Problem p02539 (C++) — buggy submission, verdict: Time Limit Exceeded


#include <bits/stdc++.h>
// #include "bits/stdc++.h"

#include <algorithm>
#include <array>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
  int x = 0;
  while ((1U << x) < (unsigned int)(n))
    x++;
  return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
  unsigned long index;
  _BitScanForward(&index, n);
  return index;
#else
  return __builtin_ctz(n);
#endif
}

} // namespace internal

} // namespace atcoder

#include <utility>

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
  x %= m;
  if (x < 0)
    x += m;
  return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
  unsigned int _m;
  unsigned long long im;

  // @param m `1 <= m < 2^31`
  barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

  // @return m
  unsigned int umod() const { return _m; }

  // @param a `0 <= a < m`
  // @param b `0 <= b < m`
  // @return `a * b % m`
  unsigned int mul(unsigned int a, unsigned int b) const {
    // [1] m = 1
    // a = b = im = 0, so okay

    // [2] m >= 2
    // im = ceil(2^64 / m)
    // -> im * m = 2^64 + r (0 <= r < m)
    // let z = a*b = c*m + d (0 <= c, d < m)
    // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
    // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) <
    // 2^64 * 2
    // ((ab * im) >> 64) == c or c + 1
    unsigned long long z = a;
    z *= b;
#ifdef _MSC_VER
    unsigned long long x;
    _umul128(z, im, &x);
#else
    unsigned long long x =
        (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
    unsigned int v = (unsigned int)(z - x * _m);
    if (_m <= v)
      v += _m;
    return v;
  }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
  if (m == 1)
    return 0;
  unsigned int _m = (unsigned int)(m);
  unsigned long long r = 1;
  unsigned long long y = safe_mod(x, m);
  while (n) {
    if (n & 1)
      r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  long long d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  constexpr long long bases[3] = {2, 7, 61};
  for (long long a : bases) {
    long long t = d;
    long long y = pow_mod_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1) {
      y = y * y % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0) {
      return false;
    }
  }
  return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
  a = safe_mod(a, b);
  if (a == 0)
    return {b, 0};

  // Contracts:
  // [1] s - m0 * a = 0 (mod b)
  // [2] t - m1 * a = 0 (mod b)
  // [3] s * |m1| + t * |m0| <= b
  long long s = b, t = a;
  long long m0 = 0, m1 = 1;

  while (t) {
    long long u = s / t;
    s -= t * u;
    m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b

    // [3]:
    // (s - t * u) * |m1| + t * |m0 - m1 * u|
    // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
    // = s * |m1| + t * |m0| <= b

    auto tmp = s;
    s = t;
    t = tmp;
    tmp = m0;
    m0 = m1;
    m1 = tmp;
  }
  // by [3]: |m0| <= b/g
  // by g != b: |m0| < b/g
  if (m0 < 0)
    m0 += b / s;
  return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
  if (m == 2)
    return 1;
  if (m == 167772161)
    return 3;
  if (m == 469762049)
    return 3;
  if (m == 754974721)
    return 11;
  if (m == 998244353)
    return 3;
  int divs[20] = {};
  divs[0] = 2;
  int cnt = 1;
  int x = (m - 1) / 2;
  while (x % 2 == 0)
    x /= 2;
  for (int i = 3; (long long)(i)*i <= x; i += 2) {
    if (x % i == 0) {
      divs[cnt++] = i;
      while (x % i == 0) {
        x /= i;
      }
    }
  }
  if (x > 1) {
    divs[cnt++] = x;
  }
  for (int g = 2;; g++) {
    bool ok = true;
    for (int i = 0; i < cnt; i++) {
      if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
        ok = false;
        break;
      }
    }
    if (ok)
      return g;
  }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral =
    typename std::conditional<std::is_integral<T>::value ||
                                  is_signed_int128<T>::value ||
                                  is_unsigned_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_signed_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_signed<T>::value) ||
                                  is_signed_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value, make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using to_unsigned =
    typename std::conditional<is_signed_int<T>::value, std::make_unsigned<T>,
                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

} // namespace internal

template <int m, std::enable_if_t<(1 <= m)> * = nullptr>
struct static_modint : internal::static_modint_base {
  using mint = static_modint;

public:
  static constexpr int mod() { return m; }
  static mint raw(int v) {
    mint x;
    x._v = v;
    return x;
  }

  static_modint() : _v(0) {}
  template <class T, internal::is_signed_int_t<T> * = nullptr>
  static_modint(T v) {
    long long x = (long long)(v % (long long)(umod()));
    if (x < 0)
      x += umod();
    _v = (unsigned int)(x);
  }
  template <class T, internal::is_unsigned_int_t<T> * = nullptr>
  static_modint(T v) {
    _v = (unsigned int)(v % umod());
  }
  static_modint(bool v) { _v = ((unsigned int)(v) % umod()); }

  unsigned int val() const { return _v; }

  mint &operator++() {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint &operator--() {
    if (_v == 0)
      _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int) {
    mint result = *this;
    ++*this;
    return result;
  }
  mint operator--(int) {
    mint result = *this;
    --*this;
    return result;
  }

