr bool od(int a) { return (a & 1); }
//@拡張系 こう出来るべきというもの

// 埋め込み 存在を意識せずに機能を増やされているもの
namespace std {
template <> class hash<std::pair<signed, signed>> {
public:
  size_t operator()(const std::pair<signed, signed> &x) const {
    return hash<ll>()(((ll)x.first << 32) + x.second);
  }
};
template <> class hash<std::pair<ll, ll>> {
public:
  // 大きいllが渡されると、<<32でオーバーフローするがとりあえず問題ないと判断
  size_t operator()(const std::pair<ll, ll> &x) const {
    return hash<ll>()(((ll)x.first << 32) + x.second);
  }
};
} // namespace std
istream &operator>>(istream &iss, P &a) {
  iss >> a.first >> a.second;
  return iss;
}
template <typename T> istream &operator>>(istream &iss, vector<T> &vec) {
  for (T &x : vec)
    iss >> x;
  return iss;
}
template <class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {
  os << p.fi << " " << p.se << endl;
  return os;
}
ostream &operator<<(ostream &os, T p) {
  os << p.f << " " << p.s << " " << p.t;
  return os;
}
ostream &operator<<(ostream &os, F p) {
  os << p.a << " " << p.b << " " << p.c << " " << p.d;
  return os;
}
template <typename T> ostream &operator<<(ostream &os, vector<T> &vec) {
  for (int i = 0; i < vec.size(); i++)
    os << vec[i] << (i + 1 == vec.size() ? "" : " ");
  return os;
}
template <typename T> ostream &operator<<(ostream &os, vector<vector<T>> &vec) {
  for (int i = 0; i < vec.size(); i++) {
    for (int j = 0; j < vec[0].size(); j++) {
      os << vec[i][j];
    }
    os << endl;
  }
  return os;
}
template <typename V, typename H> void resize(vector<V> &vec, const H head) {
  vec.resize(head);
}
template <typename V, typename H, typename... T>
void resize(vector<V> &vec, const H &head, const T... tail) {
  vec.resize(head);
  for (auto &v : vec)
    resize(v, tail...);
}
template <typename T, typename _Pr>
bool all_of(const vector<T> &vec, _Pr pred) {
  return std::all_of(vec.begin(), vec.end(), pred);
}
template <typename T, typename _Pr>
bool any_of(const vector<T> &vec, _Pr pred) {
  return std::any_of(vec.begin(), vec.end(), pred);
}
template <typename T, typename _Pr>
bool none_of(const vector<T> &vec, _Pr pred) {
  return std::none_of(vec.begin(), vec.end(), pred);
}
template <typename T, typename _Pr>
const typename vector<T>::const_iterator find_if(const vector<T> &vec,
                                                 _Pr pred) {
  return std::find_if(vec.begin(), vec.end(), pred);
}
template <typename T> bool contains(const vector<T> &vec, const T &val) {
  return std::find(vec.begin(), vec.end(), val) != vec.end();
}
template <typename T, typename _Pr>
bool contains_if(const vector<T> &vec, _Pr pred) {
  return std::find_if(vec.begin(), vec.end(), pred) != vec.end();
}
template <class T, class U> void replace(vector<T> &a, T key, U v) {
  replace(a.begin(), a.end(), key, v);
}
template <class T> void replace(str &a, T key, T v) {
  a.replace(all(a), key, v);
}
// keyと同じかどうか01で置き換える
template <class T, class U> void replace(T &a, U k) {
  rep(i, sz(a)) a[i] = a[i] == k;
}
template <class T, class U> void replace(T &a) { replace(a, '#'); }

template <class T> bool includes(vector<T> &a, vector<T> &b) {
  vi c = a;
  vi d = b;
  sort(all(c));
  sort(all(d));
  return includes(all(c), all(d));
}
template <class T> bool is_permutation(vector<T> &a, vector<T> &b) {
  return is_permutation(all(a), all(b));
}
template <class T> bool next_permutation(vector<T> &a) {
  return next_permutation(all(a));
}

template <class T> T pop(set<T> &set) {
  T res = *set.begin();
  set.erase(set.find(res));
  return res;
}
template <class T> T pop(mset<T> &set) {
  T res = *set.begin();
  set.erase(set.find(res));
  return res;
}
template <class T> T popBack(set<T> &set) {
  T res = *set.rbegin();
  set.erase(set.find(res));
  return res;
}
template <class T> T popBack(mset<T> &set) {
  T res = *set.rbegin();
  set.erase(set.find(res));
  return res;
}
inline void sort(string &a) { sort(a.begin(), a.end()); }
template <class T> inline void sort(vector<T> &a) { sort(a.begin(), a.end()); }
template <class T> inline void sort(vector<T> &a, int len) {
  sort(a.begin(), a.begin() + len);
}
template <class T, class F> inline void sort(vector<T> &a, F f) {
  sort(a.begin(), a.end(), [&](T a, T b) { return f(a) < f(b); });
}
enum ___pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
inline void sort(vector<P> &a, ___pcomparator type) {
  switch (type) {
  case fisi:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });
    break;
  case fisd:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });
    break;
  case fdsi:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });
    break;
  case fdsd:
    sort(all(a),
         [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });
    break;
  case sifi:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });
    break;
  case sifd:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });
    break;
  case sdfi:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });
    break;
  case sdfd:
    sort(all(a),
         [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });
    break;
  }
};
inline void sort(vector<T> &a, ___pcomparator type) {
  switch (type) {
  case fisi:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });
    break;
  case fisd:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });
    break;
  case fdsi:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });
    break;
  case fdsd:
    sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });
    break;
  case sifi:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });
    break;
  case sifd:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });
    break;
  case sdfi:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });
    break;
  case sdfd:
    sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });
    break;
  }
};
template <class T> inline void rsort(vector<T> &a) {
  sort(a.begin(), a.end(), greater<T>());
};
template <class T> inline void rsort(vector<T> &a, int len) {
  sort(a.begin(), a.begin() + len, greater<T>());
};
template <class U, class F> inline void rsort(vector<U> &a, F f) {
  sort(a.begin(), a.end(), [&](U l, U r) { return f(l) > f(r); });
};
template <class U> inline void sortp(vector<U> &a, vector<U> &b) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  sort(c);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
    ;
  }
};
// F = T<T>
// 例えばreturn p.fi + p.se;
template <class U, class F> inline void sortp(vector<U> &a, vector<U> &b, F f) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  sort(c, f);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U, class F>
inline void sortp(vector<U> &a, vector<U> &b, char type) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  sort(c, type);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U> inline void rsortp(vector<U> &a, vector<U> &b) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  rsort(c);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U, class F>
inline void rsortp(vector<U> &a, vector<U> &b, F f) {
  vp c;
  int n = sz(a);
  assert(n == sz(b));
  rep(i, n) c.eb(a[i], b[i]);
  rsort(c, f);
  rep(i, n) {
    a[i] = c[i].first;
    b[i] = c[i].second;
  }
};
template <class U> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c) {
  vt r;
  int n = sz(a);
  assert(n == sz(b));
  assert(n == sz(c));
  rep(i, n) r.eb(a[i], b[i], c[i]);
  sort(r);
  rep(i, n) {
    a[i] = r[i].f;
    b[i] = r[i].s;
    c[i] = r[i].t;
  }
};
template <class U, class F>
inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c, F f) {
  vt r;
  int n = sz(a);
  assert(n == sz(b));
  assert(n == sz(c));
  rep(i, n) r.eb(a[i], b[i], c[i]);
  sort(r, f);
  rep(i, n) {
    a[i] = r[i].f;
    b[i] = r[i].s;
    c[i] = r[i].t;
  }
};
template <class U, class F>
inline void rsortt(vector<U> &a, vector<U> &b, vector<U> &c, F f) {
  vt r;
  int n = sz(a);
  assert(n == sz(b));
  assert(n == sz(c));
  rep(i, n) r.eb(a[i], b[i], c[i]);
  rsort(r, f);
  rep(i, n) {
    a[i] = r[i].f;
    b[i] = r[i].s;
    c[i] = r[i].t;
  }
};
// indexをつけてsortする (second)
template <class T> auto sorti(vector<T> &a) {
  int n = sz(a);
  vp ret(n);
  rep(i, n) ret[i] = P(a[i], i);
  sort(ret);
  return ret;
};
template <class T> auto rsorti(vector<T> &a) {
  int n = sz(a);
  vp ret(n);
  rep(i, n) ret[i] = P(a[i], i);
  rsort(ret);
  return ret;
};
template <class T> inline void sort2(vector<vector<T>> &a) {
  for (int i = 0, n = a.size(); i < n; i++)
    sort(a[i]);
}
template <class T> inline void rsort2(vector<vector<T>> &a) {
  for (int i = 0, n = a.size(); i < n; i++)
    rsort(a[i]);
}
template <typename A, size_t N, typename T>
inline void fill(A (&a)[N], const T &v) {
  rep(i, N) a[i] = v;
}
template <typename A, size_t N, size_t O, typename T>
inline void fill(A (&a)[N][O], const T &v) {
  rep(i, N) rep(j, O) a[i][j] = v;
}
template <typename A, size_t N, size_t O, size_t P, typename T>
inline void fill(A (&a)[N][O][P], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) a[i][j][k] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, typename T>
inline void fill(A (&a)[N][O][P][Q], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) a[i][j][k][l] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          typename T>
inline void fill(A (&a)[N][O][P][Q][R], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) a[i][j][k][l][m] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S, typename T>
inline void fill(A (&a)[N][O][P][Q][R][S], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S)
      a[i][j][k][l][m][n] = v;
}

template <typename A, size_t N> A sum(A (&a)[N]) {
  A res = 0;
  rep(i, N) res += a[i];
  return res;
}
template <typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {
  A res = 0;
  rep(i, N) rep(j, O) res += a[i][j];
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A sum(A (&a)[N][O][P]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) res += a[i][j][k];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A sum(A (&a)[N][O][P][Q]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) res += a[i][j][k][l];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A sum(A (&a)[N][O][P][Q][R]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) res += a[i][j][k][l][m];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A sum(A (&a)[N][O][P][Q][R][S]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S) res +=
      a[i][j][k][l][m][n];
  return res;
}

template <typename V, typename T> void fill(V &xx, const T vall) { xx = vall; }
template <typename V, typename T> void fill(vector<V> &vecc, const T vall) {
  for (auto &&vx : vecc)
    fill(vx, vall);
}

//@汎用便利関数 入力
inline int in() {
  int ret;
  //    scanf("%lld", &ret);
  cin >> ret;
  return ret;
}
inline string sin() {
  string ret;
  cin >> ret;
  return ret;
}
template <class T> inline void in(T &head) { cin >> head; }
template <class T, class... U> inline void in(T &head, U &...tail) {
  cin >> head;
  in(tail...);
}

