// #include <bits/stdc++.h>
#include <algorithm>
#include <atcoder/all>
#include <bitset>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <iomanip>
#include <iostream>
#include <limits>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <vector>

using namespace std;
using namespace atcoder;
using mint1 = modint1000000007;
using mint2 = modint998244353;
using ll = long long;
using VI = vector<int>;
using VL = vector<ll>;
using VD = vector<double>;
using VS = vector<string>;
using VB = vector<bool>;
using VVB = vector<vector<bool>>;
using VVI = vector<VI>;
using VVL = vector<VL>;
using VVD = vector<VD>;
using VVVI = vector<VVI>;
using VVVL = vector<VVL>;
using VVVD = vector<VVD>;
using PII = std::pair<int, int>;
using VPII = std::vector<std::pair<int, int>>;
using PLL = std::pair<ll, ll>;
using VPLL = std::vector<std::pair<ll, ll>>;
using TI3 = std::tuple<int, int, int>;
using TI4 = std::tuple<int, int, int, int>;
using TL3 = std::tuple<ll, ll, ll>;
using TL4 = std::tuple<ll, ll, ll, ll>;

#define rep(i, n) for (int i = 0; i < (int)(n); i++)
#define repr(i, n) for (int i = (int)(n)-1; i >= 0; i--)
#define rep2(i, s, n) for (int i = (s); i < (int)(n); i++)
#define repr2(i, g, n) for (int i = (int)(n)-1; i >= (g); i--)
#define rep3(i, s, n, d) for (int i = (s); i < (int)(n); i += (d))
#define repr3(i, g, n, d) for (int i = (int)(n)-1; i >= (g); i -= (d))
#define allpt(v) (v).begin(), (v).end()
#define allpt_c(v) (v).cbegin(), (v).cend()
#define allpt_r(v) (v).rbegin(), (v).rend()
#define allpt_cr(v) (v).crbegin(), (v).crend()

constexpr int mod1 = 1e9 + 7, mod2 = 998244353, mod3 = 1e9 + 9;
constexpr int mod = mod2;
constexpr ll inf = 1e18;
constexpr int infint = 1e9;
constexpr double pi = 3.14159265359;

bool test;
const string wsp = " ";
const string tb = "\t";
const string rt = "\n";
const string alphabets = "abcdefghijklmnopqrstuvwxyz";
std::random_device seed_gen;
std::mt19937 engine(seed_gen());
// 乱数生成用メルセンヌ・ツイスタエンジン

template <typename T>
void operator++(vector<T> &v) {
  for (auto &x : v) ++x;
}

template <typename T>
void operator--(vector<T> &v) {
  for (auto &x : v) --x;
}

template <typename T, typename S>
void operator++(pair<T, S> &v) {
  ++v.first, ++v.second;
}

template <typename T, typename S>
void operator--(pair<T, S> &v) {
  --v.first, --v.second;
}

int get1dcoodinate(int w, int i, int j) { return i * w + j; }

template <typename T, typename S>
istream &operator>>(istream &is, pair<T, S> &v) {
  is >> v.first >> v.second;
  return is;
}

template <typename T, typename S>
ostream &operator<<(ostream &os, pair<T, S> &v) {
  os << v.first << wsp << v.second << rt;
  return os;
}

ostream &operator<<(ostream &os, modint &v) {
  os << v.val();
  return os;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
  if (v.size() == 0) return os;
  int n = (int)v.size() - 1;
  rep(i, n) os << v[i] << wsp;
  os << v[n] << rt;
  return os;
}

template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
  if (v.size() == 0) return is;
  int n = v.size();
  rep(i, n) is >> v[i];
  return is;
}

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

template <typename T>
istream &operator>>(istream &is, vector<vector<T>> &v) {
  if (v.size() == 0) return is;
  int n = v.size();
  rep(i, n) is >> v[i];
  return is;
}

template <typename T, typename S>
istream &operator>>(istream &is, vector<pair<T, S>> &v) {
  if (v.size() == 0) return is;
  int n = v.size();
  rep(i, n) is >> v[i].first >> v[i].second;
  return is;
}

template <typename T, typename S>
ostream &operator<<(ostream &os, const vector<pair<T, S>> &v) {
  if (v.size() == 0) return os;
  int n = v.size();
  rep(i, n) os << v[i].first << wsp << v[i].second << rt;
  return os;
}