  mint &operator+=(const mint &rhs) {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint &operator-=(const mint &rhs) {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint &operator*=(const mint &rhs) {
    unsigned long long z = _v;
    z *= rhs._v;
    _v = (unsigned int)(z % umod());
    return *this;
  }
  mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }

  mint pow(long long n) const {
    assert(0 <= n);
    mint x = *this, r = 1;
    while (n) {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const {
    if (prime) {
      assert(_v);
      return pow(umod() - 2);
    } else {
      auto eg = internal::inv_gcd(_v, m);
      assert(eg.first == 1);
      return eg.second;
    }
  }

  friend mint operator+(const mint &lhs, const mint &rhs) {
    return mint(lhs) += rhs;
  }
  friend mint operator-(const mint &lhs, const mint &rhs) {
    return mint(lhs) -= rhs;
  }
  friend mint operator*(const mint &lhs, const mint &rhs) {
    return mint(lhs) *= rhs;
  }
  friend mint operator/(const mint &lhs, const mint &rhs) {
    return mint(lhs) /= rhs;
  }
  friend bool operator==(const mint &lhs, const mint &rhs) {
    return lhs._v == rhs._v;
  }
  friend bool operator!=(const mint &lhs, const mint &rhs) {
    return lhs._v != rhs._v;
  }

private:
  unsigned int _v;
  static constexpr unsigned int umod() { return m; }
  static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
  using mint = dynamic_modint;

public:
  static int mod() { return (int)(bt.umod()); }
  static void set_mod(int m) {
    assert(1 <= m);
    bt = internal::barrett(m);
  }
  static mint raw(int v) {
    mint x;
    x._v = v;
    return x;
  }

  dynamic_modint() : _v(0) {}
  template <class T, internal::is_signed_int_t<T> * = nullptr>
  dynamic_modint(T v) {
    long long x = (long long)(v % (long long)(mod()));
    if (x < 0)
      x += mod();
    _v = (unsigned int)(x);
  }
  template <class T, internal::is_unsigned_int_t<T> * = nullptr>
  dynamic_modint(T v) {
    _v = (unsigned int)(v % mod());
  }
  dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); }

  unsigned int val() const { return _v; }

  mint &operator++() {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint &operator--() {
    if (_v == 0)
      _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int) {
    mint result = *this;
    ++*this;
    return result;
  }
  mint operator--(int) {
    mint result = *this;
    --*this;
    return result;
  }

  mint &operator+=(const mint &rhs) {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint &operator-=(const mint &rhs) {
    _v += mod() - rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint &operator*=(const mint &rhs) {
    _v = bt.mul(_v, rhs._v);
    return *this;
  }
  mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }

  mint pow(long long n) const {
    assert(0 <= n);
    mint x = *this, r = 1;
    while (n) {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const {
    auto eg = internal::inv_gcd(_v, mod());
    assert(eg.first == 1);
    return eg.second;
  }

  friend mint operator+(const mint &lhs, const mint &rhs) {
    return mint(lhs) += rhs;
  }
  friend mint operator-(const mint &lhs, const mint &rhs) {
    return mint(lhs) -= rhs;
  }
  friend mint operator*(const mint &lhs, const mint &rhs) {
    return mint(lhs) *= rhs;
  }
  friend mint operator/(const mint &lhs, const mint &rhs) {
    return mint(lhs) /= rhs;
  }
  friend bool operator==(const mint &lhs, const mint &rhs) {
    return lhs._v == rhs._v;
  }
  friend bool operator!=(const mint &lhs, const mint &rhs) {
    return lhs._v != rhs._v;
  }

private:
  unsigned int _v;
  static internal::barrett bt;
  static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt = 998244353;

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {

namespace internal {

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly(std::vector<mint> &a) {
  static constexpr int g = internal::primitive_root<mint::mod()>;
  int n = int(a.size());
  int h = internal::ceil_pow2(n);

  static bool first = true;
  static mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
  if (first) {
    first = false;
    mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
    int cnt2 = bsf(mint::mod() - 1);
    mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
    for (int i = cnt2; i >= 2; i--) {
      // e^(2^i) == 1
      es[i - 2] = e;
      ies[i - 2] = ie;
      e *= e;
      ie *= ie;
    }
    mint now = 1;
    for (int i = 0; i <= cnt2 - 2; i++) {
      sum_e[i] = es[i] * now;
      now *= ies[i];
    }
  }
  for (int ph = 1; ph <= h; ph++) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    mint now = 1;
    for (int s = 0; s < w; s++) {
      int offset = s << (h - ph + 1);
      for (int i = 0; i < p; i++) {
        auto l = a[i + offset];
        auto r = a[i + offset + p] * now;
        a[i + offset] = l + r;
        a[i + offset + p] = l - r;
      }
      now *= sum_e[bsf(~(unsigned int)(s))];
    }
  }
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly_inv(std::vector<mint> &a) {
  static constexpr int g = internal::primitive_root<mint::mod()>;
  int n = int(a.size());
  int h = internal::ceil_pow2(n);

  static bool first = true;
  static mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
  if (first) {
    first = false;
    mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
    int cnt2 = bsf(mint::mod() - 1);
    mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
    for (int i = cnt2; i >= 2; i--) {
      // e^(2^i) == 1
      es[i - 2] = e;
      ies[i - 2] = ie;
      e *= e;
      ie *= ie;
    }
    mint now = 1;
    for (int i = 0; i <= cnt2 - 2; i++) {
      sum_ie[i] = ies[i] * now;
      now *= es[i];
    }
  }

  for (int ph = h; ph >= 1; ph--) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    mint inow = 1;
    for (int s = 0; s < w; s++) {
      int offset = s << (h - ph + 1);
      for (int i = 0; i < p; i++) {
        auto l = a[i + offset];
        auto r = a[i + offset + p];
        a[i + offset] = l + r;
        a[i + offset + p] =
            (unsigned long long)(mint::mod() + l.val() - r.val()) * inow.val();
      }
      inow *= sum_ie[bsf(~(unsigned int)(s))];
    }
  }
}