#define _overloaddin(_1, _2, _3, _4, name, ...) name
#define din1(a)                                                                \
  int a;                                                                       \
  cin >> a
#define din2(a, b)                                                             \
  int a, b;                                                                    \
  cin >> a >> b
#define din3(a, b, c)                                                          \
  int a, b, c;                                                                 \
  cin >> a >> b >> c
#define din4(a, b, c, d)                                                       \
  int a, b, c, d;                                                              \
  cin >> a >> b >> c >> d
#define din(...) _overloaddin(__VA_ARGS__, din4, din3, din2, din1)(__VA_ARGS__)

#define _overloaddind(_1, _2, _3, _4, name, ...) name
#define din1d(a)                                                               \
  din1(a);                                                                     \
  a--
#define din2d(a, b)                                                            \
  din2(a, b);                                                                  \
  a--, b--
#define din3d(a, b, c)                                                         \
  din3(a, b, c);                                                               \
  a--, b--, c--
#define din4d(a, b, c, d)                                                      \
  din4(a, b, c, d);                                                            \
  a--, b--, c--, d--
#define dind(...)                                                              \
  _overloaddind(__VA_ARGS__, din4d, din3d, din2d, din1d)(__VA_ARGS__)

#define _overloadout(_1, _2, _3, _4, name, ...) name
#define out1(a) cout << a << endl
#define out2(a, b) cout << a << " " << b << endl
#define out3(a, b, c) cout << a << " " << b << " " << c << endl
#define out4(a, b, c, d) cout << a << " " << b << " " << c << " " << d << endl
#define out(...) _overloadout(__VA_ARGS__, out4, out3, out2, out1)(__VA_ARGS__)

template <class T> void outl(vector<T> &a) { fora(v, a) cout << v << endl; }
#define na(a, n)                                                               \
  a.resize(n);                                                                 \
  rep(_i, n) cin >> a[_i];
#define nao(a, n)                                                              \
  a.resize(n + 1);                                                             \
  rep(_i, n) cin >> a[_i + 1];
#define nad(a, n)                                                              \
  a.resize(n);                                                                 \
  rep(_i, n) {                                                                 \
    cin >> a[_i];                                                              \
    a[_i]--;                                                                   \
  }
#define na2(a, b, n)                                                           \
  a.resize(n), b.resize(n);                                                    \
  rep(_i, n) cin >> a[_i] >> b[_i];
#define nao2(a, b, n)                                                          \
  a.resize(n + 1), b.resize(n + 1);                                            \
  rep(_i, n) cin >> a[_i + 1] >> b[_i + 1];
#define na2d(a, b, n)                                                          \
  a.resize(n), b.resize(n);                                                    \
  rep(_i, n) {                                                                 \
    cin >> a[_i] >> b[_i];                                                     \
    a[_i]--, b[_i]--;                                                          \
  }
#define na3(a, b, c, n)                                                        \
  a.resize(n), b.resize(n), c.resize(n);                                       \
  rep(_i, n) cin >> a[_i] >> b[_i] >> c[_i];
#define na3d(a, b, c, n)                                                       \
  a.resize(n), b.resize(n), c.resize(n);                                       \
  rep(_i, n) {                                                                 \
    cin >> a[_i] >> b[_i] >> c[_i];                                            \
    a[_i]--, b[_i]--, c[_i]--;                                                 \
  }
#define nt(a, h, w)                                                            \
  resize(a, h, w);                                                             \
  rep(_hi, h) rep(_wi, w) cin >> a[_hi][_wi];
#define ntd(a, h, w)                                                           \
  rs(a, h, w);                                                                 \
  rep(_hi, h) rep(_wi, w) cin >> a[_hi][_wi], a[_hi][_wi]--;
#define ntp(a, h, w)                                                           \
  fill(a, '#');                                                                \
  rep(_hi, 1, h + 1) rep(_wi, 1, w + 1) cin >> a[_hi][_wi];

// デバッグ
#define sp << " " <<

#define debugName(VariableName) #VariableName

#define _deb1(x) debugName(x) << " = " << x
#define _deb2(x, ...) _deb1(x) << ", " << _deb1(__VA_ARGS__)
#define _deb3(x, ...) _deb1(x) << ", " << _deb2(__VA_ARGS__)
#define _deb4(x, ...) _deb1(x) << ", " << _deb3(__VA_ARGS__)
#define _deb5(x, ...) _deb1(x) << ", " << _deb4(__VA_ARGS__)
#define _deb6(x, ...) _deb1(x) << ", " << _deb5(__VA_ARGS__)
#define _deb7(x, ...) _deb1(x) << ", " << _deb6(__VA_ARGS__)
#define _deb8(x, ...) _deb1(x) << ", " << _deb7(__VA_ARGS__)
#define _deb9(x, ...) _deb1(x) << ", " << _deb8(__VA_ARGS__)
#define _deb10(x, ...) _deb1(x) << ", " << _deb9(__VA_ARGS__)

#define _overloadebug(_1, _2, _3, _4, _5, _6, _7, _8, _9, _10, name, ...) name

#ifdef _DEBUG
#define deb(...)                                                               \
  cerr << _overloadebug(__VA_ARGS__, _deb10, _deb9, _deb8, _deb7, _deb6,       \
                        _deb5, _deb4, _deb3, _deb2, _deb1)(__VA_ARGS__)        \
       << endl
#else
#define deb(...) ;
#endif

#define debugline(x)                                                           \
  cerr << x << " "                                                             \
       << "(L:" << __LINE__ << ")" << '\n'

// よく使うクラス、構造体
class UnionFind {
public:
  vi par, rank, sizes;
  int n, trees;
  UnionFind(int n) : n(n), trees(n) {
    par.resize(n), rank.resize(n), sizes.resize(n);
    rep(i, n) par[i] = i, sizes[i] = 1;
  }
  int root(int x) {
    if (par[x] == x)
      return x;
    else
      return par[x] = root(par[x]);
  }
  int find(int x) { return root(x); }
  void unite(int x, int y) {
    x = root(x);
    y = root(y);
    if (x == y)
      return;
    if (rank[x] < rank[y])
      swap(x, y);
    trees--;
    par[y] = x;
    sizes[x] += sizes[y];
    if (rank[x] == rank[y])
      rank[x]++;
  }
  bool same(int x, int y) { return root(x) == root(y); }
  int size(int x) { return sizes[root(x)]; }
  // 順不同 umapなので
  vec<vi> sets() {
    vec<vi> res(trees);
    umap<int, vi> map;
    rep(i, n) map[root(i)].push_back(i);
    int i = 0;
    for (auto &&p : map) {
      int r = p.fi;
      res[i].push_back(r);
      for (auto &&v : p.se) {
        if (r == v)
          continue;
        res[i].push_back(v);
      }
      i++;
    }
    return res;
  }
};

using bint = __int128;
using u32 = unsigned int;
using u64 = unsigned long long;
using u128 = __uint128_t;
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;
}

__int128 toi128(string &s) {
  __int128 ret = 0;
  for (int i = 0; i < s.length(); i++)
    if ('0' <= s[i] && s[i] <= '9')
      ret = 10 * ret + s[i] - '0';
  return ret;
}

template <typename T> T minv(T a, T m);
template <typename T> T minv(T a);

template <typename T> class Modular {
public:
  using Type = typename decay<decltype(T::value)>::type;

  constexpr Modular() : value() {}
  template <typename U> Modular(const U &x) { value = normalize(x); }

  template <typename U> static Type normalize(const U &x) {
    Type v;
    if (-mod() <= x && x < mod())
      v = static_cast<Type>(x);
    else
      v = static_cast<Type>(x % mod());
    if (v < 0)
      v += mod();
    return v;
  }

  const Type &operator()() const { return value; }
  template <typename U> explicit operator U() const {
    return static_cast<U>(value);
  }
  constexpr static Type mod() { return T::value; }

  Modular &operator+=(const Modular &other) {
    if ((value += other.value) >= mod())
      value -= mod();
    return *this;
  }
  Modular &operator-=(const Modular &other) {
    if ((value -= other.value) < 0)
      value += mod();
    return *this;
  }
  template <typename U> Modular &operator+=(const U &other) {
    return *this += Modular(other);
  }
  template <typename U> Modular &operator-=(const U &other) {
    return *this -= Modular(other);
  }
  Modular &operator++() { return *this += 1; }
  Modular &operator--() { return *this -= 1; }
  Modular operator++(signed) {
    Modular result(*this);
    *this += 1;
    return result;
  }
  Modular operator--(signed) {
    Modular result(*this);
    *this -= 1;
    return result;
  }
  Modular operator-() const { return Modular(-value); }

  template <typename U = T>
  typename enable_if<is_same<typename Modular<U>::Type, signed>::value,
                     Modular>::type &
  operator*=(const Modular &rhs) {
#ifdef _WIN32
    uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);
    uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x),
             d, m;
    asm("divl %4; \n\t" : "=a"(d), "=d"(m) : "d"(xh), "a"(xl), "r"(mod()));
    value = m;
#else
    value = normalize(static_cast<int64_t>(value) *
                      static_cast<int64_t>(rhs.value));
#endif
    return *this;
  }
  template <typename U = T>
  typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value,
                     Modular>::type &
  operator*=(const Modular &rhs) {
    int64_t q =
        static_cast<int64_t>(static_cast<double>(value) * rhs.value / mod());
    value = normalize(value * rhs.value - q * mod());
    return *this;
  }
  template <typename U = T>
  typename enable_if<!is_integral<typename Modular<U>::Type>::value,
                     Modular>::type &
  operator*=(const Modular &rhs) {
    value = normalize(value * rhs.value);
    return *this;
  }
  Modular &operator/=(const Modular &other) {
    return *this *= Modular(minv(other.value));
  }
  template <typename U>
  friend bool operator==(const Modular<U> &lhs, const Modular<U> &rhs);
  template <typename U>
  friend bool operator<(const Modular<U> &lhs, const Modular<U> &rhs);
  template <typename U>
  friend std::istream &operator>>(std::istream &stream, Modular<U> &number);
  operator int() { return normalize(value); }

private:
  Type value;
};

template <typename T>
bool operator==(const Modular<T> &lhs, const Modular<T> &rhs) {
  return lhs.value == rhs.value;
}
template <typename T, typename U>
bool operator==(const Modular<T> &lhs, U rhs) {
  return lhs == Modular<T>(rhs);
}
template <typename T, typename U>
bool operator==(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) == rhs;
}

template <typename T>
bool operator!=(const Modular<T> &lhs, const Modular<T> &rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U>
bool operator!=(const Modular<T> &lhs, U rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U>
bool operator!=(U lhs, const Modular<T> &rhs) {
  return !(lhs == rhs);
}

template <typename T>
bool operator<(const Modular<T> &lhs, const Modular<T> &rhs) {
  return lhs.value < rhs.value;
}

template <typename T>
Modular<T> operator+(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) += rhs;
}
template <typename T, typename U>
Modular<T> operator+(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) += rhs;
}
template <typename T, typename U>
Modular<T> operator+(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) += rhs;
}

