#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/template.hpp"

#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/alias.hpp"

#include <boost/multiprecision/cpp_int.hpp>

#include <bits/stdc++.h>
using namespace std;
// using namespace std::views;
// using namespace boost::multiprecision;

// --- 型エイリアス ---
using ll = long long;
using ull = unsigned long long;
template <typename T> using vec = vector<T>;
template <typename T> using vvec = vector<vector<T>>;
template <typename T> using p_queue = priority_queue<T>;
template <typename T> using rp_queue = priority_queue<T, vec<T>, greater<T>>;
using bint = boost::multiprecision::cpp_int;

// --- 黒魔術 ---
#define int ll

// --- 制御マクロ ---
#define rep(i, n) for (ll i = 0; i < n; ++i)
#define all(v) begin(v), end(v)
// #define BIT(n) (1LL << (n))
#define MAX(type) numeric_limits<type>::max()
#define MIN(type) numeric_limits<type>::min()
#define yes cout << "Yes" << endl
#define no cout << "No" << endl
#define pb push_back
#define mp make_pair
#define fir first
#define sec second

// --- 定数 ---
constexpr ll INF = 1LL << 60;

inline signed bit_width(ll x) { return bit_width((ull)x); }
inline ull bit_ceil(ll x) { return bit_ceil((ull)x); }
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/concepts.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/concepts.hpp"

template <class T>
concept addable = requires(T a, T b) {
  { a + b } -> convertible_to<T>;
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/debug.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/debug.hpp"

// --- デバッグ ---
#ifndef ONLINE_JUDGE
// #define printd(x) cerr << #x << ": " << x << endl;
const string _TERM_ESC = "\033";
const string _TERM_BOLD = _TERM_ESC + "[1m";
const string _TERM_DECO_RESET = _TERM_ESC + "[0m";
const string _TERM_FORE_RED = _TERM_ESC + "[31m";
const string _TERM_FORE_RESET = _TERM_ESC + "[39m";
const string _TERM_BACK_RED = _TERM_ESC + "[41m";
const string _TERM_BACK_RESET = _TERM_ESC + "[49m";

#define line_debug() cerr << "line: " << __LINE__ << endl;
#define coutd(x) cerr << "[debug] " << x;
#define printd(x)                                                              \
  cerr << _TERM_BOLD << _TERM_BACK_RED << "[debug] " << (#x) << " = " << (x)   \
       << _TERM_BACK_RESET << _TERM_DECO_RESET << endl;

#else

#define line_debug() ;
#define coutd(x) ;
#define printd(x) ;

#endif
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/int128_t.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/int128_t.hpp"

using int128_t = __int128_t;
using uint128_t = __uint128_t;
using lll = int128_t;
using ulll = uint128_t;

// https://kenkoooo.hatenablog.com/entry/2016/11/30/163533

ostream &operator<<(ostream &dest, lll value) {
  ostream::sentry s(dest);
  if (s) {
    lll tmp = value < 0 ? -value : value;
    char buffer[128];
    char *d = end(buffer);
    do {
      --d;
      *d = "0123456789"[tmp % 10];
      tmp /= 10;
    } while (tmp != 0);
    if (value < 0) {
      --d;
      *d = '-';
    }
    int len = end(buffer) - d;
    if (dest.rdbuf()->sputn(d, len) != len) {
      dest.setstate(ios_base::badbit);
    }
  }
  return dest;
}

lll string_to_lll(string &s) {
  lll 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;
}

istream &operator>>(istream &src, lll &value) {
  string s;
  src >> s;
  value = string_to_lll(s);
  return src;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/print.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/print.hpp"

// ---------------
// ----- TOC -----
template <typename T1, typename T2>
ostream &operator<<(ostream &os, const pair<T1, T2> &p);
template <typename T1, typename T2>
istream &operator>>(istream &is, pair<T1, T2> &p);
template <typename T> ostream &operator<<(ostream &os, const vector<T> &v);
template <typename T>
ostream &operator<<(ostream &os, const vector<vector<T>> &v);
template <typename T>
ostream &operator<<(ostream &os, const vector<vector<vector<T>>> &v);
template <typename T> istream &operator>>(istream &is, vector<T> &v);
template <typename T, typename S>
ostream &operator<<(ostream &os, const map<T, S> &mp);
template <typename T> ostream &operator<<(ostream &os, const set<T> &st);
template <typename T> ostream &operator<<(ostream &os, const multiset<T> &st);
template <typename T> ostream &operator<<(ostream &os, queue<T> q);
template <typename T> ostream &operator<<(ostream &os, deque<T> q);
template <typename T> ostream &operator<<(ostream &os, stack<T> st);
template <class T, class Container, class Compare>
ostream &operator<<(ostream &os, priority_queue<T, Container, Compare> pq);
// --- END TOC ---
// ---------------

template <typename T1, typename T2>
ostream &operator<<(ostream &os, const pair<T1, T2> &p) {
  os << "(" << p.first << "," << p.second << ")";
  return os;
}

template <typename T1, typename T2>
istream &operator>>(istream &is, pair<T1, T2> &p) {
  is >> p.first >> p.second;
  return is;
}

template <typename T> ostream &operator<<(ostream &os, const vector<T> &v) {
  os << "[";
  for (int i = 0; i < (int)v.size(); i++) {
    os << v[i] << (i + 1 != (int)v.size() ? ", " : "]");
  }
  return os;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<vector<T>> &v) {
  for (int i = 0; i < (int)v.size(); i++) {
    os << v[i] << endl;
  }
  return os;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<vector<vector<T>>> &v) {
  for (int i = 0; i < (int)v.size(); i++) {
    os << "i = " << i << endl;
    os << v[i];
  }
  return os;
}

template <typename T> istream &operator>>(istream &is, vector<T> &v) {
  for (T &in : v)
    is >> in;
  return is;
}

template <typename T, typename S>
ostream &operator<<(ostream &os, const map<T, S> &mp) {
  for (auto &[key, val] : mp) {
    os << key << ":" << val << " ";
  }
  return os;
}

template <typename T> ostream &operator<<(ostream &os, const set<T> &st) {
  os << "{";
  auto itr = st.begin();
  for (int i = 0; i < (int)st.size(); i++) {
    os << *itr << (i + 1 != (int)st.size() ? ", " : "}");
    itr++;
  }
  return os;
}

template <typename T> ostream &operator<<(ostream &os, const multiset<T> &st) {
  auto itr = st.begin();
  for (int i = 0; i < (int)st.size(); i++) {
    os << *itr << (i + 1 != (int)st.size() ? " " : "");
    itr++;
  }
  return os;
}

template <typename T> ostream &operator<<(ostream &os, queue<T> q) {
  while (q.size()) {
    os << q.front() << " ";
    q.pop();
  }
  return os;
}

template <typename T> ostream &operator<<(ostream &os, deque<T> q) {
  while (q.size()) {
    os << q.front() << " ";
    q.pop_front();
  }
  return os;
}

template <typename T> ostream &operator<<(ostream &os, stack<T> st) {
  while (st.size()) {
    os << st.top() << " ";
    st.pop();
  }
  return os;
}

template <class T, class Container, class Compare>
ostream &operator<<(ostream &os, priority_queue<T, Container, Compare> pq) {
  while (pq.size()) {
    os << pq.top() << " ";
    pq.pop();
  }
  return os;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/util.hpp"