} // namespace internal

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
  int n = int(a.size()), m = int(b.size());
  if (!n || !m)
    return {};
  if (std::min(n, m) <= 60) {
    if (n < m) {
      std::swap(n, m);
      std::swap(a, b);
    }
    std::vector<mint> ans(n + m - 1);
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        ans[i + j] += a[i] * b[j];
      }
    }
    return ans;
  }
  int z = 1 << internal::ceil_pow2(n + m - 1);
  a.resize(z);
  internal::butterfly(a);
  b.resize(z);
  internal::butterfly(b);
  for (int i = 0; i < z; i++) {
    a[i] *= b[i];
  }
  internal::butterfly_inv(a);
  a.resize(n + m - 1);
  mint iz = mint(z).inv();
  for (int i = 0; i < n + m - 1; i++)
    a[i] *= iz;
  return a;
}

template <unsigned int mod = 998244353, class T,
          std::enable_if_t<internal::is_integral<T>::value> * = nullptr>
std::vector<T> convolution(const std::vector<T> &a, const std::vector<T> &b) {
  int n = int(a.size()), m = int(b.size());
  if (!n || !m)
    return {};

  using mint = static_modint<mod>;
  std::vector<mint> a2(n), b2(m);
  for (int i = 0; i < n; i++) {
    a2[i] = mint(a[i]);
  }
  for (int i = 0; i < m; i++) {
    b2[i] = mint(b[i]);
  }
  auto c2 = convolution(move(a2), move(b2));
  std::vector<T> c(n + m - 1);
  for (int i = 0; i < n + m - 1; i++) {
    c[i] = c2[i].val();
  }
  return c;
}

std::vector<long long> convolution_ll(const std::vector<long long> &a,
                                      const std::vector<long long> &b) {
  int n = int(a.size()), m = int(b.size());
  if (!n || !m)
    return {};

  static constexpr unsigned long long MOD1 = 754974721; // 2^24
  static constexpr unsigned long long MOD2 = 167772161; // 2^25
  static constexpr unsigned long long MOD3 = 469762049; // 2^26
  static constexpr unsigned long long M2M3 = MOD2 * MOD3;
  static constexpr unsigned long long M1M3 = MOD1 * MOD3;
  static constexpr unsigned long long M1M2 = MOD1 * MOD2;
  static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

  static constexpr unsigned long long i1 =
      internal::inv_gcd(MOD2 * MOD3, MOD1).second;
  static constexpr unsigned long long i2 =
      internal::inv_gcd(MOD1 * MOD3, MOD2).second;
  static constexpr unsigned long long i3 =
      internal::inv_gcd(MOD1 * MOD2, MOD3).second;

  auto c1 = convolution<MOD1>(a, b);
  auto c2 = convolution<MOD2>(a, b);
  auto c3 = convolution<MOD3>(a, b);

  std::vector<long long> c(n + m - 1);
  for (int i = 0; i < n + m - 1; i++) {
    unsigned long long x = 0;
    x += (c1[i] * i1) % MOD1 * M2M3;
    x += (c2[i] * i2) % MOD2 * M1M3;
    x += (c3[i] * i3) % MOD3 * M1M2;
    // B = 2^63, -B <= x, r(real value) < B
    // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
    // r = c1[i] (mod MOD1)
    // focus on MOD1
    // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
    // r = x,
    //     x - M' + (0 or 2B),
    //     x - 2M' + (0, 2B or 4B),
    //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
    // (r - x) = 0, (0)
    //           - M' + (0 or 2B), (1)
    //           -2M' + (0 or 2B or 4B), (2)
    //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
    // we checked that
    //   ((1) mod MOD1) mod 5 = 2
    //   ((2) mod MOD1) mod 5 = 3
    //   ((3) mod MOD1) mod 5 = 4
    long long diff =
        c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
    if (diff < 0)
      diff += MOD1;
    static constexpr unsigned long long offset[5] = {0, 0, M1M2M3, 2 * M1M2M3,
                                                     3 * M1M2M3};
    x -= offset[diff % 5];
    c[i] = x;
  }

  return c;
}

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

// Implement (union by size) + (path compression)
// Reference:
// Zvi Galil and Giuseppe F. Italiano,
// Data structures and algorithms for disjoint set union problems
struct dsu {
public:
  dsu() : _n(0) {}
  dsu(int n) : _n(n), parent_or_size(n, -1) {}