template <typename T>
Modular<T> operator-(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) -= rhs;
}
template <typename T, typename U>
Modular<T> operator-(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) -= rhs;
}
template <typename T, typename U>
Modular<T> operator-(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) -= rhs;
}

template <typename T>
Modular<T> operator*(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) *= rhs;
}
template <typename T, typename U>
Modular<T> operator*(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) *= rhs;
}
template <typename T, typename U>
Modular<T> operator*(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) *= rhs;
}

template <typename T>
Modular<T> operator/(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) /= rhs;
}
template <typename T, typename U>
Modular<T> operator/(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) /= rhs;
}
template <typename T, typename U>
Modular<T> operator/(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) /= rhs;
}

constexpr signed MOD = 1e9 + 7; // MOD
using mint = Modular<std::integral_constant<decay<decltype(MOD)>::type, MOD>>;

constexpr int mint_len = 1400001;
vi fac, finv, inv, p2;
void setmod() {
  fac.resize(mint_len);
  finv.resize(mint_len);
  inv.resize(mint_len);
  p2.resize(mint_len);
  inv[1] = fac[0] = finv[0] = p2[0] = 1;
  for (int i = 2; i < mint_len; ++i)
    inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
  for (int i = 1; i < mint_len; ++i)
    fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD,
    p2[i] = p2[i - 1] * 2 % MOD;
}
#define smod setmod
inline mint com(int n, int r) {
  if (r < 0 || r > n)
    return 0;
  return mint(finv[r] * fac[n] % MOD * finv[n - r]);
}
inline mint pom(int n, int r) {
  //    if (!sz(fac)) com(0, -1);
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - 1]);
}
inline mint npr(int n, int r) {
  //    if (!sz(fac)) com(0, -1);
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - r]);
}
inline int nprin(int n, int r) {
  //    if (!sz(fac)) com(0, -1);
  if (r < 0 || r > n)
    return 0;
  return fac[n] * finv[n - r] % MOD;
}

inline int icom(int n, int r) {
  const int NUM_ = 1400001;
  static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];
  if (fac[0] == 0) {
    inv[1] = fac[0] = finv[0] = 1;
    for (int i = 2; i <= NUM_; ++i)
      inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
    for (int i = 1; i <= NUM_; ++i)
      fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
  }
  if (r < 0 || r > n)
    return 0;
  return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;
}
#define ncr com
#define ncri icom
inline mint nhr(int n, int r) { return com(n + r - 1, r); }
inline mint hom(int n, int r) { return com(n + r - 1, r); }
inline int nhri(int n, int r) { return icom(n + r - 1, r); }

template <typename T> inline T minv(T a, T m) {
  T u = 0, v = 1;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}
template <typename T> inline T minv(T a) {
  if (a < mint_len)
    return inv[a];
  T u = 0, v = 1;
  T m = MOD;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}

template <typename T, typename U>
inline Modular<T> mpow(const Modular<T> &a, const U &b) {
  assert(b >= 0);
  int x = a(), res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= MOD;
    (x *= x) %= MOD;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U, typename V>
inline mint mpow(const T a, const U b, const V m = MOD) {
  assert(b >= 0);
  int x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= m;
    (x *= x) %= m;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U> inline mint mpow(const T a, const U b) {
  assert(b >= 0);
  int x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= MOD;
    (x *= x) %= MOD;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U, typename V>
inline int mpowi(const T &a, const U &b, const V &m = MOD) {
  assert(b >= 0);
  int x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      (res *= x) %= m;
    (x *= x) %= m;
    p >>= 1;
  }
  return res;
}
template <typename T> inline string to_string(const Modular<T> &number) {
  return to_string(number());
}

template <typename T>
std::ostream &operator<<(std::ostream &stream, const Modular<T> &number) {
  return stream << number();
}

template <typename T>
std::istream &operator>>(std::istream &stream, Modular<T> &number) {
  typename common_type<typename Modular<T>::Type, int64_t>::type x;
  stream >> x;
  number.value = Modular<T>::normalize(x);
  return stream;
}

using PM = pair<mint, mint>;
using vm = vector<mint>;
using mapm = map<int, mint>;
using umapm = umap<int, mint>;
#define _overloadvvm(_1, _2, _3, _4, name, ...) name
#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)

#define vvm1(a) vec<vm> a
#define vvm2(a, b) vec<vm> a(b)
#define vvm3(a, b, c) vec<vm> a(b, vm(c))
#define vvm4(a, b, c, d) vec<vm> a(b, vm(c, d))
#define vvm(...) _overloadvvm(__VA_ARGS__, vvm4, vvm3, vvm2, vvm1)(__VA_ARGS__)

// エラー
void ole() {
#ifdef _DEBUG
  debugline("ole");
  exit(0);
#endif
  string a = "a";
  rep(i, 30) a += a;
  rep(i, 1 << 17) cout << a << endl;
  cout << "OLE 出力長制限超過" << endl;
  exit(0);
}
void tle() {
  while (inf)
    cout << inf << endl;
}

// 便利関数

// テスト用
char ranc() { return (char)('a' + rand() % 26); }
int rand(int min, int max) {
  assert(min <= max);
  if (min >= 0 && max >= 0) {
    return rand() % (max + 1 - min) + min;
  } else if (max < 0) {
    return -rand(-max, -min);
  } else {
    //+
    if (rand() % 2) {
      return rand(0, max);
      //-
    } else {
      return -rand(0, -min);
    }
  }
}
vi ranv(int n, int min, int max) {
  vi v(n);
  rep(i, n) v[i] = rand(min, max);
  return v;
}
// 単調増加
vi ranvi(int n, int min, int max) {
  vi v(n);
  bool bad = 1;
  while (bad) {
    bad = 0;
    v.resize(n);
    rep(i, n) {
      if (i && min > max - v[i - 1]) {
        bad = 1;
        break;
      }
      if (i)
        v[i] = v[i - 1] + rand(min, max - v[i - 1]);
      else
        v[i] = rand(min, max);
    }
  }
  return v;
}
void ranvlr(int n, int min, int max, vi &l, vi &r) {
  l.resize(n);
  r.resize(n);
  rep(i, n) {
    l[i] = rand(min, max);
    r[i] = l[i] + rand(0, max - l[i]);
  }
}
// 便利 汎用
vp run_length(vi &a) {
  vp ret;
  ret.eb(a[0], 1);
  rep(i, 1, sz(a)) {
    if (ret.back().fi == a[i]) {
      ret.back().se++;
    } else {
      ret.eb(a[i], 1);
    }
  }
  return ret;
}
vector<pair<char, int>> run_length(string &a) {
  vector<pair<char, int>> ret;
  ret.eb(a[0], 1);
  rep(i, 1, sz(a)) {
    if (ret.back().fi == a[i]) {
      ret.back().se++;
    } else {
      ret.eb(a[i], 1);
    }
  }
  return ret;
}
template <class F> inline int mgr(int ok, int ng, F f) {
#define _mgrbody                                                               \
  int mid = (ok + ng) / 2;                                                     \
  if (f(mid))                                                                  \
    ok = mid;                                                                  \
  else                                                                         \
    ng = mid;
  if (ok < ng)
    while (ng - ok > 1) {
      _mgrbody
    }
  else
    while (ok - ng > 1) {
      _mgrbody
    }
  return ok;
}

template <class F> inline int mgr(int ok, int ng, int second, F f) {
#define _mgrbody2                                                              \
  int mid = (ok + ng) / 2;                                                     \
  if (f(mid, second))                                                          \
    ok = mid;                                                                  \
  else                                                                         \
    ng = mid;
  if (ok < ng)
    while (ng - ok > 1) {
      _mgrbody2
    }
  else
    while (ok - ng > 1) {
      _mgrbody2
    }
  return ok;
}
// strを整数として比較
string smax(str &a, str b) {
  if (sz(a) < sz(b)) {
    return b;
  } else if (sz(a) > sz(b)) {
    return a;
  } else {
    rep(i, sz(a)) {
      if (a[i] < b[i]) {
        return b;
      } else if (a[i] > b[i])
        return a;
    }
  }
  return a;
}
// strを整数として比較
string smin(str &a, str b) {
  if (sz(a) < sz(b)) {
    return a;
  } else if (sz(a) > sz(b)) {
    return b;
  } else {
    rep(i, sz(a)) {
      if (a[i] < b[i]) {
        return a;
      } else if (a[i] > b[i])
        return b;
    }
  }
  return a;
}
template <typename V, typename T> int find(vector<V> &a, const T key) {
  rep(i, sz(a)) if (a[i] == key) return i;
  return -1;
}
template <typename V, typename T> P find(vector<vector<V>> &a, const T key) {
  rep(i, sz(a)) rep(j, sz(a[0])) if (a[i][j] == key) return mp(i, j);
  return mp(-1, -1);
}
template <typename V, typename U>
T find(vector<vector<vector<V>>> &a, const U key) {
  rep(i, sz(a)) rep(j, sz(a[0]))
      rep(k, sz(a[0][0])) if (a[i][j][k] == key) return mt(i, j, k);
  return mt(-1, -1, -1);
}

template <typename V, typename T> int count(V &a, const T k) { return a == k; }
template <typename V, typename T> int count(vector<V> &a, const T k) {
  int ret = 0;
  fora(v, a) ret += count(v, k);
  return ret;
}
template <typename T> int count(char &a, const T k) { return a == k; }
template <typename T> int count(str &a, const T k) {
  int ret = 0;
  rep(i, sz(a)) ret += count(a[i], k);
  return ret;
}
template <typename V> int count_odd(V &a) { return a % 2; }
template <typename V> int count_odd(vector<V> &a) {
  int ret = 0;
  fora(v, a) ret += count_odd(v);
  return ret;
}
template <typename V> int count_even(V &a) { return a % 2 == 0; }
template <typename V> int count_even(vector<V> &a) {
  int ret = 0;
  fora(v, a) ret += count_even(v);
  return ret;
}
// algorythm
void iota(vector<int> &ve, int s, int n) {
  ve.resize(n);
  iota(all(ve), s);
}
vi iota(int s, int n) {
  vi ve(n);
  iota(all(ve), s);
  return ve;
}

inline ll rev(ll a) {
  ll res = 0;
  while (a) {
    res *= 10;
    res += a % 10;
    a /= 10;
  }
  return res;
}
template <class T> inline void rev(vector<T> &a) { reverse(all(a)); }
void inline rev(string &a) { reverse(all(a)); }

int get(int a, int keta) { return (a / (int)pow(10, keta)) % 10; }
int keta(int v) {
  int cou = 0;
  while (v) {
    cou++, v %= 10;
  }
  return cou;
}
int dsum(int v) {
  int ret = 0;
  for (; v; v /= 10)
    ret += v % 10;
  return ret;
}
inline int sumd(int v) { return dsum(v); }
// 変換系
template <class T, class U> inline vector<T> keys(vector<pair<T, U>> a) {
  vector<T> res;
  for (auto &&k : a)
    res.pb(k.fi);
  return res;
}
template <class T, class U> inline vector<U> keys(map<T, U> a) {
  vector<U> res;
  for (auto &&k : a)
    res.pb(k.fi);
  return res;
}