#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/util.hpp"

// --- Utils ---

// 二分探索
template <typename T>
inline T bin_search(T ok, T ng, const function<bool(T)> is_ok) {
  assert(is_ok(ok));
  assert(!is_ok(ng));
  assert(ok < ng);
  while (abs(ok - ng) > 1) {
    T mid = (ok + ng) / 2;
    if (is_ok(mid))
      ok = mid;
    else
      ng = mid;
  }
  return ok;
}

// 順列全探索
inline void rep_perm(int n, function<void(vec<int> &)> f) {
  vec<int> v(n);
  iota(v.begin(), v.end(), 0);
  do {
    f(v);
  } while (next_permutation(v.begin(), v.end()));
}

inline void rep_bit(int n, function<void(int)> f) { rep(i, 1LL << n) f(i); }

// 配列 to string
template <typename T> inline string join(const vec<T> &v, string sep = " ") {
  string res = "";
  rep(i, v.size()) res += to_string(v[i]) + (i == v.size() - 1 ? "" : sep);
  return res;
}
template <typename T>
inline void join_out(ostream &os, const vec<T> &v, string sep = " ",
                     string end = "\n") {
  int n = v.size();
  rep(i, n) os << v[i] << (i == n - 1 ? end : sep);
}

template <typename T>
inline void transform(vec<T> &src, function<void(T &)> f) {
  for (auto &val : src)
    f(val);
}

// ベース指定ceil
inline ll ceil(ll x, ll base) { return (x + base - 1) / base * base; }
// ベース指定floor
inline ll floor(ll x, ll base) { return x / base * base; }

// 合計値を求める
// ll sum(const vec<ll> &v) { return accumulate(all(v), 0LL); }
template <addable T> T sum(const vec<T> &v) { return accumulate(all(v), T()); }

// 可変引数min
template <class... T> auto min(T... a) {
  return min(initializer_list<common_type_t<T...>>{a...});
}

// 可変引数max
template <class... T> auto max(T... a) {
  return max(initializer_list<common_type_t<T...>>{a...});
}

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

// 3項間不等式
// 広義単調増加
inline bool is_increasing(int x, int y, int z) { return x <= y && y <= z; }

// 半開区間[x, z)にyが含まれるか?
inline bool is_contained(int x, int y, int z) { return x <= y && y < z; }

// 頂点(x, y)が範囲に含まれるか
inline bool is_contained(int H, int W, int x, int y) {
  return is_contained(0, x, H) && is_contained(0, y, W);
}

// rootに対し、aとbが同じ側にあるか (=は含まない)
inline bool is_same_side(int root, int a, int b) {
  return (root < a) == (root < b);
}

// vector継承にする?
template <typename T> struct vec_accumulate : public vec<T> {
  explicit vec_accumulate(vec<T> &v) : vec<T>(v.size()) {
    assert(v.size() > 0);
    this->at(0) = v[0];
    for (int i = 1; i < v.size(); ++i)
      this->at(i) = this->at(i - 1) + v[i];
  }

  T operator[](int i) {
    assert(is_contained(-1, i, this->size()));
    if (i == -1)
      return T();
    return this->at(i);
  }
};

// vector func
template <typename T> inline void unique_erase(vec<T> &v) {
  sort(all(v));
  v.erase(unique(all(v)), v.end());
}

void io_setup() {
  cin.tie(nullptr);
  ios::sync_with_stdio(false);
  cout << std::fixed << std::setprecision(15);
}
#line 12 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/common/template.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/pow.hpp"

#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/pow.hpp"

// 繰り返し2乗法
template <integral T, integral F> constexpr T pow(T a, F n) {
  T ans = 1;
  while (n > 0) {
    if (n & 1)
      ans *= a;
    a *= a;
    n >>= 1;
  }
  return ans;
}

constexpr ll powm(ll a, ll n, const ll mod) {
  lll ans = 1;
  while (n > 0) {
    if (n & 1)
      ans = ans * a % mod;
    a = ((lll)a) * a % mod;
    n >>= 1;
  }
  return ans;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/all.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/2d/all.hpp"

#line 3 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/2d/point.hpp"

class Point {
public:
  int x, y;
  Point() : x(0), y(0) {}
  Point(int x, int y) : x(x), y(y) {}

  Point up() const { return Point(x - 1, y); }
  Point down() const { return Point(x + 1, y); }
  Point left() const { return Point(x, y - 1); }
  Point right() const { return Point(x, y + 1); }
  Point upper_left() const { return Point(x - 1, y - 1); }
  Point upper_right() const { return Point(x - 1, y + 1); }
  Point lower_left() const { return Point(x + 1, y - 1); }
  Point lower_right() const { return Point(x + 1, y + 1); }

  vec<Point> around4() const { return {up(), down(), left(), right()}; }
  vec<Point> around8() const {
    return {up(),   up().right(),  right(), right().down(),
            down(), down().left(), left(),  left().up()};
  }

  int manhattan() const { return std::abs(x) + std::abs(y); }
  int eucurid2() const { return x * x + y * y; }

  bool operator==(const Point &p) const { return x == p.x && y == p.y; }
  bool operator!=(const Point &p) const { return !(*this == p); }
  void operator+=(const Point &p) {
    x += p.x;
    y += p.y;
  }
  void operator-=(const Point &p) {
    x -= p.x;
    y -= p.y;
  }

  Point operator+(const Point &p) const { return Point(x + p.x, y + p.y); }
  Point operator-(const Point &p) const { return Point(x - p.x, y - p.y); }
};
ostream &operator<<(ostream &os, const Point &p) {
  os << p.x << p.y;
  return os;
}
istream &operator>>(istream &is, Point &p) {
  is >> p.x >> p.y;
  return is;
}

inline bool is_contained(int H, int W, Point p) {
  return is_contained(H, W, p.x, p.y);
}
#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/2d/area.hpp"

class Area {
public:
  int H, W;
  bool allow_outside = false;
  char out_val;
  Area(int H, int W) : H(H), W(W), data(vector(H, string(W, '.'))) {}
  Area(int H, int W, char out_val)
      : H(H), W(W), data(vector(H, string(W, '.'))), allow_outside(true),
        out_val(out_val) {}
  string &operator[](int i) { return data[i]; }
  // 時計回りで90度回転
  Area rotated90() const {
    Area ret(W, H);
    rep(i, H) rep(j, W) ret[j][H - i - 1] = data[i][j];
    return ret;
  }
  void rotate90() {
    assert(H == W);
    data = rotated90().data;
  }
  char at(int x, int y) {
    if (is_contained(H, W, x, y))
      return data[x][y];
    else if (allow_outside)
      return out_val;
    else {
      cerr << "[Area::at] out of range" << endl;
      assert(false);
    }
  }
  void set(int x, int y, char val) {
    if (is_contained(H, W, x, y))
      data[x][y] = val;
    else if (allow_outside)
      return;
    else {
      cerr << "[Area::set] out of range" << endl;
      assert(false);
    }
  }