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

template <typename T>
void range_sort(vector<T> &arr, int l, int r) {
  sort(arr.begin() + l, arr.begin() + r);
}

template <typename T, typename S>
void pairzip(const vector<pair<T, S>> &v, vector<T> &t, vector<T> &s) {
  int n = v.size();
  rep(i, n) {
    t.push_back(v[i].first);
    s.push_back(v[i].second);
  }
  return;
}

template <typename T, typename S>
bool myfind(T t, S s) {
  return find(t.cbegin(), t.cend(), s) != t.cend();
}

bool check(int y, int x, int h, int w) {
  return 0 <= y && y < h && 0 <= x && x < w;
}

bool check(int y, int x, int h1, int h2, int w1, int w2) {
  return h1 <= y && y < h2 && w1 <= x && x < w2;
}

bool iskadomatsu(int a, int b, int c) {
  return (a != b && b != c && c != a) && ((a > b && b < c) || (a < b && b > c));
}

template <typename T>
bool iskadomatsu(vector<T> v) {
  T a = v[0], b = v[1], c = v[2];
  return (a != b && b != c && c != a) && ((a > b && b < c) || (a < b && b > c));
}

double euc_dist(PII a, PII b) {
  return sqrt(pow(a.first - b.first, 2) + pow(a.second - b.second, 2));
}

// VS split(string s, char c) {
//     VS ret;
//     string part;
//     s += c;
//     rep(i, s.length()) {
//         if (s[i] == c) {
//             if (part != "") ret.emplace_back(part);
//             part = "";
//         } else if (s[i] != c) {
//             part += s[i];
//         }
//     }
//     return ret;
// }

VS split(string s, char c, char d = ' ') {
  VS ret;
  // reverse(allpt(s));
  string part;
  s += c;
  rep(i, s.length()) {
    if (s[i] == c) {
      if (part != "") {
        // string t;
        // t += c;
        // ret.emplace_back(t);
        ret.emplace_back(part);
      }
      part = "";
    } else {
      part += s[i];
    }
  }
  return ret;
}

template <typename T, typename S, typename R>
ll pow_mod(T p, S q, R _mod = 1ll) {
  ll ret = 1, r = p % _mod;
  while (q) {
    if (q % 2) ret *= r, ret %= _mod;
    r = (r * r) % _mod, q /= 2;
  }
  return ret % _mod;
}

template <typename T, typename S>
ll pow_no_mod(T p, S q) {
  ll ret = 1, r = p;
  while (q) {
    if (q % 2) ret *= r;
    r = (r * r), q /= 2;
  }
  return ret;
}

pair<VL, VL> make_frac_tables(int n) {
  VL frac_list(n + 1, 1), frac_inv_list(n + 1, 1);
  rep(i, n) {
    frac_list[i + 1] *= frac_list[i] * (i + 1);
    frac_list[i + 1] %= mod;
    frac_inv_list[i + 1] *= frac_inv_list[i] * pow_mod(i + 1, mod - 2, mod);
    frac_inv_list[i + 1] %= mod;
  }
  return make_pair(frac_list, frac_inv_list);
}

ll perm(int a, int b, const VL &frac_list, const VL &frac_inv_list) {
  if (a < b) return 0;
  if (b < 0) return 0;
  ll ret = frac_list[a];
  ret *= frac_inv_list[a - b];
  ret %= mod;
  return ret;
}

ll comb(int a, int b, const VL &frac_list, const VL &frac_inv_list) {
  if (a < b) return 0;
  if (b < 0) return 0;
  ll ret = frac_list[a];
  ret *= frac_inv_list[b];
  ret %= mod;
  ret *= frac_inv_list[a - b];
  ret %= mod;
  return ret;
}

struct vec2d {
  ll x;
  ll y;
  vec2d(ll _x, ll _y) {
    x = _x;
    y = _y;
  }
  ll dot(vec2d p) { return x * p.x + y * p.y; }
  vec2d diff(vec2d p) { return vec2d(x - p.x, y - p.y); }
};