template <class T, class U> inline vector<U> keys(umap<T, U> a) {
  vector<U> res;
  for (auto &&k : a)
    res.pb(k.fi);
  return res;
}

template <class T, class U> vector<U> inline values(vector<pair<T, U>> a) {
  vector<U> res;
  for (auto &&k : a)
    res.pb(k.se);
  return res;
}

template <class T, class U> inline vector<T> values(map<T, U> a) {
  vector<T> res;
  for (auto &&k : a)
    res.pb(k.se);
  return res;
}

template <class T, class U> inline vector<T> values(umap<T, U> a) {
  vector<T> res;
  for (auto &&k : a)
    res.pb(k.se);
  return res;
}

template <class T, class U> inline bool chma(T &a, const U &b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}
template <class U> inline bool chma(const U &b) { return chma(ma, b); }
template <class T, class U> inline bool chmi(T &a, const U &b) {
  if (b < a) {
    a = b;
    return true;
  }
  return false;
}
template <class U> inline bool chmi(const U &b) { return chmi(mi, b); }
template <class T> inline T min(T a, signed b) { return a < b ? a : b; }
template <class T> inline T max(T a, signed b) { return a < b ? b : a; }
template <class T> inline T min(T a, T b, T c) {
  return a >= b ? b >= c ? c : b : a >= c ? c : a;
}
template <class T> inline T max(T a, T b, T c) {
  return a <= b ? b <= c ? c : b : a <= c ? c : a;
}
template <class T> inline T min(vector<T> a) { return *min_element(all(a)); }
template <class T> inline T mini(vector<T> a) {
  return min_element(all(a)) - a.begin();
}
template <class T> inline T min(vector<T> a, int n) {
  return *min_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> inline T min(vector<T> a, int s, int n) {
  return *min_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <class T> inline T max(vector<T> a) { return *max_element(all(a)); }
template <class T> inline T maxi(vector<T> a) {
  return max_element(all(a)) - a.begin();
}
template <class T> inline T max(vector<T> a, int n) {
  return *max_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> inline T max(vector<T> a, int s, int n) {
  return *max_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <typename A, size_t N> inline A max(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = max(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> inline A max(A (&a)[N][O]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P>
inline A max(A (&a)[N][O][P]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
inline A max(A (&a)[N][O][P][Q], const T &v) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
inline A max(A (&a)[N][O][P][Q][R]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
inline A max(A (&a)[N][O][P][Q][R][S]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N> inline A min(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = min(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> inline A min(A (&a)[N][O]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P>
inline A min(A (&a)[N][O][P]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
inline A min(A (&a)[N][O][P][Q], const T &v) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
inline A min(A (&a)[N][O][P][Q][R]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
inline A min(A (&a)[N][O][P][Q][R][S]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}

template <class T> inline T sum(T &v) { return v; }
template <class T> inline auto sum(vector<T> &v) {
  auto ret = sum(v[0]);
  rep(i, 1, sz(v)) ret += sum(v[i]);
  return ret;
}
template <class T, class U, class... V>
inline auto sum(vector<T> &v, U head, V... tail) {
  auto ret = sum(v[0], tail...);
  rep(i, 1, min(sz(v), head)) ret += sum(v[i], tail...);
  return ret;
}

inline P sump(vp &v, int len = -1) {
  if (len == -1)
    len = v.size();
  P res = {0, 0};
  chmi(len, v.size());
  rep(i, len) {
    res.fi += v[i].fi;
    res.se += v[i].se;
  }
  return res;
}

/// 要素が0の時、返り値は0か1か
template <class T> inline T mul(vector<T> &v, int len = -1) {
  if (len == -1)
    len = v.size();
  T res = 1;
  chmi(len, v.size());
  rep(i, len) res *= v[i];
  return res;
}
inline void clear(PQ &q) {
  while (q.size())
    q.pop();
}
template <class T> inline void clear(queue<T> &q) {
  while (q.size())
    q.pop();
}
template <class T> inline T *negarr(int size) {
  T *body = (T *)malloc((size * 2 + 1) * sizeof(T));
  return body + size;
}
template <class T> inline T *negarr2(int h, int w) {
  double **dummy1 = new double *[2 * h + 1];
  double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];
  dummy1[0] = dummy2 + w;
  for (int i = 1; i <= 2 * h + 1; i++) {
    dummy1[i] = dummy1[i - 1] + 2 * w + 1;
  }
  double **a = dummy1 + h;
  return a;
}
// imoは0-indexed
// ruiは1-indexed
template <class T> inline vector<T> imo(vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}
// 漸化的に最小を持つ
template <class T> inline vector<T> imi(vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) chmi(ret[i + 1], ret[i]);
  return ret;
}
template <class T> struct ruiC {
  const vector<T> rui;
  ruiC(vector<T> &ru) : rui(ru) {}
  inline T operator()(int l, int r) {
    assert(l <= r);
    return rui[r] - rui[l];
  }
  inline T operator[](int i) { return rui[i]; }
};

template <class T> struct rruic {
  const T *rrui;
  rruic(T *ru) : rrui(ru) {}
  // n-1から-1へ
  inline T operator()(int l, int r) {
    assert(l >= r);
    return rrui[r] - rrui[l];
  }
  inline T operator[](int i) { return rrui[i]; }
};
template <class T> inline vector<T> ruiv(vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + a[i];
  return ret;
}
template <class T> inline ruiC<T> ruic(vector<T> &a) {
  vector<T> ret = ruiv(a);
  return ruiC<T>(ret);
}
// kと同じものの数
template <class T, class U> inline vi ruiv(T &a, U k) {
  vi ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + (a[i] == k);
  return ret;
}
template <class T, class U> inline ruiC<int> ruic(T &a, U k) {
  vi ret = ruiv(a, k);
  return ruiC<int>(ret);
}

// xor
template <class T> inline vector<T> ruix(vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] ^ a[i];
  return ret;
}
template <class T> inline vector<T> ruim(vector<T> &a) {
  vector<T> res(a.size() + 1, 1);
  rep(i, a.size()) res[i + 1] = res[i] * a[i];
  return res;
}
// 漸化的に最小を1indexで持つ
template <class T> inline vector<T> ruimi(vector<T> &a) {
  int n = sz(a);
  vector<T> ret(n + 1);
  rep(i, 1, n) {
    ret[i] = a[i - 1];
    chmi(ret[i + 1], ret[i]);
  }
  return ret;
}
// template<class T> T *rrui(vector<T> &a) {
// 右から左にかけての半開区間 (-1 n-1]
template <class T> rruic<T> rrui(vector<T> &a) {
  int len = a.size();
  T *body = (T *)malloc((len + 1) * sizeof(T));
  T *res = body + 1;
  rer(i, len - 1) res[i - 1] = res[i] + a[i];
  return rruic<T>(res);
}
// 掛け算
template <class T> T *rruim(vector<T> &a) {
  int len = a.size();
  T *body = (T *)malloc((len + 1) * sizeof(T));
  T *res = body + 1;
  res[len - 1] = 1;
  rer(i, len - 1) res[i - 1] = res[i] * a[i];
  return res;
}
template <class T, class U> void inc(T &a, U v = 1) { a += v; }
template <class T, class U> void inc(vector<T> &a, U v = 1) {
  for (auto &u : a)
    inc(u, v);
}
template <class T> void inc(vector<T> &a) {
  for (auto &u : a)
    inc(u, 1);
}
template <class T, class U> void plus(T &a, U v = 1) { a += v; }
template <class T, class U> void plus(vector<T> &a, U v = 1) {
  for (auto &u : a)
    inc(u, v);
}
template <class T> void plus(vector<T> &a) {
  for (auto &u : a)
    inc(u, 1);
}
template <class T, class U> void dec(T &a, U v = 1) { a -= v; }
template <class T, class U> void dec(vector<T> &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class T> void dec(vector<T> &a) {
  for (auto &u : a)
    dec(u, 1);
}
template <class T, class U> void minu(T &a, U v = 1) { a -= v; }
template <class T, class U> void minu(vector<T> &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class T> void minu(vector<T> &a) {
  for (auto &u : a)
    dec(u, 1);
}
template <class U> void minu(string &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
void minu(string &a) {
  for (auto &u : a)
    dec(u, 1);
}
inline bool inside(int h, int w, int H, int W) {
  return h >= 0 && w >= 0 && h < H && w < W;
}
inline bool inside(int l, int v, int r) { return l <= v && v < r; }
#define ins inside
ll u(ll a) { return a < 0 ? 0 : a; }
template <class T> vector<T> u(const vector<T> &a) {
  vector<T> ret = a;
  fora(v, ret) v = u(v);
  return ret;
}
#define MIN(a) numeric_limits<a>::min()
#define MAX(a) numeric_limits<a>::max()
ll goldd(ll left, ll right, function<ll(ll)> calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl < fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  ll minScore = MAX(ll);
  ll resIndex = left;
  for (ll i = left; i < right + 1; i++) {
    ll score = calc(i);
    if (minScore > score) {
      minScore = score;
      resIndex = i;
    }
  }
  return resIndex;
}
ll goldt(ll left, ll right, function<ll(ll)> calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl > fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  if (left > right) {
    ll l = left;
    left = right;
    right = l;
  }
  ll maxScore = MIN(ll);
  ll resIndex = left;
  for (ll i = left; i < right + 1; i++) {
    ll score = calc(i);
    if (maxScore < score) {
      maxScore = score;
      resIndex = i;
    }
  }
  return resIndex;
}
template <class T> T min(vector<vector<T>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, *min_element(all(a[i])));
  return res;
}
template <class T> T max(vector<vector<T>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, *max_element(all(a[i])));
  return res;
}

constexpr bool bget(ll m, int keta) {
  assert(keta < 64);
  return (m >> keta) & 1;
}
int bget(ll m, int keta, int sinsuu) {
  m /= (ll)pow(sinsuu, keta);
  return m % sinsuu;
}
inline ll bit(int n) { return (1LL << (n)); }
inline ll bit(int n, int sinsuu) { return (ll)pow(sinsuu, n); }
// int bcou(ll m) { return __builtin_popcount(m & 0xFFFFFFFF) +
// __builtin_popcount(m >> 32); }
#define bcou __builtin_popcountll
// 最下位ビット
int lbit(int n) { return n & -n; }
// 最上位ビット
int hbit(int n) {
  n |= (n >> 1);
  n |= (n >> 2);
  n |= (n >> 4);
  n |= (n >> 8);
  n |= (n >> 16);
  n |= (n >> 32);
  return n - (n >> 1);
}
// 初期化は0を渡す
ll nextComb(ll &mask, int n, int r) {
  if (!mask)
    return mask = (1LL << r) - 1;
  ll x = mask & -mask; // 最下位の1
  ll y = mask + x;     // 連続した下の1を繰り上がらせる
  ll res = ((mask & ~y) / x >> 1) | y;
  if (bget(res, n))
    return mask = 0;
  else
    return mask = res;
}