  int merge(int a, int b) {
    assert(0 <= a && a < _n);
    assert(0 <= b && b < _n);
    int x = leader(a), y = leader(b);
    if (x == y)
      return x;
    if (-parent_or_size[x] < -parent_or_size[y])
      std::swap(x, y);
    parent_or_size[x] += parent_or_size[y];
    parent_or_size[y] = x;
    return x;
  }

  bool same(int a, int b) {
    assert(0 <= a && a < _n);
    assert(0 <= b && b < _n);
    return leader(a) == leader(b);
  }

  int leader(int a) {
    assert(0 <= a && a < _n);
    if (parent_or_size[a] < 0)
      return a;
    return parent_or_size[a] = leader(parent_or_size[a]);
  }

  int size(int a) {
    assert(0 <= a && a < _n);
    return -parent_or_size[leader(a)];
  }

  std::vector<std::vector<int>> groups() {
    std::vector<int> leader_buf(_n), group_size(_n);
    for (int i = 0; i < _n; i++) {
      leader_buf[i] = leader(i);
      group_size[leader_buf[i]]++;
    }
    std::vector<std::vector<int>> result(_n);
    for (int i = 0; i < _n; i++) {
      result[i].reserve(group_size[i]);
    }
    for (int i = 0; i < _n; i++) {
      result[leader_buf[i]].push_back(i);
    }
    result.erase(
        std::remove_if(result.begin(), result.end(),
                       [&](const std::vector<int> &v) { return v.empty(); }),
        result.end());
    return result;
  }

private:
  int _n;
  // root node: -1 * component size
  // otherwise: parent
  std::vector<int> parent_or_size;
};

} // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

// Reference: https://en.wikipedia.org/wiki/Fenwick_tree
template <class T> struct fenwick_tree {
  using U = internal::to_unsigned_t<T>;

public:
  fenwick_tree() : _n(0) {}
  fenwick_tree(int n) : _n(n), data(n) {}

  void add(int p, T x) {
    assert(0 <= p && p < _n);
    p++;
    while (p <= _n) {
      data[p - 1] += U(x);
      p += p & -p;
    }
  }

  T sum(int l, int r) {
    assert(0 <= l && l <= r && r <= _n);
    return sum(r) - sum(l);
  }

private:
  int _n;
  std::vector<U> data;

  U sum(int r) {
    U s = 0;
    while (r > 0) {
      s += data[r - 1];
      r -= r & -r;
    }
    return s;
  }
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
namespace atcoder {

template <class S, S (*op)(S, S), S (*e)(), class F, S (*mapping)(F, S),
          F (*composition)(F, F), F (*id)()>
struct lazy_segtree {
public:
  lazy_segtree() : lazy_segtree(0) {}
  lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
  lazy_segtree(const std::vector<S> &v) : _n(int(v.size())) {
    log = internal::ceil_pow2(_n);
    size = 1 << log;
    d = std::vector<S>(2 * size, e());
    lz = std::vector<F>(size, id());
    for (int i = 0; i < _n; i++)
      d[size + i] = v[i];
    for (int i = size - 1; i >= 1; i--) {
      update(i);
    }
  }

  void set(int p, S x) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--)
      push(p >> i);
    d[p] = x;
    for (int i = 1; i <= log; i++)
      update(p >> i);
  }

  S get(int p) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--)
      push(p >> i);
    return d[p];
  }

  S prod(int l, int r) {
    assert(0 <= l && l <= r && r <= _n);
    if (l == r)
      return e();

    l += size;
    r += size;

    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l)
        push(l >> i);
      if (((r >> i) << i) != r)
        push(r >> i);
    }

    S sml = e(), smr = e();
    while (l < r) {
      if (l & 1)
        sml = op(sml, d[l++]);
      if (r & 1)
        smr = op(d[--r], smr);
      l >>= 1;
      r >>= 1;
    }

    return op(sml, smr);
  }

  S all_prod() { return d[1]; }

  void apply(int p, F f) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--)
      push(p >> i);
    d[p] = mapping(f, d[p]);
    for (int i = 1; i <= log; i++)
      update(p >> i);
  }
  void apply(int l, int r, F f) {
    assert(0 <= l && l <= r && r <= _n);
    if (l == r)
      return;

    l += size;
    r += size;

    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l)
        push(l >> i);
      if (((r >> i) << i) != r)
        push((r - 1) >> i);
    }

    {
      int l2 = l, r2 = r;
      while (l < r) {
        if (l & 1)
          all_apply(l++, f);
        if (r & 1)
          all_apply(--r, f);
        l >>= 1;
        r >>= 1;
      }
      l = l2;
      r = r2;
    }

    for (int i = 1; i <= log; i++) {
      if (((l >> i) << i) != l)
        update(l >> i);
      if (((r >> i) << i) != r)
        update((r - 1) >> i);
    }
  }

  template <bool (*g)(S)> int max_right(int l) {
    return max_right(l, [](S x) { return g(x); });
  }
  template <class G> int max_right(int l, G g) {
    assert(0 <= l && l <= _n);
    assert(g(e()));
    if (l == _n)
      return _n;
    l += size;
    for (int i = log; i >= 1; i--)
      push(l >> i);
    S sm = e();
    do {
      while (l % 2 == 0)
        l >>= 1;
      if (!g(op(sm, d[l]))) {
        while (l < size) {
          push(l);
          l = (2 * l);
          if (g(op(sm, d[l]))) {
            sm = op(sm, d[l]);
            l++;
          }
        }
        return l - size;
      }
      sm = op(sm, d[l]);
      l++;
    } while ((l & -l) != l);
    return _n;
  }