struct node {
  int parent = -1;
  ll weight = 0;
  int depth = 0;
  int degree = 0;
  int subtree = 0;
  int check = 0;
  int scc = -1;
  VPLL children;
  VI parent_list;
  VPLL connect;
  ll black_from_children;
  ll black_from_parent;

  node() {
    parent = -1;
    weight = 0;
    depth = 0;
    degree = 0;
    subtree = 0;
    check = 0;
    black_from_children = 1;
    black_from_parent = 1;
  }
};

struct graph {
  int _n;
  int root = 0;
  vector<node> nodes;
  graph(int n) {
    _n = n;
    rep(i, _n) nodes.emplace_back(node());
  }
  void getconnect_nocost() {
    ll a, b;
    cin >> a >> b;
    nodes[a].connect.emplace_back(b, 1);
    nodes[b].connect.emplace_back(a, 1);
  }
  void getconnect_cost() {
    ll a, b, c;
    cin >> a >> b >> c;
    nodes[a].connect.emplace_back(b, c);
    nodes[b].connect.emplace_back(a, c);
  }
  void getconnect_increment_nocost() {
    ll a, b;
    cin >> a >> b;
    a--, b--;
    nodes[a].connect.emplace_back(b, 1);
    nodes[b].connect.emplace_back(a, 1);
  }
  void getconnect_increment_cost() {
    ll a, b, c;
    cin >> a >> b >> c;
    a--, b--;
    nodes[a].connect.emplace_back(b, c);
    nodes[b].connect.emplace_back(a, c);
  }
  void showparent() {
    rep(i, _n - 1) cout << nodes[i].parent << wsp;
    cout << nodes[_n - 1].parent << rt;
  }
  void showweight() {
    rep(i, _n - 1) cout << nodes[i].weight << wsp;
    cout << nodes[_n - 1].weight << rt;
  }
  void showsubtree() {
    rep(i, _n - 1) cout << nodes[i].subtree << wsp;
    cout << nodes[_n - 1].subtree << rt;
  }
  void showdepth() {
    rep(i, _n - 1) cout << nodes[i].depth << wsp;
    cout << nodes[_n - 1].depth << rt;
  }
  // pair<VL, VI> djcstra(int s = 0) {
  //     VL shortest(_n, inf);
  //     VI prev(_n, -1);
  //     shortest[s] = 0;
  //     priority_queue<pair<ll, int>> pq;
  //     pq.push(make_pair(0, s));
  //     while (!pq.empty()) {
  //         auto [t, u] = pq.top();
  //         pq.pop();
  //         t = -t;
  //         if (shortest[u] != t) continue;
  //         for (auto [v, c] : nodes[u].connect) {
  //             if (t + c < shortest[v]) {
  //                 shortest[v] = t + c;
  //                 prev[v] = u;
  //                 pq.push(make_pair(-(t + c), v));
  //             }
  //         }
  //     }
  //     return make_pair(shortest, prev);
  // }
  pair<VL, VI> djcstra(int s = 0) {
    VL shortest(_n, inf);
    VI prev(_n, -1);
    shortest[s] = 0;
    priority_queue<pair<ll, int>> pq;
    pq.push(make_pair(0, s));
    while (!pq.empty()) {
      auto [t, u] = pq.top();
      pq.pop();
      t = -t;
      if (shortest[u] != t) continue;
      for (auto [v, c] : nodes[u].connect) {
        if (t + c < shortest[v]) {
          shortest[v] = t + c;
          prev[v] = u;
          pq.push(make_pair(-(t + c), v));
        }
      }
    }
    return make_pair(shortest, prev);
  }

  // pair<VL, VI> djcstra(VI s) {
  //   VL shortest(_n, inf);
  //   VI prev(_n, -1);
  //   priority_queue<pair<ll, int>> pq;
  //   for (auto x : s) {
  //     shortest[x] = 0;
  //     pq.push(make_pair(0, x));
  //   }
  //   while (!pq.empty()) {
  //     auto [t, u] = pq.top();
  //     pq.pop();
  //     t = -t;
  //     if (shortest[u] != t) continue;
  //     for (auto [v, c] : nodes[u].connect) {
  //       if (t + c < shortest[v]) {
  //         shortest[v] = t + c;
  //         prev[v] = u;
  //         pq.push(make_pair(-(t + c), v));
  //       }
  //     }
  //   }
  //   return make_pair(shortest, prev);
  // }