// n桁以下でビットがr個立っているもののvectorを返す
vl bitCombList(int n, int r) {
  vl res;
  int m = 0;
  while (nextComb(m, n, r)) {
    res.pb(m);
  }
  return res;
}
// 大文字小文字を区別する
int altoiaZ(char c) {
  if ('A' <= c && c <= 'Z')
    return c - 'A';
  return c - 'a' + 26;
}
char itoalaZ(int i) {
  if (i < 26)
    return 'A' + i;
  return 'a' + i - 26;
}
// aもAも0を返す 基本小文字
int altoi(char c) {
  if ('A' <= c && c <= 'Z')
    return c - 'A';
  return c - 'a';
}
char itoal(int i) { return 'a' + i; }
int ctoi(char c) { return c - '0'; }
char itoc(int i) { return i + '0'; }
int vtoi(vi &v) {
  int res = 0;
  if (sz(v) > 18) {
    debugline("vtoi");
    deb(sz(v));
    ole();
  }
  rep(i, sz(v)) {
    res *= 10;
    res += v[i];
  }
  return res;
}
vi itov(int i) {
  vi res;
  while (i) {
    res.pb(i % 10);
    i /= 10;
  }
  rev(res);
  return res;
}
vi stov(string &a) {
  int n = sz(a);
  vi ret(n);
  rep(i, n) { ret[i] = a[i] - '0'; }
  return ret;
}
vector<vector<int>> ctoi(vector<vector<char>> s, char c) {
  int n = sz(s), m = sz(s[0]);
  vector<vector<int>> res(n, vector<int>(m));
  rep(i, n) rep(j, m) res[i][j] = s[i][j] == c;
  return res;
}
#define unique(v) v.erase(unique(v.begin(), v.end()), v.end());

void compress(vi &a) {
  vi b;
  int len = a.size();
  for (int i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (int i = 0; i < len; ++i) {
    a[i] = lower_bound(all(b), a[i]) - b.begin();
  }
}

void compress(int a[], int len) {
  vi b;
  for (int i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (int i = 0; i < len; ++i) {
    a[i] = lower_bound(all(b), a[i]) - b.begin();
  }
}
// 要素が見つからなかったときに困る
#define binarySearch(a, v) (binary_search(all(a), v))
#define lowerIndex(a, v) (lower_bound(all(a), v) - a.begin())
#define lowerBound(a, v) (*lower_bound(all(a), v))
#define upperIndex(a, v) (upper_bound(all(a), v) - a.begin())
#define upperBound(a, v) (*upper_bound(all(a), v))
#define ans(a)                                                                 \
  cout << a << endl;                                                           \
  continue;
#define poll(a)                                                                \
  q.front();                                                                   \
  q.pop()
#define dpoll(a)                                                               \
  q.front();                                                                   \
  q.pop_front()
#define pollLast(a)                                                            \
  q.back();                                                                    \
  q.pop_back()
#define pollBack(a)                                                            \
  q.back();                                                                    \
  q.pop_back()
template <class T> inline void fin(T s) { cout << s << endl, exit(0); }

// 便利 数学 math
int mod(int a, int m) { return (a % m + m) % m; }
int pow(int a) { return a * a; };
ll fact(int v) { return v <= 1 ? 1 : v * fact(v - 1); }
ll comi(int n, int r) {
  assert(n < 100);
  static vvi(pas, 100, 100);
  if (pas[0][0])
    return pas[n][r];
  pas[0][0] = 1;
  rep(i, 1, 100) {
    pas[i][0] = 1;
    rep(j, 1, i + 1) pas[i][j] = pas[i - 1][j - 1] + pas[i - 1][j];
  }
  return pas[n][r];
}
double comd(int n, int r) {
  assert(n < 2020);
  static vvd(comb, 2020, 2020);
  if (comb[0][0] == 0) {
    comb[0][0] = 1;
    rep(i, 2000) {
      comb[i + 1][0] = 1;
      rep(j, 1, i + 2) { comb[i + 1][j] = comb[i][j] + comb[i][j - 1]; }
    }
  }
  return comb[n][r];
}
int gcd(int a, int b) {
  while (b)
    a %= b, swap(a, b);
  return abs(a);
}
int gcd(vi b) {
  ll res = b[0];
  rep(i, 1, sz(b)) res = gcd(b[i], res);
  return res;
}
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
ll ceil(ll a, ll b) {
  if (b == 0) {
    debugline("ceil");
    deb(a, b);
    ole();
    return -1;
  } else if (a < 0) {
    return 0;
  } else {
    return (a + b - 1) / b;
  }
}
// v * v >= aとなる最小のvを返す
ll sqrt(ll a) {
  if (a < 0) {
    debugline("sqrt");
    deb(a);
    ole();
  }
  ll res = (ll)std::sqrt(a);
  while (res * res < a)
    res++;
  return res;
}
double log(double e, double x) { return log(x) / log(e); }
ll sig(ll t) { return (1 + t) * t / 2; }
ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; }

namespace pnumber_tools {
#define Eratosthenes30_2                                                       \
  class Eratosthenes {                                                         \
  public:                                                                      \
    virtual void initialize() {}                                               \
    virtual void generate(const int64_t x) = 0;                                \
    virtual void generate(const int64_t from, const int64_t to) = 0;           \
    virtual int64_t count() = 0;                                               \
    virtual bool prime(int x) = 0;                                             \
    virtual int32_t version() = 0;                                             \
    inline uint64_t PopCnt(uint64_t);                                          \
    inline int BitToIndex(uint8_t);                                            \
  };                                                                           \
  uint64_t Eratosthenes::PopCnt(uint64_t bits) {                               \
    bits = (bits & 0x5555555555555555ULL) +                                    \
           ((bits >> 1) & 0x5555555555555555ULL);                              \
    bits = (bits & 0x3333333333333333ULL) +                                    \
           ((bits >> 2) & 0x3333333333333333ULL);                              \
    bits = (bits & 0x0f0f0f0f0f0f0f0fULL) +                                    \
           ((bits >> 4) & 0x0f0f0f0f0f0f0f0fULL);                              \
    return (bits * 0x0101010101010101ULL) >> (64 - 8);                         \
  }                                                                            \
  int Eratosthenes::BitToIndex(uint8_t b) {                                    \
    switch (b) {                                                               \
    case 1 << 0:                                                               \
      return 0;                                                                \
    case 1 << 1:                                                               \
      return 1;                                                                \
    case 1 << 2:                                                               \
      return 2;                                                                \
    case 1 << 3:                                                               \
      return 3;                                                                \
    case 1 << 4:                                                               \
      return 4;                                                                \
    case 1 << 5:                                                               \
      return 5;                                                                \
    case 1 << 6:                                                               \
      return 6;                                                                \
    case 1 << 7:                                                               \
      return 7;                                                                \
    }                                                                          \
    return -1;                                                                 \
  }                                                                            \
  namespace {                                                                  \
  const uint64_t kSegmentSize = 1000000;                                       \
  const int32_t kMod30[] = {1, 7, 11, 13, 17, 19, 23, 29};                     \
  const int C1[] = {6, 4, 2, 4, 2, 4, 6, 2};                                   \
  const int C0[][8] = {                                                        \
      {0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 0, 1, 1, 1, 1},                      \
      {2, 2, 0, 2, 0, 2, 2, 1}, {3, 1, 1, 2, 1, 1, 3, 1},                      \
      {3, 3, 1, 2, 1, 3, 3, 1}, {4, 2, 2, 2, 2, 2, 4, 1},                      \
      {5, 3, 1, 4, 1, 3, 5, 1}, {6, 4, 2, 4, 2, 4, 6, 1},                      \
  };                                                                           \
  const uint8_t kMask[][8] = {                                                 \
      {0xfe, 0xfd, 0xfb, 0xf7, 0xef, 0xdf, 0xbf, 0x7f},                        \
      {0xfd, 0xdf, 0xef, 0xfe, 0x7f, 0xf7, 0xfb, 0xbf},                        \
      {0xfb, 0xef, 0xfe, 0xbf, 0xfd, 0x7f, 0xf7, 0xdf},                        \
      {0xf7, 0xfe, 0xbf, 0xdf, 0xfb, 0xfd, 0x7f, 0xef},                        \
      {0xef, 0x7f, 0xfd, 0xfb, 0xdf, 0xbf, 0xfe, 0xf7},                        \
      {0xdf, 0xf7, 0x7f, 0xfd, 0xbf, 0xfe, 0xef, 0xfb},                        \
      {0xbf, 0xfb, 0xf7, 0x7f, 0xfe, 0xef, 0xdf, 0xfd},                        \
      {0x7f, 0xbf, 0xdf, 0xef, 0xf7, 0xfb, 0xfd, 0xfe},                        \
  };                                                                           \
  const int64_t kSievedPrimes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23};            \
  const int64_t kInitialSize = 7 * 11 * 13 * 17 * 19 * 23;                     \
  class Eratosthenes2 final : public Eratosthenes {                            \
  public:                                                                      \
    void generate(const int64_t x) override;                                   \
    void generate(const int64_t, const int64_t) override { assert(false); }    \
    bool prime(int x) override;                                                \
    int64_t count() override;                                                  \
    int32_t version() override { return 2; }                                   \
    std::vector<uint8_t> flags_;                                               \
  };                                                                           \
  void Eratosthenes2::generate(const int64_t x) {                              \
    flags_.clear();                                                            \
    if (x > 10000000000)                                                       \
      return;                                                                  \
    int64_t size = x / 30 + (x % 30 != 0);                                     \
    flags_.resize(size, 0xff);                                                 \
    if (int r = x % 30) {                                                      \
      if (r <= 1)                                                              \
        flags_[size - 1] = 0x0;                                                \
      else if (r <= 7)                                                         \
        flags_[size - 1] = 0x1;                                                \
      else if (r <= 11)                                                        \
        flags_[size - 1] = 0x3;                                                \
      else if (r <= 13)                                                        \
        flags_[size - 1] = 0x7;                                                \
      else if (r <= 17)                                                        \
        flags_[size - 1] = 0xf;                                                \
      else if (r <= 19)                                                        \
        flags_[size - 1] = 0x1f;                                               \
      else if (r <= 23)                                                        \
        flags_[size - 1] = 0x3f;                                               \
      else if (r <= 29)                                                        \
        flags_[size - 1] = 0x7f;                                               \
    }                                                                          \
    flags_[0] = 0xfe;                                                          \
    const uint64_t sqrt_x = std::ceil(std::sqrt(x) + 0.1);                     \
    const uint64_t sqrt_xi = sqrt_x / 30 + 1;                                  \
    for (uint64_t i = 0; i < sqrt_xi; ++i) {                                   \
      for (uint8_t flags = flags_[i]; flags; flags &= flags - 1) {             \
        uint8_t lsb = flags & (-flags);                                        \
        int ibit = BitToIndex(lsb);                                            \
        const int32_t m = kMod30[ibit];                                        \
        int32_t pm = 30 * i + 2 * m;                                           \
        for (uint64_t j = i * pm + (m * m) / 30, k = ibit; j < flags_.size();  \
             j += i * C1[k] + C0[ibit][k], k = (k + 1) & 7) {                  \
          flags_[j] &= kMask[ibit][k];                                         \
        }                                                                      \
      }                                                                        \
    }                                                                          \
  }                                                                            \
  int64_t Eratosthenes2::count() {                                             \
    if (flags_.empty())                                                        \
      return -1;                                                               \
    int64_t ret = 3;                                                           \
    for (uint8_t f : flags_) {                                                 \
      ret += PopCnt(f);                                                        \
    }                                                                          \
    return ret;                                                                \
  }                                                                            \
  inline bool Eratosthenes2::prime(int x) {                                    \
    static unordered_map<int, int> imap = {                                    \
        {1, 0}, {7, 1}, {11, 2}, {13, 3}, {17, 4}, {19, 5}, {23, 6}, {29, 7}}; \
    if (x == 2 || x == 3 || x == 5)                                            \
      return true;                                                             \
    else if (!(x & 1) || x % 3 == 0 || x % 5 == 0)                             \
      return false;                                                            \
    return (flags_[x / 30] >> imap[x % 30]) & 1;                               \
  }                                                                            \
  }
Eratosthenes30_2 Eratosthenes2 era;
int prime_len = 3e6;
vector<int> primes;
void setprime(int size = 3e6) {
  prime_len = size;
  era.generate(size);
  primes = {2, 3, 5};
  int nums[129] = {};
  nums[1] = 1;
  nums[2] = 7;
  nums[4] = 11;
  nums[8] = 13;
  nums[16] = 17;
  nums[32] = 19;
  nums[64] = 23;
  nums[128] = 29;
  // p = 30i + j
  int j = 0;
  int sflim = era.flags_.size();
  int i30 = 0;
  for (int i = 0; i < sflim; i++, i30 += 30) {
    uint8_t m = era.flags_[i];
    while (m) {
      j = m & -m;
      primes.push_back(i30 + nums[j]);
      m ^= j;
    }
  }
}
using u32 = unsigned int; // longlongになっていると思うが放置していいと思う
using u64 = unsigned long long;
using u128 = __uint128_t;