  template <bool (*g)(S)> int min_left(int r) {
    return min_left(r, [](S x) { return g(x); });
  }
  template <class G> int min_left(int r, G g) {
    assert(0 <= r && r <= _n);
    assert(g(e()));
    if (r == 0)
      return 0;
    r += size;
    for (int i = log; i >= 1; i--)
      push((r - 1) >> i);
    S sm = e();
    do {
      r--;
      while (r > 1 && (r % 2))
        r >>= 1;
      if (!g(op(d[r], sm))) {
        while (r < size) {
          push(r);
          r = (2 * r + 1);
          if (g(op(d[r], sm))) {
            sm = op(d[r], sm);
            r--;
          }
        }
        return r + 1 - size;
      }
      sm = op(d[r], sm);
    } while ((r & -r) != r);
    return 0;
  }

private:
  int _n, size, log;
  std::vector<S> d;
  std::vector<F> lz;

  void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
  void all_apply(int k, F f) {
    d[k] = mapping(f, d[k]);
    if (k < size)
      lz[k] = composition(f, lz[k]);
  }
  void push(int k) {
    all_apply(2 * k, lz[k]);
    all_apply(2 * k + 1, lz[k]);
    lz[k] = id();
  }
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>

namespace atcoder {

long long pow_mod(long long x, long long n, int m) {
  assert(0 <= n && 1 <= m);
  if (m == 1)
    return 0;
  internal::barrett bt((unsigned int)(m));
  unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
  while (n) {
    if (n & 1)
      r = bt.mul(r, y);
    y = bt.mul(y, y);
    n >>= 1;
  }
  return r;
}

long long inv_mod(long long x, long long m) {
  assert(1 <= m);
  auto z = internal::inv_gcd(x, m);
  assert(z.first == 1);
  return z.second;
}

// (rem, mod)
std::pair<long long, long long> crt(const std::vector<long long> &r,
                                    const std::vector<long long> &m) {
  assert(r.size() == m.size());
  int n = int(r.size());
  // Contracts: 0 <= r0 < m0
  long long r0 = 0, m0 = 1;
  for (int i = 0; i < n; i++) {
    assert(1 <= m[i]);
    long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
    if (m0 < m1) {
      std::swap(r0, r1);
      std::swap(m0, m1);
    }
    if (m0 % m1 == 0) {
      if (r0 % m1 != r1)
        return {0, 0};
      continue;
    }
    // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)

    // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
    // r2 % m0 = r0
    // r2 % m1 = r1
    // -> (r0 + x*m0) % m1 = r1
    // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
    // -> x = (r1 - r0) / g * inv(u0) (mod u1)

    // im = inv(u0) (mod u1) (0 <= im < u1)
    long long g, im;
    std::tie(g, im) = internal::inv_gcd(m0, m1);

    long long u1 = (m1 / g);
    // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
    if ((r1 - r0) % g)
      return {0, 0};

    // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
    long long x = (r1 - r0) / g % u1 * im % u1;

    // |r0| + |m0 * x|
    // < m0 + m0 * (u1 - 1)
    // = m0 + m0 * m1 / g - m0
    // = lcm(m0, m1)
    r0 += x * m0;
    m0 *= u1; // -> lcm(m0, m1)
    if (r0 < 0)
      r0 += m0;
  }
  return {r0, m0};
}

long long floor_sum(long long n, long long m, long long a, long long b) {
  long long ans = 0;
  if (a >= m) {
    ans += (n - 1) * n * (a / m) / 2;
    a %= m;
  }
  if (b >= m) {
    ans += n * (b / m);
    b %= m;
  }

  long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
  if (y_max == 0)
    return ans;
  ans += (n - (x_max + a - 1) / a) * y_max;
  ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
  return ans;
}

} // namespace atcoder

#include <algorithm>

#include <vector>

namespace atcoder {

namespace internal {

template <class T> struct simple_queue {
  std::vector<T> payload;
  int pos = 0;
  void reserve(int n) { payload.reserve(n); }
  int size() const { return int(payload.size()) - pos; }
  bool empty() const { return pos == int(payload.size()); }
  void push(const T &t) { payload.push_back(t); }
  T &front() { return payload[pos]; }
  void clear() {
    payload.clear();
    pos = 0;
  }
  void pop() { pos++; }
};