  bool contains(int x, int y) const { return is_contained(H, W, x, y); }
  bool contains(Point p) const { return is_contained(H, W, p.x, p.y); }

private:
  vec<string> data;
};
ostream &operator<<(ostream &os, Area &a) {
  rep(i, a.H) os << a[i] << endl;
  return os;
}
istream &operator>>(istream &is, Area &a) {
  rep(i, a.H) is >> a[i];
  return is;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/2d/rect.hpp"

class Rect {
public:
  int x, y, w, h;
  Rect(int x, int y, int w, int h) : x(x), y(y), w(w), h(h) {}
  int area() const { return w * h; }
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/3d/all.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/3d/cuboid.hpp"

class Cuboid {
public:
  int x, y, z;
  int w, h, d;
  Cuboid(int x, int y, int z, int w, int h, int d)
      : x(x), y(y), z(z), w(w), h(h), d(d) {}
  Cuboid(int w, int h, int d) : x(0), y(0), z(0), w(w), h(h), d(d) {}
  int volume() { return w * h * d; }
  Cuboid intersect(Cuboid c) {
    int x1 = max(x, c.x);
    int y1 = max(y, c.y);
    int z1 = max(z, c.z);
    int x2 = min(x + w, c.x + c.w);
    int y2 = min(y + h, c.y + c.h);
    int z2 = min(z + d, c.z + c.d);
    if (x2 <= x1 || y2 <= y1 || z2 <= z1) {
      return Cuboid(0, 0, 0, 0, 0, 0);
    }
    return Cuboid(x1, y1, z1, x2 - x1, y2 - y1, z2 - z1);
  }
  void move(int dx, int dy, int dz) {
    x += dx;
    y += dy;
    z += dz;
  }
  static Cuboid intersect(Cuboid &a, Cuboid &b) { return a.intersect(b); }
  static Cuboid intersect(initializer_list<Cuboid> cuboids) {
    auto it = cuboids.begin();
    Cuboid c = *it;
    it++;
    for (; it != cuboids.end(); it++) {
      c = c.intersect(*it);
    }
    return c;
  }
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/convolution/all.hpp"

// #include "convolution.hpp"
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/all.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/comp.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/comp.hpp"

// https://onlinejudge.u-aizu.ac.jp/courses/library/3/DSL/4/DSL_4_A

template <typename T> struct comp {
private:
  const int _n;
  int _cmp_n;
  vec<T> _raw, _comp;

public:
  explicit comp(const vec<T> &src)
      : _n(src.size()), _cmp_n(0), _raw(src), _comp(_n, 0) {
    sort(_raw.begin(), _raw.end());
    _raw.erase(unique(_raw.begin(), _raw.end()), _raw.end());
    _cmp_n = _raw.size();
    rep(i, _n) _comp[i] =
        lower_bound(_raw.begin(), _raw.end(), src[i]) - _raw.begin();
  }
  inline int size() const { return _n; }
  inline int cmp_size() const { return _cmp_n; }
  inline T get_raw(int complessed) {
    assert(0 <= complessed && complessed < _cmp_n);
    return _raw[complessed];
  }
  inline T operator[](int i) {
    assert(0 <= i && i < _n);
    return _comp[i];
  }
  inline T get_comp(int raw) {
    return lower_bound(_raw.begin(), _raw.end(), raw) - _raw.begin();
  }
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/fenwick-tree.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/fenwick-tree.hpp"

template <typename T> struct FenwickTree {
private:
  T _n;
  vec<T> _data;

public:
  explicit FenwickTree(T n) : _n(n), _data(n + 1, 0) {}
  explicit FenwickTree(const vec<T> &src) : _n(src.size()), _data(_n + 1, 0) {
    rep(i, _n) add(i, src[i]);
  }

  void add(int i, T val) {
    i++;
    while (i <= _n) {
      _data[i] += val;
      i += i & -i;
    }
  }
  // [0, i]
  int sum(int i) const {
    i++;
    T ans = 0;
    while (i > 0) {
      ans += _data[i];
      i -= i & -i;
    }
    return ans;
  }
  // [l, r)
  int sum(int l, int r) const { return sum(r - 1) - sum(l - 1); }
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/imos.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/imos.hpp"

template <typename T> struct imos {
private:
  const int _n;
  vec<T> _data;
  bool _is_build = false;

public:
  explicit imos(int n) : _n(n), _data(_n + 1, 0) {}
  explicit imos(vec<T> src) : _n(src.size()), _data(_n + 1, 0) {
    rep(i, _n) add(i, src[i]);
  }
  inline int size() const { return _n; }
  inline void add(int l, int r, T x) {
    assert(!_is_build);
    assert(l < r);
    assert(0 <= l && r <= _n);
    _data[l] += x;
    _data[r] -= x;
  }
  inline void add(int i, T x) {
    assert(!_is_build);
    assert(0 <= i && i < _n);
    add(i, i + 1, x);
  }
  inline void build() {
    rep(i, _n) _data[i + 1] += _data[i];
    _is_build = true;
  }
  inline T get(int i) {
    assert(0 <= i && i < _n);
    if (!_is_build)
      build();
    return _data[i];
  }

  friend ostream &operator<<(ostream &os, const imos<T> &a) {
    if (!a._is_build) {
      os << "not builded";
      return os;
    }
    os << a._data;
    return os;
  }
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/imos2d.hpp"

#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/imos2d.hpp"

template <typename T> struct imos2d {
private:
  const int _H, _W;
  vvec<T> _data;
  bool _is_build = false;

public:
  explicit imos2d(int H, int W)
      : _H(H), _W(W), _data(_H + 1, vec<T>(_W + 1, 0)) {}
  explicit imos2d(vvec<T> src)
      : _H(src.size()), _W(src[0].size()), _data(_H + 1, vec<T>(_W + 1, 0)) {
    rep(x, _H) rep(y, _W) add({x, y}, src[x][y]);
  }
  inline pair<int, int> size() const { return {_H, _W}; }
  // 半開区間
  inline void add(Point a, Point b, T x) {
    assert(!_is_build);
    assert(a.x < b.x && a.y < b.y);
    assert(is_contained(_H, _W, a) && is_contained(_H, _W, b.upper_left()));
    _data[a.x][a.y] += x;
    _data[a.x][b.y] -= x;
    _data[b.x][a.y] -= x;
    _data[b.x][b.y] += x;
  }
  inline void add(Point a, T x) {
    assert(!_is_build);
    assert(is_contained(_H, _W, a));
    add(a, a.lower_right(), x);
  }
  inline void build() {
    rep(x, _H) rep(y, _W) _data[x][y + 1] += _data[x][y];
    rep(y, _W) rep(x, _H) _data[x + 1][y] += _data[x][y];
    _is_build = true;
  }
  inline T get(Point p) const {
    assert(is_contained(_H, _W, p));
    assert(_is_build);
    return _data[p.x][p.y];
  }
};

template <typename T> ostream &operator<<(ostream &os, const imos2d<T> &im) {
  rep(x, im.size().first) {
    rep(y, im.size().second) { os << im.get({x, y}) << " "; }
    os << endl;
  }
  return os;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/lazy-segment-tree.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/monoid/monofunc.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/monoid/monoid.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/monoid/monoid.hpp"