// O(k log(n)^3)
// 32bit -> 270回程度
// 64bit -> 1260回程度
// 64bit以下で精度100%
template <class BinOp>
bool miller_rabin(const uint64_t &n, const uint64_t *witness, BinOp modmul) {
  if (n == 2)
    return true;
  if (n < 2 || n % 2 == 0)
    return false;
  const int64_t m = n - 1, d = m / (m & -m);
  auto modpow = [&](int64_t a, int64_t b) {
    int64_t res = 1;
    for (; b; b /= 2) {
      if (b & 1)
        res = modmul(res, a);
      a = modmul(a, a);
    }
    return res;
  };
  auto suspect = [&](uint64_t a, uint64_t t) {
    a = modpow(a, t);
    while (t != n - 1 && a != 1 && a != n - 1) {
      a = modmul(a, a);
      t = modmul(t, 2);
    }
    return a == n - 1 || t % 2 == 1;
  };
  for (const uint64_t *w = witness; *w; w++) {
    if (*w % n != 0 && !suspect(*w, d))
      return false;
  }
  return true;
}

// miller_rabin & エラトステネス
bool prime(const int &n) {
  assert(n < 9223372036854775808ULL);
  if (!sz(primes))
    setprime();
  // エラトステネスの篩 偶数省略ver
  //[i] = i*2+1
  if (n < prime_len)
    return era.prime(n);
  if (n < 4294967296) {
    rep(i, 30) if (!(n % primes[i])) return false;
    constexpr uint64_t witness[] = {2, 7, 61, 0};
    auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t { return a * b % n; };
    return miller_rabin(n, witness, modmul);
  } else {
    rep(i, 50) if (!(n % primes[i])) return false;
    constexpr uint64_t witness[] = {2,      325,     9375,       28178,
                                    450775, 9780504, 1795265022, 0};
    auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t {
      return (uint64_t)((__uint128_t)a * b % n);
    };
    return miller_rabin(n, witness, modmul);
  }
}
vi divisors(int v) {
  vi res;
  double lim = std::sqrt(v);
  for (int i = 1; i <= lim; ++i) {
    if (v % i == 0) {
      res.pb(i);
      if (i != v / i)
        res.pb(v / i);
    }
  }
  return res;
}
vp factorization(int v) {
  assert(v);
  int tv = v;
  vp res;
  if (!sz(primes))
    setprime();
  for (auto &&p : primes) {
    if (v % p == 0)
      res.eb(p, 0);
    while (v % p == 0) {
      v /= p;
      res.back().second++;
    }
    if (v == 1 || p * p > tv)
      break;
  }
  if (v > 1)
    res.eb(v, 1);
  return res;
}

// 注意 gcdを取る間隔 i%10 が場合によってはtleを引き起こす
// うまくいかなかったら%1で動かすように変更した
vp rho(int v) {
  assert(v);
  if (v == 1)
    return vp();
  vi ret;
  if (!sz(primes))
    setprime();
  int lim = 300; // 64bit
  if (v < 4294967296)
    lim = 80; // 32bit
  else if (v < 281474976710656LL)
    lim = 120; // 48bit
  for (int i = 0; i < lim; i += 5) {
    // v<=1の判定回数を減らせる分高速化できる事を期待している
    while (v % primes[i] == 0)
      ret.pb(primes[i]), v /= primes[i];
    while (v % primes[i + 1] == 0)
      ret.pb(primes[i + 1]), v /= primes[i + 1];
    while (v % primes[i + 2] == 0)
      ret.pb(primes[i + 2]), v /= primes[i + 2];
    while (v % primes[i + 3] == 0)
      ret.pb(primes[i + 3]), v /= primes[i + 3];
    while (v % primes[i + 4] == 0)
      ret.pb(primes[i + 4]), v /= primes[i + 4];
    if (v <= 1) {
      sort(ret); // 試し割りだけで終わった時
      return run_length(ret);
    }
  }
  int lower_prime_bound = primes[lim - 1] * primes[lim - 1];

  // 試し割りするか
  mt19937 mr;
  mr.seed(clock());
  int c;
  auto fac = [&](int v, const int loop = 10) {
    //        if (v == 4)return 2ll; //試し割りで消えた
    c = mr() % v;
    int x = mr() % v;
    int y = x;
    int d = 1;
    int lim = std::sqrt(v) + 2;
    // brentによる改良 差分を一操作で求められる
    for (int i = 1, mul = 1, next_2pow = 2; d == 1 && i <= lim; i++) {
      if (i == next_2pow) {
        y = x;
        next_2pow <<= 1;
      }
      x = ((u128)x * x + c) % v;
      mul = ((u128)mul * abs(x - y)) %
            v; // ユークリッドの互除法と同じ原理で因数は保たれる
      if (i % loop == 0) { // 10とかなんとか iによっては死ぬ
        d = gcd(mul, v);
        mul = 1;
      }
    }
    if (d == v)
      return -1ll;
    return d;
  };

  queue<int> q;
  q.push(v);
  while (!q.empty()) {
    int v = q.front();
    q.pop();
    if (prime(v)) {
      ret.pb(v);
      continue;
    }
    int d = -1;
    // 基本loop回数10でやり、駄目なら1で
    for (int i = 0; d == -1 && i < 5; i++)
      d = fac(v, 10);
    if (d == -1)
      while (d == -1)
        d = fac(v, 1);
    int d2 = v / d;
    rep(_, 2) {
      if (d <= lower_prime_bound)
        ret.pb(d);
      else
        q.push(d);
      d = d2;
    }
  }
  sort(ret);
  return run_length(ret);
}
} // namespace pnumber_tools
using namespace pnumber_tools;
// 幾何 Pをcomplexとして扱う
template <class T, class U> bool eq(T a, U b) { return fabs(a - b) < eps; }
dou atan2(pd a) { return atan2(a.se, a.fi); }
dou angle(pd f, pd t) { return atan2(t.se - f.se, t.fi - f.fi); }
dou distance(pd a, pd b) { return hypot(a.fi - b.fi, a.se - b.se); }
// bを中心とするabcのtheta aからcにかけて時計回り
dou angle(pd a, pd b, pd c) {
  dou ax = a.fi - b.fi;
  dou ay = a.se - b.se;
  dou cx = c.fi - b.fi;
  dou cy = c.se - b.se;
  double ret = atan2(cy, cx) - atan2(ay, ax);
  if (ret < 0)
    ret += 2 * PI;
  return ret;
}

dou dot(pd a, pd b) { return a.fi * b.fi + a.se + b.se; }
dou cro(pd a, pd b) { return a.fi * b.se - a.se + b.fi; }

template <class T> struct edge {
  int f, t;
  T c;
  //    int id;
  //    int type;
  edge(int f, int t, T c = 1) : f(f), t(t), c(c) {}
  bool operator<(const edge &b) const { return c < b.c; }
  bool operator>(const edge &b) const { return c > b.c; }
};
template <class T> ostream &operator<<(ostream &os, edge<T> &e) {
  os << e.f << " " << e.t << " " << e.c;
  return os;
}
template <typename T> class graph {
protected:
  vector<bool> _used;

public:
  vector<vector<edge<T>>> g;
  vector<edge<T>> edges;
  int n;
  graph(int n) : n(n) { g.resize(n), _used.resize(n); }
  void clear() { g.clear(), edges.clear(); }
  void resize(int n) {
    this->n = n;
    g.resize(n);
    _used.resize(n);
  }
  int size() { return g.size(); }
  vector<edge<T>> &operator[](int i) { return g[i]; }
  virtual void add(int f, int t, T c) = 0;
};
template <typename T = ll> class digraph : public graph<T> {
public:
  using graph<T>::g;
  using graph<T>::n;
  using graph<T>::edges;
  using graph<T>::_used;
  int eid = 0;