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

template <class Cap> struct mf_graph {
public:
  mf_graph() : _n(0) {}
  mf_graph(int n) : _n(n), g(n) {}

  int add_edge(int from, int to, Cap cap) {
    assert(0 <= from && from < _n);
    assert(0 <= to && to < _n);
    assert(0 <= cap);
    int m = int(pos.size());
    pos.push_back({from, int(g[from].size())});
    int from_id = int(g[from].size());
    int to_id = int(g[to].size());
    if (from == to)
      to_id++;
    g[from].push_back(_edge{to, to_id, cap});
    g[to].push_back(_edge{from, from_id, 0});
    return m;
  }

  struct edge {
    int from, to;
    Cap cap, flow;
  };

  edge get_edge(int i) {
    int m = int(pos.size());
    assert(0 <= i && i < m);
    auto _e = g[pos[i].first][pos[i].second];
    auto _re = g[_e.to][_e.rev];
    return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
  }
  std::vector<edge> edges() {
    int m = int(pos.size());
    std::vector<edge> result;
    for (int i = 0; i < m; i++) {
      result.push_back(get_edge(i));
    }
    return result;
  }
  void change_edge(int i, Cap new_cap, Cap new_flow) {
    int m = int(pos.size());
    assert(0 <= i && i < m);
    assert(0 <= new_flow && new_flow <= new_cap);
    auto &_e = g[pos[i].first][pos[i].second];
    auto &_re = g[_e.to][_e.rev];
    _e.cap = new_cap - new_flow;
    _re.cap = new_flow;
  }

  Cap flow(int s, int t) { return flow(s, t, std::numeric_limits<Cap>::max()); }
  Cap flow(int s, int t, Cap flow_limit) {
    assert(0 <= s && s < _n);
    assert(0 <= t && t < _n);
    assert(s != t);

    std::vector<int> level(_n), iter(_n);
    internal::simple_queue<int> que;

    auto bfs = [&]() {
      std::fill(level.begin(), level.end(), -1);
      level[s] = 0;
      que.clear();
      que.push(s);
      while (!que.empty()) {
        int v = que.front();
        que.pop();
        for (auto e : g[v]) {
          if (e.cap == 0 || level[e.to] >= 0)
            continue;
          level[e.to] = level[v] + 1;
          if (e.to == t)
            return;
          que.push(e.to);
        }
      }
    };
    auto dfs = [&](auto self, int v, Cap up) {
      if (v == s)
        return up;
      Cap res = 0;
      int level_v = level[v];
      for (int &i = iter[v]; i < int(g[v].size()); i++) {
        _edge &e = g[v][i];
        if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0)
          continue;
        Cap d = self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
        if (d <= 0)
          continue;
        g[v][i].cap += d;
        g[e.to][e.rev].cap -= d;
        res += d;
        if (res == up)
          break;
      }
      return res;
    };

    Cap flow = 0;
    while (flow < flow_limit) {
      bfs();
      if (level[t] == -1)
        break;
      std::fill(iter.begin(), iter.end(), 0);
      while (flow < flow_limit) {
        Cap f = dfs(dfs, t, flow_limit - flow);
        if (!f)
          break;
        flow += f;
      }
    }
    return flow;
  }

  std::vector<bool> min_cut(int s) {
    std::vector<bool> visited(_n);
    internal::simple_queue<int> que;
    que.push(s);
    while (!que.empty()) {
      int p = que.front();
      que.pop();
      visited[p] = true;
      for (auto e : g[p]) {
        if (e.cap && !visited[e.to]) {
          visited[e.to] = true;
          que.push(e.to);
        }
      }
    }
    return visited;
  }

private:
  int _n;
  struct _edge {
    int to, rev;
    Cap cap;
  };
  std::vector<std::pair<int, int>> pos;
  std::vector<std::vector<_edge>> g;
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

template <class Cap, class Cost> struct mcf_graph {
public:
  mcf_graph() {}
  mcf_graph(int n) : _n(n), g(n) {}

  int add_edge(int from, int to, Cap cap, Cost cost) {
    assert(0 <= from && from < _n);
    assert(0 <= to && to < _n);
    int m = int(pos.size());
    pos.push_back({from, int(g[from].size())});
    int from_id = int(g[from].size());
    int to_id = int(g[to].size());
    if (from == to)
      to_id++;
    g[from].push_back(_edge{to, to_id, cap, cost});
    g[to].push_back(_edge{from, from_id, 0, -cost});
    return m;
  }

  struct edge {
    int from, to;
    Cap cap, flow;
    Cost cost;
  };

  edge get_edge(int i) {
    int m = int(pos.size());
    assert(0 <= i && i < m);
    auto _e = g[pos[i].first][pos[i].second];
    auto _re = g[_e.to][_e.rev];
    return edge{
        pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,
    };
  }
  std::vector<edge> edges() {
    int m = int(pos.size());
    std::vector<edge> result(m);
    for (int i = 0; i < m; i++) {
      result[i] = get_edge(i);
    }
    return result;
  }