/*
template <class M>
concept MonoidConcept = requires(M &x, M::T a, M::T b) {
  typename M::T;
  { M::e() } -> std::same_as<typename M::T>;
  { M::op(a, b) } -> std::same_as<typename M::T>;
};
*/

template <class Type> class Monoid {
public:
  Monoid() = default;
  using T = Type;
  virtual T e() const = 0;
  virtual T op(T a, T b) const = 0;
};

template <class M>
concept MonoidConcept = requires(M &x, typename M::T a, typename M::T b) {
  // derived_from<M, Monoid<typename M::T>>;
  typename M::T;
  { x.e() } -> std::same_as<typename M::T>;
  { x.op(a, b) } -> std::same_as<typename M::T>;
};
#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/monoid/monofunc.hpp"

template <class Type, MonoidConcept Mono> class Monofunc {
public:
  Monofunc() = default;
  using T = Type;
  using M = Mono;
  using MT = typename M::T;
  virtual T id() const = 0;
  virtual MT apply(T a, MT b) const = 0;
  virtual T merge(T before, T after) const = 0;
};

template <class M>
concept MonofuncConcept = requires(M &x, typename M::T before,
                                   typename M::T after, typename M::MT mono) {
  typename M::T;
  typename M::M;
  typename M::MT;
  { x.id() } -> std::same_as<typename M::T>;
  { x.apply(before, mono) } -> std::same_as<typename M::MT>;
  { x.merge(before, after) } -> std::same_as<typename M::T>;
};
#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/lazy-segment-tree.hpp"

// 参考 https://qiita.com/ningenMe/items/bf66de877e3b97d35862

// T : Nodeの型, F : 操作情報
template <MonofuncConcept F> struct LazySegmentTree {
private:
  using M = typename F::M;
  using FT = typename F::T;
  using MT = typename M::T;
  // 単位元 デフォルトでは、min(a, e) = min(e, a) = a
  const MT e;
  const FT id;
  // Monoids
  const M _monoid;
  const F _monofunc;
  // 演算
  // const function<T(T, T)> op;
  // const function<T(F, T)> update;
  // const function<F(F, F)> merge;
  // 要素数
  int32_t _n;
  int32_t _height;
  // SefTreeのデータ
  vec<MT> _data;
  vec<FT> _lazy;

  // 自分の遅延を解消して子に伝搬する
  inline void eval(int32_t k) {
    if (_lazy[k] == id)
      return;
    _data[k] = _monofunc.apply(_lazy[k], _data[k]);
    if (k < _n) {
      _lazy[k << 1 | 0] = _monofunc.merge(_lazy[k << 1 | 0], _lazy[k]);
      _lazy[k << 1 | 1] = _monofunc.merge(_lazy[k << 1 | 1], _lazy[k]);
    }
    _lazy[k] = id;
  }

public:
  // n: 要素数, e: 単位元, id: 操作情報の単位元, op: 演算, update:
  // ノードの更新(fn,val->val), merge:
  // 操作aがすでに行われている状態に操作bを適用したときの操作
  // TODO: 2つの実装が別れているのをどうにかする
  explicit inline LazySegmentTree(const int32_t n)
      : _monoid(M()), _monofunc(F()), e(_monoid.e()), id(_monofunc.id()) {
    _n = 1;
    _height = 1;
    while (_n < n)
      _n <<= 1, _height++;
    _data = vec<MT>(_n << 1, e);
    _lazy = vec<FT>(_n << 1, id);
  }
  explicit inline LazySegmentTree(vec<MT> &data)
      : _monoid(M()), _monofunc(F()), e(_monoid.e()), id(_monofunc.id()) {
    _n = 1;
    _height = 1;
    while (_n < data.size())
      _n <<= 1, _height++;
    _data = vec<MT>(_n << 1, e);
    _lazy = vec<FT>(_n << 1, id);
    rep(i, data.size()) _data[i + _n] = data[i];
    for (int i = _n - 1; i > 0; --i)
      _data[i] = _monoid.op(_data[i << 1], _data[i << 1 | 1]);
  }
  /*
  inline LazySegmentTree(const T e, const F id,
                         const function<T(T, T)> op,
                         const function<T(F, T)> update,
                         const function<F(F, F)> merge,
                         vec<T> &v)
      : e(e), id(id), op(op), update(update), merge(merge) {
    _n = 1;
    _height = 1;
    while (_n < v.size())
      _n <<= 1, _height++;
    _data.resize(_n << 1, e);
    for(int i=0; i<v.size(); ++i) _data[i + _n] = v[i];
    _lazy = vec<F>(_n << 1, id);
  }
  */

  inline void set(int32_t l, int32_t r, FT f) {
    /*
    eval(k);
    if (l <= a && b <= r) { // 完全に内側
      _lazy[k] = merge(f, _lazy[k]);
      eval(k);
    } else if (a < r && l < b) { // 一部区間が被る
      _set(l, r, f, (k << 1) + 1, a, (a + b) >> 1);
      _set(l, r, f, (k << 1) + 2, (a + b) >> 1, b);
      _data[k] = op(_data[(k << 1) + 1], _data[(k << 1) + 2]);
    }
    */
    // a, bはseg-index [a,b]
    int32_t a = l + _n, b = r + _n - 1;

    // 上から下へとlazyを伝搬させる
    for (int32_t i = _height; 0 < i; --i)
      eval(a >> i), eval(b >> i);
    // bを [a, b]から[a, b)にする
    for (b++; a < b; a >>= 1, b >>= 1) {
      if (a & 1)
        _lazy[a] = _monofunc.merge(_lazy[a], f), eval(a), a++;
      if (b & 1)
        --b, _lazy[b] = _monofunc.merge(_lazy[b], f), eval(b);
    }
    // a, bをseg-index [a,b]に戻す
    a = l + _n, b = r + _n - 1;
    // a,bを親に移しつつ、aが実在頂点の間
    while ((a >>= 1), (b >>= 1), a) {
      if (_lazy[a] == id) {
        // _data[(a << 1) | 0] = update(_lazy[(a << 1) | 0], _data[(a << 1) |
        // 0]); _data[(a << 1) | 1] = update(_lazy[(a << 1) | 1], _data[(a << 1)
        // | 1]); _data[a] = op(_data[(a << 1) | 0], _data[(a << 1) | 1]);
        _data[a] = _monoid.op(
            _monofunc.apply(_lazy[(a << 1) | 0], _data[(a << 1) | 0]),
            _monofunc.apply(_lazy[(a << 1) | 1], _data[(a << 1) | 1]));
      }
      if (_lazy[b] == id) {
        // _data[(b << 1) | 0] = update(_lazy[(b << 1) | 0], _data[(b << 1) |
        // 0]); _data[(b << 1) | 1] = update(_lazy[(b << 1) | 1], _data[(b << 1)
        // | 1]); _data[b] = op(_data[(b << 1) | 0], _data[(b << 1) | 1]);
        _data[b] = _monoid.op(
            _monofunc.apply(_lazy[(b << 1) | 0], _data[(b << 1) | 0]),
            _monofunc.apply(_lazy[(b << 1) | 1], _data[(b << 1) | 1]));
      }
    }
  }