  // VI djcstra(VI s, const VI x_list) {
  //   VL shortest(_n, inf);
  //   VI prev(_n, -1);
  //   VI new_infection;
  //   priority_queue<pair<ll, PII>> pq;
  //   for (auto si : s) {
  //     shortest[si] = 0;
  //     pq.push(make_pair(0, make_pair(si, -x_list[si])));
  //   }
  //   while (!pq.empty()) {
  //     auto [t, p] = pq.top();
  //     auto [u, x] = p;
  //     pq.pop();
  //     t = -t, x = -x;
  //     if (shortest[u] != t) continue;
  //     for (auto [v, d] : nodes[u].connect) {
  //       if (t + d < shortest[v] && x >= d) {
  //         shortest[v] = t + d;
  //         prev[v] = u;
  //         pq.push(make_pair(-(t + d), make_pair(v, -(x - d))));
  //       }
  //     }
  //   }
  //   return new_infection;
  // }
};

struct point {
  int x;
  int y;
  point() {
    x = 0;
    y = 0;
  }
  point(int _x, int _y) {
    x = _x;
    y = _y;
  }
  void pointinput() {
    int _x, _y;
    cin >> _x >> _y;
    x = _x;
    y = _y;
  }
  void pointinv() { swap(x, y); }
};

istream &operator>>(istream &is, point &p) {
  is >> p.x >> p.y;
  return is;
}

ostream &operator<<(ostream &os, point &p) {
  os << p.x << wsp << p.y << rt;
  return os;
}

double pointseuc(point a, point b) {
  ll ax = a.x, bx = b.x, ay = a.y, by = b.y;
  return sqrt(pow(ax - bx, 2) + pow(ay - by, 2));
}

ll pointseucsquare(point a, point b) {
  ll ax = a.x, bx = b.x, ay = a.y, by = b.y;
  return (ax - bx) * (ax - bx) + (ay - by) * (ay - by);
}

int pointsmanhattan(point a, point b) {
  return abs(a.x - b.x) + abs(a.y - b.y);
}

double dist_segment_point(TL3 segment, point p) {
  double a = get<0>(segment);
  double b = get<1>(segment);
  double c = get<2>(segment);
  return abs(a * p.x + b * p.y - c) / sqrt(a * a + b * b);
}

TL3 segment_parameter(point p, point q) {
  ll a, b, c;
  a = q.y - p.y;
  b = p.x - q.x;
  c = a * p.x + b * p.y;
  TL3 ret = (TL3){a, b, c};
  // cout << a << b << c << rt;
  return ret;
}

int cross_check(TL3 segment, point p) {
  ll a = get<0>(segment);
  ll b = get<1>(segment);
  ll c = get<2>(segment);

  auto f = a * p.x + b * p.y - c;
  int ret;
  if (f > 0) ret = 1;
  if (f == 0) ret = 0;
  if (f < 0) ret = -1;
  return ret;
}

VI shave(int n) {
  VI v;
  if (n <= 1) return v;
  vector<bool> w(n + 1, true);
  int x;
  w[0] = w[1] = false;
  rep2(i, 2, n + 1) {
    if (w[i]) {
      x = i * 2;
      while (x <= n) {
        w[x] = false;
        x += i;
      }
    }
  }
  rep(i, n + 1) if (w[i]) v.emplace_back(i);
  return v;
}

bool iscross(point p1, point p2, point p3, point p4) {
  auto segment_1 = segment_parameter(p1, p2);
  auto segment_2 = segment_parameter(p3, p4);
  const double eps = 1e-6;
  bool ans{false};
  if (cross_check(segment_1, p3) * cross_check(segment_1, p4) == 0 &&
      cross_check(segment_2, p1) * cross_check(segment_2, p2) == 0) {
    auto xmin_1 = min(p1.x, p2.x) - eps;
    auto xmin_2 = min(p3.x, p4.x) - eps;
    auto xmax_1 = max(p1.x, p2.x) + eps;
    auto xmax_2 = max(p3.x, p4.x) + eps;
    auto ymin_1 = min(p1.y, p2.y) - eps;
    auto ymin_2 = min(p3.y, p4.y) - eps;
    auto ymax_1 = max(p1.y, p2.y) + eps;
    auto ymax_2 = max(p3.y, p4.y) + eps;