  std::pair<Cap, Cost> flow(int s, int t) {
    return flow(s, t, std::numeric_limits<Cap>::max());
  }
  std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
    return slope(s, t, flow_limit).back();
  }
  std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
    return slope(s, t, std::numeric_limits<Cap>::max());
  }
  std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
    assert(0 <= s && s < _n);
    assert(0 <= t && t < _n);
    assert(s != t);
    // variants (C = maxcost):
    // -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0
    // reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge
    std::vector<Cost> dual(_n, 0), dist(_n);
    std::vector<int> pv(_n), pe(_n);
    std::vector<bool> vis(_n);
    auto dual_ref = [&]() {
      std::fill(dist.begin(), dist.end(), std::numeric_limits<Cost>::max());
      std::fill(pv.begin(), pv.end(), -1);
      std::fill(pe.begin(), pe.end(), -1);
      std::fill(vis.begin(), vis.end(), false);
      struct Q {
        Cost key;
        int to;
        bool operator<(Q r) const { return key > r.key; }
      };
      std::priority_queue<Q> que;
      dist[s] = 0;
      que.push(Q{0, s});
      while (!que.empty()) {
        int v = que.top().to;
        que.pop();
        if (vis[v])
          continue;
        vis[v] = true;
        if (v == t)
          break;
        // dist[v] = shortest(s, v) + dual[s] - dual[v]
        // dist[v] >= 0 (all reduced cost are positive)
        // dist[v] <= (n-1)C
        for (int i = 0; i < int(g[v].size()); i++) {
          auto e = g[v][i];
          if (vis[e.to] || !e.cap)
            continue;
          // |-dual[e.to] + dual[v]| <= (n-1)C
          // cost <= C - -(n-1)C + 0 = nC
          Cost cost = e.cost - dual[e.to] + dual[v];
          if (dist[e.to] - dist[v] > cost) {
            dist[e.to] = dist[v] + cost;
            pv[e.to] = v;
            pe[e.to] = i;
            que.push(Q{dist[e.to], e.to});
          }
        }
      }
      if (!vis[t]) {
        return false;
      }

      for (int v = 0; v < _n; v++) {
        if (!vis[v])
          continue;
        // dual[v] = dual[v] - dist[t] + dist[v]
        //         = dual[v] - (shortest(s, t) + dual[s] - dual[t]) +
        //         (shortest(s, v) + dual[s] - dual[v]) = - shortest(s, t) +
        //         dual[t] + shortest(s, v) = shortest(s, v) - shortest(s, t) >=
        //         0 - (n-1)C
        dual[v] -= dist[t] - dist[v];
      }
      return true;
    };
    Cap flow = 0;
    Cost cost = 0, prev_cost_per_flow = -1;
    std::vector<std::pair<Cap, Cost>> result;
    result.push_back({flow, cost});
    while (flow < flow_limit) {
      if (!dual_ref())
        break;
      Cap c = flow_limit - flow;
      for (int v = t; v != s; v = pv[v]) {
        c = std::min(c, g[pv[v]][pe[v]].cap);
      }
      for (int v = t; v != s; v = pv[v]) {
        auto &e = g[pv[v]][pe[v]];
        e.cap -= c;
        g[v][e.rev].cap += c;
      }
      Cost d = -dual[s];
      flow += c;
      cost += c * d;
      if (prev_cost_per_flow == d) {
        result.pop_back();
      }
      result.push_back({flow, cost});
      prev_cost_per_flow = d;
    }
    return result;
  }

private:
  int _n;

  struct _edge {
    int to, rev;
    Cap cap;
    Cost cost;
  };

  std::vector<std::pair<int, int>> pos;
  std::vector<std::vector<_edge>> g;
};

} // namespace atcoder

#include <algorithm>

#include <algorithm>
#include <utility>
#include <vector>

namespace atcoder {
namespace internal {

template <class E> struct csr {
  std::vector<int> start;
  std::vector<E> elist;
  csr(int n, const std::vector<std::pair<int, E>> &edges)
      : start(n + 1), elist(edges.size()) {
    for (auto e : edges) {
      start[e.first + 1]++;
    }
    for (int i = 1; i <= n; i++) {
      start[i] += start[i - 1];
    }
    auto counter = start;
    for (auto e : edges) {
      elist[counter[e.first]++] = e.second;
    }
  }
};

// Reference:
// R. Tarjan,
// Depth-First Search and Linear Graph Algorithms
struct scc_graph {
public:
  scc_graph(int n) : _n(n) {}

  int num_vertices() { return _n; }

  void add_edge(int from, int to) { edges.push_back({from, {to}}); }

  // @return pair of (# of scc, scc id)
  std::pair<int, std::vector<int>> scc_ids() {
    auto g = csr<edge>(_n, edges);
    int now_ord = 0, group_num = 0;
    std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
    visited.reserve(_n);
    auto dfs = [&](auto self, int v) -> void {
      low[v] = ord[v] = now_ord++;
      visited.push_back(v);
      for (int i = g.start[v]; i < g.start[v + 1]; i++) {
        auto to = g.elist[i].to;
        if (ord[to] == -1) {
          self(self, to);
          low[v] = std::min(low[v], low[to]);
        } else {
          low[v] = std::min(low[v], ord[to]);
        }
      }
      if (low[v] == ord[v]) {
        while (true) {
          int u = visited.back();
          visited.pop_back();
          ord[u] = _n;
          ids[u] = group_num;
          if (u == v)
            break;
        }
        group_num++;
      }
    };
    for (int i = 0; i < _n; i++) {
      if (ord[i] == -1)
        dfs(dfs, i);
    }
    for (auto &x : ids) {
      x = group_num - 1 - x;
    }
    return {group_num, ids};
  }

  std::vector<std::vector<int>> scc() {
    auto ids = scc_ids();
    int group_num = ids.first;
    std::vector<int> counts(group_num);
    for (auto x : ids.second)
      counts[x]++;
    std::vector<std::vector<int>> groups(ids.first);
    for (int i = 0; i < group_num; i++) {
      groups[i].reserve(counts[i]);
    }
    for (int i = 0; i < _n; i++) {
      groups[ids.second[i]].push_back(i);
    }
    return groups;
  }

private:
  int _n;
  struct edge {
    int to;
  };
  std::vector<std::pair<int, edge>> edges;
};