  // [l, r)の区間に対するクエリを処理する
  // kはSegTree上のindex
  inline MT query(int32_t l, int32_t r) {
    // ↓再帰
    /*
    eval(k);
    if (r <= a || b <= l)
      return e;
    if (l <= a && b <= r)
      return _data[k];
    T l_val = _query(l, r, (k << 1) + 1, a, (a + b) >> 1);
    T r_val = _query(l, r, (k << 1) + 2, (a + b) >> 1, b);
    return op(l_val, r_val);
    */

    // a, bはseg-index [a,b]
    int32_t a = l + _n, b = r + _n - 1;
    // 上から下へとlazyを伝搬させる
    for (int32_t i = _height; 0 <= i; --i)
      eval(a >> i), eval(b >> i);
    // line_debug();
    // printd(_data);
    // printd(_lazy);
    MT vl = e, vr = e;
    // bを [a, b]から[a, b)にする
    for (b++; a < b; a >>= 1, b >>= 1) {
      if (a & 1)
        vl = _monoid.op(vl, _monofunc.apply(_lazy[a], _data[a])), a++;
      if (b & 1)
        b--, vr = _monoid.op(_monofunc.apply(_lazy[b], _data[b]), vr);
    }
    return _monoid.op(vl, vr);
  }
  inline MT get(int32_t i) { return query(i, i + 1); }
};
/*
template <typename T, typename F> using LazySegtree = LazySegmentTree<T, F>;

template <typename T, typename F> inline LazySegtree<T, F> RMQLazySeg(int n) {
  return LazySegmentTree<T, F>(
      n, numeric_limits<T>::max(), 0, [](T a, T b) { return min(a, b); },
      [](F a, T b) { return b + a; }, [](F a, F b) { return b + a; });
}
*/
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/segment-tree.hpp"

#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/segment-tree.hpp"

// 参考: https://qiita.com/ningenMe/items/bf66de877e3b97d35862

template <MonoidConcept M> struct SegmentTree {
  using T = typename M::T;

private:
  M _monoid;
  T e;

  // 要素数
  int _n;
  // SegTreeのノード上のデータ
  vec<T> _data;

public:
  explicit SegmentTree(int n) : _n(1), _monoid(M()), e(_monoid.e()) {
    while (_n < n)
      _n *= 2;
    _data = vec<T>(2 * _n, e);
  }
  explicit SegmentTree(const vec<T> &v) : _n(1), _monoid(M()), e(_monoid.e()) {
    while (_n < v.size())
      _n *= 2;
    _data = vec<T>(2 * _n, e);
    rep(i, v.size()) _data[i + _n] = v[i];
    for (int i = _n - 1; i > 0; --i)
      _data[i] = _monoid.op(_data[i << 1], _data[i << 1 | 1]);
  }

  void set(int i, T a) {
    // 1段目は1, 2段目は2~3, ... , 最下段はn~(2n-1)
    // より、iの最下段はi+n
    i += _n;
    _data[i] = a;
    // i >>= 1で、iを親のindexに変更
    // また、i==0となってしまうなら、i==1(頂点)であるので、終了
    // i<<1は、(更新後の)iの左の子
    // i<<1|1は、(更新後の)iの右の子 (+1を|1で表現)
    while (i >>= 1)
      _data[i] = _monoid.op(_data[i << 1], _data[i << 1 | 1]);
  }
  T get(int i) const { return _data[i + _n]; }
  T query(int l, int r) {
    T l_val = e, r_val = e;
    // l, r(index)に+_nすると、最下段のseg-indexになる
    // l < r == l,rをすべて含む区間にはたどり着いていない
    // l/r >>= 1 : l/rを親に移動させる
    for (l += _n, r += _n; l < r; l >>= 1, r >>= 1) {
      // lが左の子ならば、l<=iの結果はlの親の結果であるので、計算しない
      // lが右の子ならば、計算する必要がある
      if (l & 1)
        l_val = _monoid.op(l_val, _data[l++]);
      // l++をしても、l>>1が変わることはない
      // ただし、r=l+1のとき、答えはここのlである
      // ので、l<rに引っかかるようにして終了させている?

      // rが左の子ならば、rの親はギリギリ使われない、すなわちrの条件を満たす
      // rが右の子ならば、r-1は範囲に含まれるので、計算する必要がある
      if (r & 1)
        r_val = _monoid.op(_data[--r], r_val);
    }
    return _monoid.op(l_val, r_val);
  }
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/union-find.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/union-find.hpp"

struct UnionFind {
private:
  int _n;
  vec<int> _parents;
  vec<int> _size;

public:
  explicit UnionFind(int n) : _n(n), _parents(n), _size(n, 1) {
    iota(all(_parents), 0);
  }
  int find(int a) {
    if (_parents[a] == a)
      return a;
    else
      return _parents[a] = find(_parents[a]);
  }
  void merge(int a, int b) {
    a = find(a);
    b = find(b);
    if (a != b) {
      // merge後の親はa
      if (_size[a] < _size[b])
        swap(a, b);
      _size[a] += _size[b];
      _size[b] = 0;
      _parents[b] = a;
    }
  }
  int size(int a) { return _size[find(a)]; }
  bool same(int a, int b) { return find(a) == find(b); }
  vec<vec<int>> groups() {
    vec<int> leaders(_n);
    rep(i, _n) { leaders[i] = find(i); }
    vec<vec<int>> res(_n);
    rep(i, _n) res[i].reserve(_size[i]);
    rep(i, _n) res[leaders[i]].push_back(i);
    res.erase(remove_if(all(res), [&](const vec<int> &v) { return v.empty(); }),
              res.end());
    return res;
  }
};
using UF = UnionFind;
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/weighted-union-find.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/datastructure/weighted-union-find.hpp"

template <addable T> struct WeightedUnionFind {
private:
  int _n;
  vec<int> _parents;
  vec<int> _size;
  vec<T> _weight;

  T weight(int i) {
    find(i);
    return _weight[i];
  }

public:
  explicit WeightedUnionFind(int n)
      : _n(n), _parents(n), _size(n, 1), _weight(n) {
    iota(all(_parents), 0);
  }

  int find(int i) {
    if (_parents[i] == i)
      return i;
    const int root = find(_parents[i]);
    _weight[i] += _weight[_parents[i]];
    return _parents[i] = root;
  }

  void merge(int a, int b, T w) {
    w += weight(a);
    w -= weight(b);
    a = find(a);
    b = find(b);
    if (a != b) {
      // merge後の親はa
      if (_size[a] < _size[b])
        swap(a, b), w = -w;
      _size[a] += _size[b];
      _size[b] = 0;
      _parents[b] = a;
      _weight[b] = w;
    }
  }