    if (xmin_1 <= p3.x && p3.x <= xmax_1 && ymin_1 <= p3.y && p3.y <= ymax_1)
      ans = true;
    if (xmin_1 <= p4.x && p4.x <= xmax_1 && ymin_1 <= p4.y && p4.y <= ymax_1)
      ans = true;
    if (xmin_2 <= p1.x && p1.x <= xmax_2 && ymin_2 <= p1.y && p1.y <= ymax_2)
      ans = true;
    if (xmin_2 <= p2.x && p2.x <= xmax_2 && ymin_2 <= p2.y && p2.y <= ymax_2)
      ans = true;
  } else if (cross_check(segment_1, p3) * cross_check(segment_1, p4) <= 0 &&
             cross_check(segment_2, p1) * cross_check(segment_2, p2) <= 0) {
    ans = true;
  }
  return ans;
}

// int on_the_segment(point p, point q1, point q2) {
//     auto [a, b, c] = segment_parameter(q1, q2);

//     if (p.x < min(q1.x, q2.x) || max(q1.x, q2.x) < p.x) return 0;
//     if (p.y < min(q1.y, q2.y) || max(q1.y, q2.y) < p.y) return 0;

//     auto f = a * p.x + b * p.y - c;
//     return f == 0;
// }

void shave(vector<int> &v, int n) {
  if (n <= 1) return;
  vector<bool> w(n + 1, true);
  int x;
  w[0] = w[1] = false;
  rep2(i, 2, n + 1) {
    if (w[i]) {
      x = i * 2;
      while (x <= n) {
        w[x] = false;
        x += i;
      }
    }
  }
  rep(i, n + 1) if (w[i]) v.emplace_back(i);
}

template <typename T>
void sort_and_unique(vector<T> &x) {
  sort(x.begin(), x.end());
  x.erase(unique(x.begin(), x.end()), x.end());
}

template <typename T>
void coordinate_compress(vector<T> &x, map<T, int> &zip, int &xs) {
  sort_and_unique(x);
  xs = x.size();
  for (int i = 0; i < xs; i++) zip[x[i]] = i;
}

template <typename T>
pair<map<T, int>, int> coordinate_compress(vector<T> &x) {
  map<T, int> zip;
  sort_and_unique(x);
  int xs = x.size();
  for (int i = 0; i < xs; i++) zip[x[i]] = i;
  return make_pair(zip, xs);
}

template <typename T>
pair<map<T, int>, int> copy_coordinate_compress(vector<T> &x) {
  VI x_copy = x;
  auto [x_zip, xl] = coordinate_compress(x_copy);
  for (auto &v : x) v = x_zip[v];
  return make_pair(x_zip, xl);
}

// ll getpalindrome(int l, int r, const string &s, VL &memo) {
//     if (r - l <= 1) return 1;
//     if (memo[l] != -1) return memo[l];
//     ll ret{1};
//     int n = (r - l) / 2;
//     string sl, sr;
//     rep(i, n) {
//         sl += s[l + i];
//         sr += s[r - 1 - i];
//         string rsr(i + 1, '0');
//         reverse_copy(allpt_c(sr), rsr.begin());
//         // cout << sl << wsp << sr << wsp << rsr << rt;
//         if (sl == rsr) {
//             ret += getpalindrome(l + i + 1, r - i - 1, s, memo);
//             ret %= mod;
//         }
//     }
//     memo[l] = ret;
//     return ret;
// }