  digraph(int n) : graph<T>(n) {}
  void add(int f, int t, T c = 1) {
    g[f].emplace_back(f, t, c);
    //        edges.emplace_back(f, t, c);
  }
  //    bool used(edge<T> &e) { return _used[e.id]; }
  //    bool used(int id) { return _used[id]; }
  //    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }
  //    void del(int id) { _used[id] = _used[id ^ 1] = 1; }
  void set_edges() {
    if (sz(edges))
      return;
    rep(i, n) fora(e, g[i]) edges.pb(e);
  }
};
template <class T = int> class undigraph : public graph<T> {
public:
  using graph<T>::g;
  using graph<T>::n;
  using graph<T>::edges;
  using graph<T>::_used;
  int eid = 0;
  undigraph(int n) : graph<T>(n) {}
  void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
    if (!(0 <= f && f < n && 0 <= t && t < n)) {
      debugline("undigraph add");
      deb(f, t, c, id, ty);
      ole();
    }
    if (id == -1)
      id = eid, eid += 2;
    g[f].emplace_back(f, t, c, id, ty);
    g[t].emplace_back(t, f, c, id + 1, ty);
    edges.emplace_back(f, t, c, id, ty);
    edges.emplace_back(t, f, c, id + 1, ty);
  }
  void add(edge<T> &e) {
    int f = e.f, t = e.t, ty = e.type;
    T c = e.c;
    add(f, t, c, ty);
  }
  bool used(edge<T> &e) { return _used[e.id]; }
  bool used(int id) { return _used[id]; }
  void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }
  void del(int id) { _used[id] = _used[id ^ 1] = 1; }
  void set_edges() {
    if (sz(edges))
      return;
    rep(i, n) fora(e, g[i]) edges.pb(e);
  }
};
template <typename T> struct radixheap {
  vector<pair<u64, T>> v[65];
  u64 size, last;
  radixheap() : size(0), last(0) {}
  bool empty() const { return size == 0; }
  inline int getbit(int a) { return a ? 64 - __builtin_clzll(a) : 0; }
  void push(u64 key, const T &value) {
    ++size;
    v[getbit(key ^ last)].emplace_back(key, value);
  }
  pair<u64, T> pop() {
    if (v[0].empty()) {
      int idx = 1;
      while (v[idx].empty())
        ++idx;
      last = min_element(begin(v[idx]), end(v[idx]))->first;
      for (auto &p : v[idx])
        v[getbit(p.first ^ last)].emplace_back(p);
      v[idx].clear();
    }
    --size;
    auto ret = v[0].back();
    v[0].pop_back();
    return ret;
  }
};
vi dijkstra2(const graph<int> &g, int s, int cant_arrive = -1) {
  if (!(0 <= s && s < g.n)) {
    debugline("dijkstra");
    deb(s, g.n);
    ole();
  }
  int initValue = MAX(int);
  vi dis(g.n, initValue);
  radixheap<int> q;
  dis[s] = 0;
  q.push(0, s);
  while (!q.empty()) {
    int nowc, i;
    tie(nowc, i) = q.pop();
    if (dis[i] != nowc)
      continue;
    for (auto &&e : g.g[i]) {
      int to = e.t;
      int c = nowc + e.c;
      if (dis[to] > c) {
        dis[to] = c;
        q.push(dis[to], to);
      }
    }
  }
  // 基本、たどり着かないなら-1
  if (cant_arrive == -1)
    for (auto &&d : dis)
      if (d == initValue)
        d = -1;
  return dis;
}

template <class T>
vector<T> dijkstra(const graph<T> &g, int s, int cant_arrive = -1) {
  if (!(0 <= s && s < g.n)) {
    debugline("dijkstra");
    deb(s, g.n);
    ole();
  }
  T initValue = MAX(T);
  vector<T> dis(g.n, initValue);
  priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>> q;
  dis[s] = 0;
  q.emplace(0, s);
  while (q.size()) {
    T nowc = q.top().fi;
    int i = q.top().se;
    q.pop();
    if (dis[i] != nowc)
      continue;
    for (auto &&e : g.g[i]) {
      int to = e.t;
      T c = nowc + e.c;
      if (dis[to] > c) {
        dis[to] = c;
        q.emplace(dis[to], to);
      }
    }
  }
  // 基本、たどり着かないなら-1
  if (cant_arrive == -1)
    for (auto &&d : dis)
      if (d == initValue)
        d = -1;
  return dis;
}
template <class T>
vector<vector<T>> warshall(const graph<T> &g, int cant_arrive = -1) {
  int n = g.n;
  vector<vector<T>> dis(n, vector<T>(n, linf));
  rep(i, n) fora(e, g.g[i]) chmi(dis[e.f][e.t], e.c);
  rep(i, n) dis[i][i] = 0;
  rep(k, n) rep(i, n) rep(j, n) chmi(dis[i][j], dis[i][k] + dis[k][j]);
  // 基本、たどり着かないなら-1
  if (cant_arrive == -1)
    rep(i, n) rep(j, n) if (dis[i][j] == linf) dis[i][j] = -1;
  return dis;
}

template <class T = int> class tree : public undigraph<T> {
public:
  using undigraph<T>::g;
  using undigraph<T>::n;
  using undigraph<T>::edges;
  using undigraph<T>::_used;
  vi disv;

private:
  bool never = 1;
  int root = -1;
  vi dep;            // 到着した順の深さ
  vi vis;            // 到着した順
  vi first;          // 初めてiに到着した時
  vector<int> bestt; // タイミングを持つ
  int sn = 0;        // seg木の底辺の長さ euler tourの長さ
  void built() {
    never = 0;
    n = g.size();
    // euler tour
    disv.resize(n);
    first.resize(n);
    dep.resize(n * 2);
    vis.resize(n * 2);
    function<void(int, int, int, int &)> dfs = [&](int i, int p, int d,
                                                   int &k) {
      first[i] = k;
      vis[k] = i;
      dep[k++] = d;
      forg(gi, g[i]) {
        if (t == p)
          continue;
        disv[t] = disv[i] + c;
        dfs(t, i, d + 1, k);
        vis[k] = i;
        dep[k++] = d;
      }
    };
    int k = 0;
    dfs(root, -1, 0, k);
    // rmq
    sn = 1;
    while (sn < n * 2)
      sn <<= 1;

    bestt.resize((sn << 1) - 1);
    rep(i, k) bestt[i + sn - 1] = i; // 時間を持つ
    rer(i, sn - 2) {
      int lt = (i << 1) + 1, rt = (i << 1) + 2;
      if (dep[bestt[lt]] < dep[bestt[rt]]) {
        bestt[i] = bestt[lt];
      } else {
        bestt[i] = bestt[rt];
      }
    }
  }
  // 最小の深さを持つタイミングが知りたい
  inline int lca_rmq(int l, int r) {
    int midv = inf;
    int midi = -1;
    l += sn - 1;
    r += sn - 1;
    while (l < r) {
      // 右なら、次に右上へ移動してしまうので
      if (!(l & 1)) {
        if (chmi(midv, dep[bestt[l]])) {
          midi = bestt[l];
        }
      }
      // 右なら、左を取らないと機会を逃す
      if (!(r & 1)) {
        if (chmi(midv, dep[bestt[r - 1]])) {
          midi = bestt[r - 1];
        }
      }
      l >>= 1;
      r = (r - 1) >> 1;
    }
    return vis[midi];
  }

public:
  tree(int n, int root = 0) : undigraph<T>(n), root(root) {}
  bool leaf(int v) { return sz(g[v]) == 1 && v != root; }
  int lca(int u, int v) {
    if (never)
      built();
    if (first[u] > first[v])
      swap(u, v);
    return lca_rmq(first[u], first[v] + 1);
  }
  int dis(int u, int v) {
    if (never)
      built();
    if (first[u] > first[v])
      swap(u, v);
    int p = lca_rmq(first[u], first[v] + 1);
    return disv[u] + disv[v] - disv[p] * 2;
  }
};
// 辺によりメモリを大量消費
//  よってedgesを消している
// 頂点10^6でメモリを190MB(制限の8割)使う

// 軽量化のため、辺を持たないbig gridクラスがあってもいいかもしれない
//
template <class T = int> class grid_k6 : public undigraph<T> {
public:
  using undigraph<T>::g;
  using undigraph<T>::n;
  using undigraph<T>::edges;
  using undigraph<T>::_used;
  int H, W;
  int eid = 0;
  void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
    if (!(0 <= f && f < n && 0 <= t && t < n)) {
      debugline("grid_k6 add");
      deb(f, t, c, id, ty);
      ole();
    }
    g[f].emplace_back(f, t, c, eid++, ty);
    g[t].emplace_back(t, f, c, eid++, ty);
  }
  int getid(int h, int w) {
    if (!inside(h, w, H, W))
      return -1;
    return W * h + w;
  }
  P get2(int id) { return mp(id / W, id % W); }
  P operator()(int id) { return get2(id); }
  int operator()(int h, int w) { return getid(h, w); }
  grid_k6(int H, int W) : H(H), W(W), undigraph<T>(H * W) {
    rep(h, H) {
      rep(w, W) {
        int f = getid(h, w);
        if (w + 1 < W)
          add(f, getid(h, w + 1));
        if (h + 1 < H)
          add(f, getid(h + 1, w));
      }
    }
  }
  grid_k6(_vvc ba, char wall = '#')
      : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])) {
    rep(h, H) {
      rep(w, W) {
        if (ba[h][w] == wall)
          con;
        int f = getid(h, w);
        if (w + 1 < W && ba[h][w + 1] != wall) {
          add(f, getid(h, w + 1));
        }
        if (h + 1 < H && ba[h + 1][w] != wall) {
          add(f, getid(h + 1, w));
        }
      }
    }
  }
  void add(int fh, int fw, int th, int tw) {
    add(getid(fh, fw), getid(th, tw));
  }
  void set_edges() { rep(i, n) fora(e, g[i]) edges.pb(e); }
};
// 左上から右下に移動できる
template <class T = int> class digrid_k6 : public digraph<T> {
public:
  using digraph<T>::g;
  using digraph<T>::n;
  using digraph<T>::edges;
  using digraph<T>::_used;
  int H, W;
  int eid = 0;
  void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
    if (!(0 <= f && f < n && 0 <= t && t < n)) {
      debugline("digrid_k6 add");
      deb(f, t, c, id, ty);
      ole();
    }
    g[f].emplace_back(f, t, c, eid++, ty);
  }
  int getid(int h, int w) {
    if (!inside(h, w, H, W))
      return -1;
    return W * h + w;
  }
  P get2(int id) { return mp(id / W, id % W); }
  P operator()(int id) { return get2(id); }
  int operator()(int h, int w) { return getid(h, w); }
  digrid_k6(int H, int W) : H(H), W(W), digraph<T>(H * W) {
    rep(h, H) {
      rep(w, W) {
        int f = getid(h, w);
        if (w + 1 < W)
          add(f, getid(h, w + 1));
        if (h + 1 < H)
          add(f, getid(h + 1, w));
      }
    }
  }
  digrid_k6(_vvc ba, char wall = '#')
      : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])) {
    rep(h, H) {
      rep(w, W) {
        if (ba[h][w] == wall)
          con;
        int f = getid(h, w);
        if (w + 1 < W && ba[h][w + 1] != wall) {
          add(f, getid(h, w + 1));
        }
        if (h + 1 < H && ba[h + 1][w] != wall) {
          add(f, getid(h + 1, w));
        }
      }
    }
  }
  void add(int fh, int fw, int th, int tw) {
    add(getid(fh, fw), getid(th, tw));
  }
  void set_edges() { rep(i, n) fora(e, g[i]) edges.pb(e); }
};
template <class T> bool nibu(const graph<T> &g) {
  int size = 0;
  rep(i, g.n) size += sz(g.g[i]);
  if (size == 0)
    return true;