  T diff(int a, int b) { return weight(b) - weight(a); }

  int size(int a) { return _size[find(a)]; }

  bool same(int a, int b) { return find(a) == find(b); }
};

template <addable T> using WUF = WeightedUnionFind<T>;
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/cycle.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/template.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/template.hpp"

// Node
template <class T = ll> struct WeightedNode {
  int id;
  T cost = numeric_limits<T>::max();
  WeightedNode(int id, T cost) : id(id), cost(cost) {}
};

struct WeightState {
public:
  int location;
  ll used_cost;
  WeightState(int location, ll used_cost)
      : location(location), used_cost(used_cost) {}
  bool operator<(const WeightState &n) const { return used_cost < n.used_cost; }
  bool operator>(const WeightState &n) const { return used_cost > n.used_cost; }
};

// Graph
struct Graph {
private:
  const int n;
  const bool directed;
  vec<vec<int>> edges;

public:
  explicit Graph(int n, bool directed = false)
      : n(n), directed(directed), edges(n) {}
  inline void add_edge(int u, int v) {
    edges[u].emplace_back(v);
    if (!directed)
      edges[v].emplace_back(u);
  }
  inline vec<int> &operator[](int i) { return edges[i]; }
  inline int size() const { return n; }
};

template <class T = ll> struct WeightedGraph {
private:
  const int n;
  const bool directed;
  vec<vec<WeightedNode<T>>> edges;

public:
  explicit WeightedGraph(int n, bool directed = false)
      : n(n), directed(directed), edges(n) {}
  inline void add_edge(int u, int v, T w) {
    edges[u].emplace_back(WeightedNode(v, w));
    if (!directed)
      edges[v].emplace_back(WeightedNode(u, w));
  }
  inline vec<WeightedNode<T>> &operator[](int i) { return edges[i]; }
  inline int size() const { return n; }
};

// template <class T = ll> using WeightedGraph = vec<vec<WeightedNode<T>>>;

template <class T = ll> using WGraph = WeightedGraph<T>;
#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/cycle.hpp"

bool has_cycle(Graph &g, int start = 0) {
  vec<int> seen(g.size(), 0);
  vec<int> finished(g.size(), 0);
  auto dfs = [&](auto fn, int index) {
    seen[index] = 1;
    for (auto next : g[index]) {
      if (finished[next])
        continue;
      if (seen[next] && !finished[next])
        return true;
      if (fn(fn, next))
        return true;
    }
    finished[index] = 1;
    return false;
  };
  return dfs(dfs, start);
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/depth.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/depth.hpp"

struct TreeNodeInfo {
public:
  TreeNodeInfo(int parent, int depth) : parent(parent), depth(depth) {}
  int parent;
  int depth;
};

vec<TreeNodeInfo> depth(Graph &graph, int root = 0) {
  vec<TreeNodeInfo> result(graph.size(), TreeNodeInfo(-1, -1));
  stack<int> st;
  st.push(root);
  result[root] = TreeNodeInfo(root, 0);
  while (!st.empty()) {
    int index = st.top();
    st.pop();
    for (auto next : graph[index]) {
      if (result[next].depth != -1)
        continue;
      result[next] = TreeNodeInfo(index, result[index].depth + 1);
      st.push(next);
    }
  }
  return result;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/diameter.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/diameter.hpp"

int diameter(Graph &graph, int start = 0) {
  vec<int> seen(graph.size(), 0);
  auto dfs = [&](auto fn, int index) -> pair<int, int> {
    // max-cost, index
    pair<int, int> result = {0, index};

    seen[index] = 1;

    for (auto next : graph[index]) {
      if (seen[next])
        continue;
      auto next_result = fn(fn, next);
      next_result.first += 1;
      result = max(result, next_result);
    }
    seen[index] = 0;
    return result;
  };
  auto result = dfs(dfs, start);
  result = dfs(dfs, result.second);
  return result.first;
}

template <typename T = ll> int diameter(WGraph<T> &graph, int start = 0) {
  auto dfs = [&](int index) -> pair<T, int> {
    vec<T> result(graph.size(), -1);

    T mx = 0;
    int mx_index = index;
    stack<int> st;
    st.push(index);
    result[index] = 0;
    while (!st.empty()) {
      int current = st.top();
      st.pop();
      for (auto next : graph[current]) {
        if (result[next.id] >= 0)
          continue;
        st.push(next.id);
        result[next.id] = result[current] + next.cost;
        if (chmax(mx, result[next.id])) {
          mx_index = next.id;
        }
      }
    }
    return {mx, mx_index};
  };
  auto result = dfs(start);
  result = dfs(result.second);
  return result.first;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/dijkstra.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/dijkstra.hpp"

template <typename T = ll> vec<ll> dijkstra(WGraph<T> &graph, int start) {
  vec<ll> way(graph.size(), INF);
  rp_queue<WeightState> q;
  q.push(WeightState(start, 0));
  way[start] = 0;
  while (!q.empty()) {
    WeightState current = q.top();
    q.pop();
    for (auto &next : graph[current.location]) {
      ll next_cost = current.used_cost + next.cost;
      if (way[next.id] <= next_cost)
        continue;
      way[next.id] = next_cost;
      q.push(WeightState(next.id, way[next.id]));
    }
  }
  return way;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/lca.hpp"

struct LCA {
private:
  Graph graph;
  int root;
  // parent[k][v] := vの2^k先の親
  vec<vec<int>> parent;
  vec<TreeNodeInfo> depth_data;

public:
  explicit LCA(Graph &graph, int root = 0)
      : graph(graph), root(root), depth_data(depth(graph, root)) {
    int n = graph.size();
    int logn = 1;
    while ((1 << logn) < n)
      logn++;
    parent = vector(logn, vector(n, -1LL));
    for (int i = 0; i < n; i++)
      parent[0][i] = depth_data[i].parent;
    // ダブリング
    for (int k = 0; k + 1 < logn; k++) {
      for (int i = 0; i < n; i++) {
        if (parent[k][i] < 0)
          parent[k + 1][i] = -1;
        else
          parent[k + 1][i] = parent[k][parent[k][i]];
      }
    }
  }

  int query(int u, int v) const {
    if (depth_data[u].depth > depth_data[v].depth)
      swap(u, v);
    int logn = parent.size();
    // uとvの深さが同じになるまで親を辿る
    rep(k, logn) {
      if ((depth_data[v].depth - depth_data[u].depth) >> k & 1) {
        v = parent[k][v];
      }
    }
    // にぶたん
    if (u == v)
      return u;
    for (int k = logn - 1; k >= 0; k--) {
      if (parent[k][u] != parent[k][v]) {
        u = parent[k][u];
        v = parent[k][v];
      }
    }
    return parent[0][u];
  }
};
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/topological-sort.hpp"