// void getpalindrome(int l, int r, int n, const string &s, VVB &memo) {
//     if (r - l <= 1 || s[l] == s[r - 1]) {
//         memo[l][r] = true;
//         if (l > 0 && r < n) {
//             getpalindrome(l - 1, r + 1, n, s, memo);
//         }
//     }
//     return;
// }

class Unionfind {
  vector<int> p;
  vector<int> w;

 public:
  Unionfind(int n, int _w = 1) {
    for (int i = 0; i < n; i++) {
      p.emplace_back(i);
      w.emplace_back(_w);
    }
  }
  int find(int x) {
    while (p[x] != x) {
      p[x] = p[p[x]];
      x = p[x];
    }
    return x;
  }
  int getval(int x) { return p[x]; }
  int getweight(int x) { return w[find(x)]; }
  void setweight(int x, int _w) { w[find(x)] += _w; }
  void unite(int x, int y) {
    x = find(x);
    y = find(y);

    if (x != y) {
      p[x] = min(x, y);
      p[y] = min(x, y);
      w[min(x, y)] += w[max(x, y)];
      w[max(x, y)] = 0;
    }
  }

  bool isunion(int x, int y) { return find(x) == find(y); }
  void showtree() { cout << p << w; }
};

template <typename T>
class RangeMinorMaxorSumQuery  // 0-index
{
  T const intmax = inf;
  T const intmin = 0;
  vector<T> sgt;
  int n;
  int k;

 public:
  RangeMinorMaxorSumQuery(int n1, T f) {
    int na = 1;
    int ka = 0;
    while (na < n1) {
      na *= 2;
      ka++;
    }
    for (int i = 0; i < 2 * na; i++) sgt.push_back(f);
    n = na;
    k = ka;
  }

  void update_min(int i, T x) {
    i += n;
    sgt[i] = x;
    while (i > 1) {
      i /= 2;
      sgt[i] = min(sgt[2 * i], sgt[2 * i + 1]);
    }
  }
  void update_max(int i, T x) {
    i += n;
    sgt[i] = x;
    while (i > 1) {
      i /= 2;
      sgt[i] = max(sgt[2 * i], sgt[2 * i + 1]);
    }
  }
  void update_sum(int i, T x) {
    i += n;
    sgt[i] = x;
    while (i > 1) {
      i /= 2;
      sgt[i] = sgt[2 * i] + sgt[2 * i + 1];
    }
  }
  void update_gcd(int i, T x) {
    i += n;
    sgt[i] = x;
    while (i > 1) {
      i /= 2;
      sgt[i] = gcd(sgt[2 * i], sgt[2 * i + 1]);
    }
  }
  void add_sum(int i, T x) {
    i += n;
    sgt[i] += x;
    while (i > 1) {
      i /= 2;
      sgt[i] = sgt[2 * i] + sgt[2 * i + 1];
    }
  }

  void add_min(int i, T x) {
    i += n;
    sgt[i] += x;
    while (i > 1) {
      i /= 2;
      sgt[i] = min(sgt[2 * i], sgt[2 * i + 1]);
    }
  }

  void add_max(int i, T x) {
    i += n;
    sgt[i] += x;
    while (i > 1) {
      i /= 2;
      sgt[i] = max(sgt[2 * i], sgt[2 * i + 1]);
    }
  }

  T get_min(int a, int b, int k = 1, int l = 0,
            int r = -1)  // 閉区間 l <= x < r とする
  {
    if (r == -1) r = n;
    if (r <= a || b <= l) return inf;
    if (a == l && b == r)
      return sgt[k];
    else
      return min(get_min(a, min(b, (l + r) / 2), 2 * k, l, (l + r) / 2),
                 get_min(max(a, (l + r) / 2), b, 2 * k + 1, (l + r) / 2, r));
  }
  T get_max(int a, int b, int k = 1, int l = 0,
            int r = -1)  // 閉区間 l <= x < r とする
  {
    if (r == -1) r = n;
    if (r <= a || b <= l) return 0;
    if (a == l && b == r)
      return sgt[k];
    else
      return max(get_max(a, min(b, (l + r) / 2), 2 * k, l, (l + r) / 2),
                 get_max(max(a, (l + r) / 2), b, 2 * k + 1, (l + r) / 2, r));
  }
  T get_sum(int a, int b, int k = 1, int l = 0,
            int r = -1)  // 閉区間 l <= x < r とする
  {
    if (r == -1) r = n;
    if (r <= a || b <= l) return intmin;
    if (a == l && b == r)
      return sgt[k];
    else
      return get_sum(a, min(b, (l + r) / 2), 2 * k, l, (l + r) / 2) +
             get_sum(max(a, (l + r) / 2), b, 2 * k + 1, (l + r) / 2, r);
  }
  T get_gcd(int a, int b, int k = 1, int l = 0,
            int r = -1)  // 閉区間 l <= x < r とする
  {
    if (r == -1) r = n;
    if (r <= a || b <= l) return 0;
    if (a == l && b == r)
      return sgt[k];
    else
      return gcd(get_gcd(a, min(b, (l + r) / 2), 2 * k, l, (l + r) / 2),
                 get_gcd(max(a, (l + r) / 2), b, 2 * k + 1, (l + r) / 2, r));
  }