  UnionFind uf(g.n * 2);
  rep(i, g.n) fora(e, g.g[i]) uf.unite(e.f, e.t + g.n),
      uf.unite(e.f + g.n, e.t);
  rep(i, g.n) if (uf.same(i, i + g.n)) return 0;
  return 1;
}
// 二部グラフを色分けした際の頂点数を返す
template <class T> vp nibug(graph<T> &g) {
  vp cg;
  if (!nibu(g)) {
    debugline("nibu");
    ole();
  }
  int _n = g.size();
  vb _was(_n);
  queue<P> q;
  rep(i, _n) {
    if (_was[i])
      continue;
    q.push(mp(i, 1));
    _was[i] = 1;
    int red = 0;
    int coun = 0;
    while (q.size()) {
      int now = q.front().fi;
      int col = q.front().se;
      red += col;
      coun++;
      q.pop();
      forg(gi, g[now]) {
        if (_was[t])
          continue;
        q.push(mp(t, col ^ 1));
        _was[t] = 1;
      }
    }
    cg.push_back(mp(red, coun - red));
  }
  return cg;
}
template <class T> ostream &operator<<(ostream &os, graph<T> &g) {
  rep(i, sz(g)) {
    forg(gi, g[i]) { os << f << " " << t << " " << c << endl; }
  }
  return os;
}

// 機能拡張
vp vtop(vi &a, vi &b) {
  vp res(sz(a));
  rep(i, sz(a)) res[i] = mp(a[i], b[i]);
  return res;
}
void ptov(vp &p, vi &a, vi &b) {
  a.resize(sz(p));
  b.resize(sz(p));
  rep(i, sz(p)) a[i] = p[i].fi, b[i] = p[i].se;
}
template <typename _CharT, typename _Traits, typename _Alloc>
basic_string<_CharT, _Traits, _Alloc>
operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {
  basic_string<_CharT, _Traits, _Alloc> __str(__lhs);
  __str.append(to_string(__rv));
  return __str;
}
template <typename _CharT, typename _Traits, typename _Alloc>
void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {
  __lhs += to_string(__rv);
}
template <typename _CharT, typename _Traits, typename _Alloc>
basic_string<_CharT, _Traits, _Alloc>
operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs,
          const signed __rv) {
  basic_string<_CharT, _Traits, _Alloc> __str(__lhs);
  __str.append(to_string(__rv));
  return __str;
}
template <typename _CharT, typename _Traits, typename _Alloc>
void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs,
                const signed __rv) {
  __lhs += to_string(__rv);
}
template <class T, class U> void operator+=(queue<T> &a, U v) { a.push(v); }
template <class T, class U> void operator+=(deque<T> &a, U v) {
  a.push_back(v);
}
template <class T, class U>
priority_queue<T, vector<T>, greater<T>> &
operator+=(priority_queue<T, vector<T>, greater<T>> &a, U v) {
  a.push(v);
  return a;
}
template <class T, class U>
priority_queue<T> &operator+=(priority_queue<T> &a, U v) {
  a.push(v);
  return a;
}
template <class T, class U> set<T> &operator+=(set<T> &a, U v) {
  a.insert(v);
  return a;
}
template <class T, class U>
set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {
  a.insert(v);
  return a;
}
template <class T, class U> vector<T> &operator+=(vector<T> &a, U v) {
  a.pb(v);
  return a;
}
template <class T, class U> vector<T> operator+(const vector<T> &a, U v) {
  vector<T> ret = a;
  ret += v;
  return ret;
}
template <class T, class U> vector<T> operator+(U v, const vector<T> &a) {
  vector<T> ret = a;
  ret.insert(ret.begin(), v);
  return ret;
}
template <class T> vector<T> operator+(vector<T> &a, vector<T> &b) {
  vector<T> ret;
  ret = a;
  fora(v, b) ret += v;
  return ret;
}
template <class T> vector<T> &operator+=(vector<T> &a, vector<T> &b) {
  fora(v, b) a += v;
  return a;
}
template <class T, class U>
vector<T> &operator+=(vector<T> &a, initializer_list<U> v) {
  for (auto &&va : v)
    a.pb(va);
  return a;
}
template <class T> vector<T> &operator-=(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-=");
    deb(a);
    deb(b);
    ole();
  }
  rep(i, sz(a)) a[i] -= b[i];
  return a;
}
template <class T> vector<T> operator-(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-");
    deb(a);
    deb(b);
    ole();
  }
  vector<T> res(sz(a));
  rep(i, sz(a)) res[i] = a[i] - b[i];
  return res;
}

template <typename T> void remove(vector<T> &v, unsigned int i) {
  v.erase(v.begin() + i);
}
template <typename T>
void remove(vector<T> &v, unsigned int s, unsigned int e) {
  v.erase(v.begin() + s, v.begin() + e);
}
template <typename T>
void removen(vector<T> &v, unsigned int s, unsigned int n) {
  v.erase(v.begin() + s, v.begin() + s + n);
}
template <typename T> void erase(vector<T> &v, unsigned int i) {
  v.erase(v.begin() + i);
}
template <typename T> void erase(vector<T> &v, unsigned int s, unsigned int e) {
  v.erase(v.begin() + s, v.begin() + e);
}
template <typename T>
void erasen(vector<T> &v, unsigned int s, unsigned int n) {
  v.erase(v.begin() + s, v.begin() + s + n);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned int i, U t) {
  v.insert(v.begin() + i, t);
}
template <typename T, typename U> void push_front(vector<T> &v, U t) {
  v.insert(v.begin(), t);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned int i, vector<T> list) {
  for (auto &&va : list)
    v.insert(v.begin() + i++, va);
}
template <typename T, typename U>
void insert(vector<T> &v, initializer_list<U> list) {
  for (auto &&va : list)
    v.pb(va);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned int i, initializer_list<U> list) {
  for (auto &&va : list)
    v.insert(v.begin() + i++, va);
}
template <typename T> void insert(set<T> &v, vector<T> list) {
  for (auto &&va : list)
    v.insert(va);
}
template <typename T> void insert(set<T> &v, initializer_list<T> list) {
  for (auto &&va : list)
    v.insert(va);
}

// 閉路がなければtrue
bool topo(vi &res, digraph<int> &g) {
  int n = g.g.size();
  vi nyu(n);
  rep(i, n) for (auto &&e : g[i]) nyu[e.t]++;
  queue<int> st;
  rep(i, n) if (nyu[i] == 0) st.push(i);
  while (st.size()) {
    int v = st.front();
    st.pop();
    res.pb(v);
    fora(e, g[v]) if (--nyu[e.t] == 0) st.push(e.t);
  }
  return res.size() == n;
}
// 辞書順最小トポロジカルソート
bool topos(vi &res, digraph<int> &g) {
  int n = g.g.size();
  vi nyu(n);
  rep(i, n) for (auto &&e : g[i]) nyu[e.t]++;
  // 小さい順
  priority_queue<int, vector<int>, greater<int>> q;
  rep(i, n) if (nyu[i] == 0) q.push(i);
  while (q.size()) {
    int i = q.top();
    q.pop();
    res.pb(i);
    fora(e, g[i]) if (--nyu[e.t] == 0) q.push(e.t);
  }
  return res.size() == n;
}
vector<string> split(const string a, const char deli) {
  string b = a + deli;
  int l = 0, r = 0, n = b.size();
  vector<string> res;
  rep(i, n) {
    if (b[i] == deli) {
      r = i;
      if (l < r)
        res.push_back(b.substr(l, r - l));
      l = i + 1;
    }
  }
  return res;
}
vector<string> split(const string a, const string deli) {
  string b = a + deli;
  int l = 0, r = 0, n = b.size(), dn = deli.size();
  vector<string> res;
  rep(i, n) {
    if (i + dn <= n && b.substr(i, i + dn) == deli) {
      r = i;
      if (l < r)
        res.push_back(b.substr(l, r - l));
      i += dn - 1;
      l = i + 1;
    }
  }
  return res;
}
void yn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
}
void Yn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
}
void YN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
}
void fyn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
  exit(0);
}
void fYn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
  exit(0);
}
void fYN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
  exit(0);
}
void Possible(bool a) {
  if (a)
    cout << "Possible" << endl;
  else
    cout << "Impossible" << endl;
  exit(0);
}

int n, m, k, d, H, W, x, y, z, q;
int cou;
vi a, b, c;
// vvi (s, 0, 0);
vvc(ba, 0, 0);
vp p;
str s;
digraph<> rg(303);
int dp[303][303][303];
void solve() {
  //    setmod();
  cin >> n >> m;
  rep(i, m) {
    int f, t, c;
    cin >> f >> t >> c;
    rg.add(t, f, c);
  }
  mint res = 0;
  dp[0][0][0] = 1;

  rep(r, n + 1) {
    rep(g, n + 1) {
      rep(b, n + 1) {

        int i = max(r, g, b);
        vi p = {r, g, b};
        sort(p);
        int &now = dp[r][g][b];
        forg(gi, rg[i]) {
          if (c == 1) {
            // p1 dame p2 ok
            if (!(p[1] < t && t <= p[2])) {
              now = 0;
              brk;
            }
          } else if (c == 2) {
            // p0 dame p1 ok
            if (!(p[0] < t && t <= p[1])) {
              now = 0;
              brk;
            }
          } else {
            // p0 ok
            if (!(t <= p[0])) {
              now = 0;
              brk;
            }
          }
        }
        if (now == 0)
          con;
        if (i == n)
          res += now;
        dp[i + 1][g][b] += now;
        if (dp[i + 1][g][b] >= MOD)
          dp[i + 1][g][b] -= MOD;
        dp[r][i + 1][b] += now;
        if (dp[r][i + 1][b] >= MOD)
          dp[r][i + 1][b] -= MOD;
        dp[r][g][i + 1] += now;
        if (dp[r][g][i + 1] >= MOD)
          dp[r][g][i + 1] -= MOD;
      }
    }
  }
  cout << res << endl;
}
int my(int n, vi &a) { return 0; }
int sister(int n, vi &a) {
  int ret = 0;

  return ret;
}

signed main() {
  solve();

#define arg n, a
#ifdef _DEBUG
  bool bad = 0;
  for (int i = 0, ok = 1; i < k5 && ok; i++) {
    int n = rand(1, 8);
    vi a = ranv(n, 1, 10);
    int myres = my(arg);
    int res = sister(arg);
    ok = myres == res;
    if (!ok) {
      out(arg);
      cerr << "正解 : " << res << endl;
      cerr << "自分  : " << myres << endl;
      bad = 1;
      break;
    }
  }
  if (!bad) {
    //        cout << "完璧 : solveを書き直そう" << endl;
    //        cout << "     : そして、solve()を呼び出すのだ" << endl;
    //        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;
  }
#endif
  return 0;
};