#line 4 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/graph/topological-sort.hpp"

vec<int> topological_sort(Graph &graph) {
  vec<int> result;
  vec<int> deleted(graph.size(), false);

  vec<int> in_count(graph.size());
  rep(i, graph.size()) {
    for (auto next : graph[i]) {
      in_count[next]++;
    }
  }
  queue<int> q;
  rep(i, graph.size()) {
    if (in_count[i] == 0) {
      q.push(i);
    }
  }

  while (!q.empty()) {
    int now = q.front();
    q.pop();
    result.emplace_back(now);
    deleted[now] = true;
    for (auto next : graph[now]) {
      if (deleted[next])
        continue;
      in_count[next]--;
      if (in_count[next] == 0) {
        q.push(next);
      }
    }
  }

  return result;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/all.hpp"

// #include "combination.hpp"
// #include "factorial.hpp"
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/modint.hpp"

#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/modint.hpp"

namespace modint_utils {
constexpr inline int32_t normalize(int val, int32_t mod) {
  return (val % mod + mod) % mod;
}
constexpr inline int32_t inv(int32_t a, int32_t mod) {
  int32_t b = mod, u = 1, v = 0;
  while (b) {
    int32_t t = a / b;
    a -= t * b;
    swap(a, b);
    u -= t * v;
    swap(u, v);
  }
  u %= mod;
  if (u < 0)
    u += mod;
  return u;
  // return mod_pow(val, mod - 2, mod);
}
} // namespace modint_utils

template <int MOD> struct dynamic_modint {
private:
  int _val;

public:
  dynamic_modint() : dynamic_modint(0) {}
  dynamic_modint(int val) : _val(modint_utils::normalize(val, MOD)) {}

  // logic
  int val() const { return _val; }
  inline dynamic_modint inv() const {
    return dynamic_modint(modint_utils::inv(_val, MOD));
  }
  inline dynamic_modint pow(const int n) const {
    return dynamic_modint(powm(_val, n, MOD));
  }
  inline static constexpr dynamic_modint pow(const dynamic_modint &a,
                                             const int n) {
    return dynamic_modint(mod_pow(a._val, n, a.MOD));
  }

  // op
  inline dynamic_modint &operator++() { return *this += 1; }
  inline dynamic_modint &operator--() { return *this -= 1; }
  inline dynamic_modint operator++(int32_t) const {
    dynamic_modint tmp = *this;
    ++*this;
    return tmp;
  }
  inline dynamic_modint operator--(int32_t) {
    dynamic_modint tmp = *this;
    --*this;
    return tmp;
  }
  inline dynamic_modint operator+(const dynamic_modint &a) const {
    return dynamic_modint(_val) += a;
  }
  inline dynamic_modint operator-(const dynamic_modint &a) const {
    return dynamic_modint(_val) -= a;
  }
  inline dynamic_modint operator*(const dynamic_modint &a) const {
    return dynamic_modint(_val) *= a;
  }
  inline dynamic_modint operator/(const dynamic_modint &a) const {
    return dynamic_modint(_val) /= a;
  }
  inline bool operator==(const dynamic_modint &a) { return _val == a._val; }
  inline bool operator!=(const dynamic_modint &a) { return _val != a._val; }
  inline dynamic_modint &operator+=(const dynamic_modint &a) {
    _val += a.val();
    if (_val >= MOD)
      _val -= MOD;
    return *this;
  }
  inline dynamic_modint &operator-=(const dynamic_modint &a) {
    _val -= a.val();
    if (_val < 0)
      _val += MOD;
    return *this;
  }
  inline dynamic_modint &operator*=(const dynamic_modint &a) {
    _val = _val * a.val() % MOD;
    return *this;
  }
  inline dynamic_modint &operator/=(const dynamic_modint &a) {
    *this *= modint_utils::inv(a.val(), MOD);
    return *this;
  }

  explicit operator int() const { return _val; }

  // io
  friend ostream &operator<<(ostream &os, const dynamic_modint &a) {
    return os << a._val;
  }
  friend istream &operator>>(istream &os, dynamic_modint &a) {
    os >> a._val;
    a._val = modint_utils::normalize(a._val, MOD);
    return os;
  }
};

// using modint998 = modint<998244353>;

template <int MOD>
ostream &operator<<(ostream &os, const dynamic_modint<MOD> &i) {
  os << i.val();
  return os;
}

template <int MOD>
ostream &operator<<(ostream &os, const vector<dynamic_modint<MOD>> &v) {
  for (int i = 0; i < (int)v.size(); i++) {
    os << v[i].val() << (i + 1 != (int)v.size() ? " " : "");
  }
  return os;
}

template <const int32_t MOD> struct modint {
private:
  using i32 = int32_t;
  i32 _val;

public:
  consteval inline modint() noexcept : _val(0) {}
  constexpr inline modint(int val) noexcept
      : _val(modint_utils::normalize(val, MOD)) {}
  constexpr inline modint(modint const &val) noexcept : _val(val._val) {}

  // logic
  constexpr i32 val() const noexcept { return _val; }
  constexpr modint pow(const int n) const { return modint(powm(_val, n, MOD)); }
  constexpr inline static modint pow(const modint &a, const int n) noexcept {
    return modint(mod_pow(a._val, n, a.MOD));
  }
  constexpr inline modint inv() const noexcept {
    return modint(modint_utils::inv(_val, MOD));
  }

  // op
  constexpr inline modint &operator++() noexcept { return *this += 1; }
  constexpr inline modint &operator--() noexcept { return *this -= 1; }
  constexpr inline modint operator++(int32_t) noexcept {
    modint tmp = *this;
    ++*this;
    return tmp;
  }
  constexpr inline modint operator--(int32_t) noexcept {
    modint tmp = *this;
    --*this;
    return tmp;
  }
  constexpr inline modint operator+(const modint &a) const noexcept {
    return modint(*this) += a;
  }
  constexpr inline modint operator-(const modint &a) const noexcept {
    return modint(*this) -= a;
  }
  constexpr inline modint operator*(const modint &a) const noexcept {
    return modint(*this) *= a;
  }
  constexpr inline modint operator/(const modint &a) const noexcept {
    return modint(*this) /= a;
  }
  constexpr inline bool operator==(const modint &a) const noexcept {
    return _val == a.val();
  }
  constexpr inline bool operator!=(const modint &a) const noexcept {
    return _val != a.val();
  }
  constexpr inline modint &operator+=(const modint &a) noexcept {
    _val += a._val;
    if (_val >= MOD)
      _val -= MOD;
    return *this;
  }
  constexpr inline modint &operator-=(const modint &a) noexcept {
    _val -= a._val;
    if (_val < 0)
      _val += MOD;
    return *this;
  }
  constexpr inline modint &operator*=(const modint &a) noexcept {
    _val = (ll)_val * a._val % MOD;
    return *this;
  }
  constexpr inline modint &operator/=(const modint &a) noexcept {
    *this *= modint_utils::inv(a.val(), MOD);
    return *this;
  }

  explicit operator int() const noexcept { return _val; }

  // io
  friend ostream &operator<<(ostream &os, const modint &a) noexcept {
    return os << a._val;
  }
  friend istream &operator>>(istream &os, modint &a) noexcept {
    os >> a._val;
    a._val = modint_utils::normalize(a._val, MOD);
    return os;
  }
};
using mint998 = modint<998244353>;
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/prime.hpp"