  void update_xor(int i, T x) {
    i += n;
    sgt[i] = x;
    while (i > 1) {
      i /= 2;
      sgt[i] = sgt[2 * i] ^ sgt[2 * i + 1];
    }
  }

  T get_xor(int a, int b, int k = 1, int l = 0,
            int r = -1)  // 閉区間 l <= x < r とする
  {
    if (r == -1) r = n;
    if (r <= a || b <= l) return 0;
    if (a == l && b == r)
      return sgt[k];
    else
      return get_xor(a, min(b, (l + r) / 2), 2 * k, l, (l + r) / 2) ^
             get_xor(max(a, (l + r) / 2), b, 2 * k + 1, (l + r) / 2, r);
  }

  T operator[](int i) { return sgt[i + n]; }

  void printsegtree() {
    for (int i = 0; i < 2 * n; i++) {
      cout << sgt[i] << " ";
    }
    cout << endl;
  }
};

template <typename T>
class Bit_Index_Tree  // 1-index
{
  int n = 0;
  vector<T> v;

 public:
  void add(int i, T x) {
    while (i <= n) {
      v[i] += x;
      i += i & -i;
    }
  }
  Bit_Index_Tree(int _n, T x = 0) {
    n = _n;
    v.resize(n + 1);
    fill(allpt(v), x);
  }
  Bit_Index_Tree(vector<T> _v) {
    n = _v.size();
    v.resize(n + 1);
    fill(allpt(v), 0);
    rep(i, n) add(i, v[i]);
  }
  T get_sum(int i) {
    T ret = 0;
    while (i > 0) {
      ret += v[i];
      i -= i & -i;
    }
    return ret;
  }
  T get_range_sum(int l, int r) {  // 半開区間　l <= x < r
    return get_sum(r) - get_sum(l);
  }
};

// #define DEBUG

VI smallest_prime_factors(int n) {
  VI ret(n + 1);
  iota(allpt(ret), 0);
  for (int i = 2; i * i <= n; i++) {
    if (ret[i] == i) {
      for (int j = i * i; j <= n; j += i) {
        if (ret[j] == j) ret[j] = i;
      }
    }
  }
  return ret;
}

// void dfs_1(graph &tree, int u)
// {
//     cout << u << rt;
//     for (auto [v, _] : tree.nodes[u].connect)
//         if (v != tree.root && tree.nodes[v].parent == -1)
//         {
//             tree.nodes[v].parent = u;
//             dfs_1(tree, v);
//             tree.nodes[u].subtree += tree.nodes[v].subtree;
//             tree.nodes[u].children.emplace_back(PII(tree.nodes[v].subtree,
//             v));
//         }
// }

// void dfs_2(graph &tree, int u, int d)
// {
//     cout << u << wsp << d << rt;

//     sort(allpt_r(tree.nodes[u].children));
//     rep(j, tree.nodes[u].children.size())
//     {
//         auto [_, v] = tree.nodes[u].children[j];
//         cout << (VI){u, (int)v, j};
//         dfs_2(tree, v, j == 0 ? d : d + 1);
//     }
// }

template <typename T>
class Matrix_n {
  int matsize;
  vector<vector<T>> mat;

 public:
  Matrix_n() {
    matsize = 0;
    vector<vector<T>> mat;
  }
  Matrix_n(int n) {
    matsize = n;
    mat.resize(n);
    rep(i, n) rep(j, n) mat[i].emplace_back(0);
  }
  Matrix_n(const vector<vector<T>> &inmat) {
    int n = inmat.size();
    matsize = n;
    mat.resize(n);
    rep(i, n) {
      mat[i].resize(n);
      rep(j, n) { mat[i][j] = inmat[i][j]; }
    }
  }
  void identify() {
    rep(i, matsize) rep(j, matsize) { mat[i][j] = i == j ? 1 : 0; }
  }
  T getone(const int &i, const int &j) { return mat[i][j]; }