#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/prime.hpp"

// ミラーラビン素数判定法
// O(k log^3 n)
// https://drken1215.hatenablog.com/entry/2023/05/23/233000
bool MillerRabin(long long N, vector<long long> A) {
  long long s = 0, d = N - 1;
  while (d % 2 == 0) {
    ++s;
    d >>= 1;
  }
  for (auto a : A) {
    if (N <= a)
      return true;
    long long t, x = powm(a, d, N);
    if (x != 1) {
      for (t = 0; t < s; ++t) {
        if (x == N - 1)
          break;
        x = __int128_t(x) * x % N;
      }
      if (t == s)
        return false;
    }
  }
  return true;
}

bool is_prime(long long N) {
  if (N <= 1)
    return false;
  if (N == 2)
    return true;
  if (N % 2 == 0)
    return false;
  if (N < 4759123141LL)
    return MillerRabin(N, {2, 7, 61});
  else
    return MillerRabin(N, {2, 325, 9375, 28178, 450775, 9780504, 1795265022});
}

// ポラード・ロー素因数分解法
// 値が大きい場合に有効、ちゃんと調べる!
// O(n^(1/4))
// https://lpha-z.hatenablog.com/entry/2023/01/15/231500
// https://algo-method.com/tasks/553/editorial
uint64_t pollard_rho(uint64_t N) {
  if (N % 2 == 0)
    return 2;
  if (is_prime(N))
    return N;

  uint64_t step = 0;
  auto f = [&](uint64_t x) -> uint64_t {
    return (__int128_t(x) * x + step) % N;
  };
  while (true) {
    ++step;
    uint64_t x = step, y = f(x);
    while (true) {
      uint64_t p = gcd(y - x + N, N);
      if (p == 0 || p == N)
        break;
      if (p != 1)
        return p;
      x = f(x);
      y = f(f(y));
    }
  }
}

// 素因数分解
// https://algo-method.com/tasks/553/editorial
struct PrimeFactor {
  uint64_t prime;
  uint64_t exp;
};
vec<PrimeFactor> prime_factorize(uint64_t N) {
  if (N == 1)
    return {};
  uint64_t p = pollard_rho(N);
  if (p == N)
    return {{p, 1}};
  auto left = prime_factorize(p);
  auto right = prime_factorize(N / p);
  left.insert(left.end(), all(right));
  sort(all(left), [](const PrimeFactor &a, const PrimeFactor &b) {
    return a.prime < b.prime;
  });
  vec<PrimeFactor> ans;
  rep(i, left.size()) {
    if (i == 0 || left[i].prime != left[i - 1].prime)
      ans.emplace_back(left[i]);
    else
      ans.back().exp += left[i].exp;
  }
  return ans;
}
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/math/rational.hpp"

template <typename T = ll> struct Rational {
  T num, den;
  Rational(T num) : num(num), den(1) {}
  Rational(T num, T den) : num(num), den(den) {
    T g = gcd(num, den);
    this->num /= g;
    this->den /= g;
  }
  Rational operator+(const Rational &rhs) const {
    return Rational(num * rhs.den + rhs.num * den, den * rhs.den);
  }
  Rational operator-(const Rational &rhs) const {
    return Rational(num * rhs.den - rhs.num * den, den * rhs.den);
  }
  Rational operator*(const Rational &rhs) const {
    return Rational(num * rhs.num, den * rhs.den);
  }
  Rational operator/(const Rational &rhs) const {
    return Rational(num * rhs.den, den * rhs.num);
  }

  bool operator<(const Rational &rhs) const {
    return num * rhs.den < rhs.num * den;
  }
  bool operator>(const Rational &rhs) const {
    return num * rhs.den > rhs.num * den;
  }
  bool operator==(const Rational &rhs) const {
    return num * rhs.den == rhs.num * den;
  }
  bool operator!=(const Rational &rhs) const { return !(*this == rhs); }
  bool operator<=(const Rational &rhs) const {
    return (*this < rhs) || (*this == rhs);
  }
  bool operator>=(const Rational &rhs) const {
    return (*this > rhs) || (*this == rhs);
  }
  friend ostream &operator<<(ostream &os, const Rational &r) {
    return os << r.num << "/" << r.den;
  }
  void operator+=(const Rational &rhs) { *this = *this + rhs; }
  void operator-=(const Rational &rhs) { *this = *this - rhs; }
  void operator*=(const Rational &rhs) { *this = *this * rhs; }
  void operator/=(const Rational &rhs) { *this = *this / rhs; }

  double val() const { return (double)num / den; }
  // operator double() const { return val(); }
  // operator string() const { return to_string(num) + "/" + to_string(den); }
  Rational pow(ll n) const {
    if (n == 0) {
      return Rational(1);
    } else if (n < 0) {
      return Rational(den, num).pow(-n);
    } else {
      Rational res = pow(n / 2);
      res *= res;
      if (n % 2 == 1) {
        res *= *this;
      }
      return res;
    }
  }
};
#line 9 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/all.hpp"
// #include "string/all.hpp"
#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/util/all.hpp"

#line 2 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/util/inversion-num.hpp"

#line 5 "/home/zoi/ghq/github.com/ZOI-dayo/atcoder-library/util/inversion-num.hpp"

int inversion_num(const vec<int> &a) {
  comp<int> c(a);

  int n = a.size();
  FenwickTree<int> bit(n);
  int ans = 0;
  rep(i, n) {
    ans += i - bit.sum(c[i]);
    bit.add(c[i], 1);
  }
  return ans;
}
#line 3 "main.cpp"

using mint = modint<998'244'353>;

void solve() {
  int N, M, W;
  cin >> N >> M >> W;
  vec<int> A(N);
  vec<int> B(M);
  vec<int> C(M);
  cin >> A >> B >> C;
  vec<pair<int, int>> objects(N+M);
  rep(i, N) {
    objects[i] = {A[i], 1};
  }
  rep(i, M) {
    objects[N+i] = {C[i], B[i]};
  }

  vvec<int> dp(N+M+1, vec<int>(W+1, -1));
  rep(i, N+M+1) {
    dp[i][0] = 0;
  }
  rep(i, N+M) {
    rep(w, W+1) {
      if (dp[i][w] == -1) continue;
      chmax(dp[i+1][w], dp[i][w]);
      if (w + objects[i].sec <= W) {
        chmax(dp[i+1][w+objects[i].sec], dp[i][w] + objects[i].fir);
      }
    }
  }
  cout << *max_element(all(dp[N+M]))<< endl;
}

void test() {
}

signed main() {
  io_setup();
  // input
  
  solve();
  // test();
}