  int getsize() { return matsize; }

  const Matrix_n operator+(const Matrix_n &b) {
    Matrix_n ret(matsize);
    rep(i, matsize) rep(j, matsize) { ret.mat[i][j] = mat[i][j] + b.mat[i][j]; }
    return ret;
  }

  const Matrix_n operator-(const Matrix_n &b) {
    Matrix_n ret(matsize);
    rep(i, matsize) rep(j, matsize) { ret.mat[i][j] = mat[i][j] - b.mat[i][j]; }
    return ret;
  }

  const Matrix_n operator*(const Matrix_n &b) {
    Matrix_n ret(matsize);
    rep(i, matsize) rep(j, matsize) rep(k, matsize) {
      ret.mat[i][j] += mat[i][k] * b.mat[k][j];
      ret.mat[i][j] %= mod;
    }
    return ret;
  }

  const Matrix_n operator%(const int &b) {
    Matrix_n ret(matsize);
    rep(i, matsize) rep(j, matsize) { ret.mat[i][j] = mat[i][j] % b; }
    return ret;
  }

  const Matrix_n operator+() { return *this; }

  const Matrix_n operator-() {
    Matrix_n ret(matsize);
    rep(i, matsize) rep(j, matsize) { ret.mat[i][j] = -mat[i][j]; }
    return ret;
  }

  Matrix_n &operator+=(const Matrix_n &b) {
    rep(i, matsize) rep(j, matsize) { mat[i][j] += b.mat[i][j]; }
    return *this;
  }

  Matrix_n &operator-=(const Matrix_n &b) {
    rep(i, matsize) rep(j, matsize) { mat[i][j] -= b.mat[i][j]; }
    return *this;
  }

  Matrix_n &operator%=(const int &b) {
    rep(i, matsize) rep(j, matsize) { mat[i][j] %= b; }
    return *this;
  }

  void showmat() { cout << mat; }
};

template <typename T>
Matrix_n<T> mat_pow_mod(Matrix_n<T> a, ll p, ll m = mod) {
  const int n = a.getsize();
  Matrix_n<T> b(n);
  b.identify();

  while (p > 0) {
    if (p % 2) {
      b = b * a;
      b %= m;
    }
    p /= 2;
    a = a * a;
    a %= m;
  }
  return b;
}

template <typename T>
Matrix_n<T> mat_pow_no_mod(Matrix_n<T> a, ll p) {
  const int n = a.getsize();
  Matrix_n<T> b(n);
  b.identify();

  while (p > 0) {
    if (p % 2) {
      b = b * a;
    }
    p /= 2;
    a = a * a;
  }
  return b;
}

VL make_frac_k(int k, int l) {
  VL ret(l + 1, 1);
  rep(i, l) {
    ret[i + 1] = ret[i] * k;
    ret[i + 1] %= mod;
  }
  return ret;
}

template <typename T, typename S>
void ans_print(T ans, T nodata, S printdata) {
  if (ans == nodata) {
    cout << printdata;
  } else {
    cout << ans;
  }
  cout << endl;
}

int mex(VI v) {
  sort_and_unique(v);
  int n = v.size();
  rep(i, n) {
    if (v[i] != i) return i;
  }
  return n;
}


int main(int argc, char *argv[]) {
  // cin.tie(0);
  // ios::sync_with_stdio(false);
  // リアクティブ問題のときはコメントアウト

  int n, a_gt_b{0};
  cin >> n;
  VI a_list(n), b_list(n);
  cin >> a_list >> b_list;
  sort(allpt(a_list));
  sort(allpt_r(b_list));
  rep(i, n) {
    if (a_list[i] > b_list[i]) ++a_gt_b;
  }
  auto [frac, frac_inv] = make_frac_tables(n);
  cout << frac[a_gt_b] * frac[n - a_gt_b] % mod << rt;


  

  return 0;
}