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

struct scc_graph {
public:
  scc_graph() : internal(0) {}
  scc_graph(int n) : internal(n) {}

  void add_edge(int from, int to) {
    int n = internal.num_vertices();
    assert(0 <= from && from < n);
    assert(0 <= to && to < n);
    internal.add_edge(from, to);
  }

  std::vector<std::vector<int>> scc() { return internal.scc(); }

private:
  internal::scc_graph internal;
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

template <class S, S (*op)(S, S), S (*e)()> struct segtree {
public:
  segtree() : segtree(0) {}
  segtree(int n) : segtree(std::vector<S>(n, e())) {}
  segtree(const std::vector<S> &v) : _n(int(v.size())) {
    log = internal::ceil_pow2(_n);
    size = 1 << log;
    d = std::vector<S>(2 * size, e());
    for (int i = 0; i < _n; i++)
      d[size + i] = v[i];
    for (int i = size - 1; i >= 1; i--) {
      update(i);
    }
  }

  void set(int p, S x) {
    assert(0 <= p && p < _n);
    p += size;
    d[p] = x;
    for (int i = 1; i <= log; i++)
      update(p >> i);
  }

  S get(int p) {
    assert(0 <= p && p < _n);
    return d[p + size];
  }

  S prod(int l, int r) {
    assert(0 <= l && l <= r && r <= _n);
    S sml = e(), smr = e();
    l += size;
    r += size;

    while (l < r) {
      if (l & 1)
        sml = op(sml, d[l++]);
      if (r & 1)
        smr = op(d[--r], smr);
      l >>= 1;
      r >>= 1;
    }
    return op(sml, smr);
  }

  S all_prod() { return d[1]; }

  template <bool (*f)(S)> int max_right(int l) {
    return max_right(l, [](S x) { return f(x); });
  }
  template <class F> int max_right(int l, F f) {
    assert(0 <= l && l <= _n);
    assert(f(e()));
    if (l == _n)
      return _n;
    l += size;
    S sm = e();
    do {
      while (l % 2 == 0)
        l >>= 1;
      if (!f(op(sm, d[l]))) {
        while (l < size) {
          l = (2 * l);
          if (f(op(sm, d[l]))) {
            sm = op(sm, d[l]);
            l++;
          }
        }
        return l - size;
      }
      sm = op(sm, d[l]);
      l++;
    } while ((l & -l) != l);
    return _n;
  }

  template <bool (*f)(S)> int min_left(int r) {
    return min_left(r, [](S x) { return f(x); });
  }
  template <class F> int min_left(int r, F f) {
    assert(0 <= r && r <= _n);
    assert(f(e()));
    if (r == 0)
      return 0;
    r += size;
    S sm = e();
    do {
      r--;
      while (r > 1 && (r % 2))
        r >>= 1;
      if (!f(op(d[r], sm))) {
        while (r < size) {
          r = (2 * r + 1);
          if (f(op(d[r], sm))) {
            sm = op(d[r], sm);
            r--;
          }
        }
        return r + 1 - size;
      }
      sm = op(d[r], sm);
    } while ((r & -r) != r);
    return 0;
  }

private:
  int _n, size, log;
  std::vector<S> d;

  void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <numeric>
#include <string>
#include <vector>

namespace atcoder {

namespace internal {

std::vector<int> sa_naive(const std::vector<int> &s) {
  int n = int(s.size());
  std::vector<int> sa(n);
  std::iota(sa.begin(), sa.end(), 0);
  std::sort(sa.begin(), sa.end(), [&](int l, int r) {
    if (l == r)
      return false;
    while (l < n && r < n) {
      if (s[l] != s[r])
        return s[l] < s[r];
      l++;
      r++;
    }
    return l == n;
  });
  return sa;
}

std::vector<int> sa_doubling(const std::vector<int> &s) {
  int n = int(s.size());
  std::vector<int> sa(n), rnk = s, tmp(n);
  std::iota(sa.begin(), sa.end(), 0);
  for (int k = 1; k < n; k *= 2) {
    auto cmp = [&](int x, int y) {
      if (rnk[x] != rnk[y])
        return rnk[x] < rnk[y];
      int rx = x + k < n ? rnk[x + k] : -1;
      int ry = y + k < n ? rnk[y + k] : -1;
      return rx < ry;
    };
    std::sort(sa.begin(), sa.end(), cmp);
    tmp[sa[0]] = 0;
    for (int i = 1; i < n; i++) {
      tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);
    }
    std::swap(tmp, rnk);
  }
  return sa;
}

// SA-IS, linear-time suffix array construction
// Reference:
// G. Nong, S. Zhang, and W. H. Chan,
// Two Efficient Algorithms for Linear Time Suffix Array Construction
template <int THRESHOLD_NAIVE = 10, int THRESHOLD_DOUBLING = 40>
std::vector<int> sa_is(const std::vector<int> &s, int upper) {
  int n = int(s.size());
  if (n == 0)
    return {};
  if (n == 1)
    return {0};
  if (n == 2) {
    if (s[0] < s[1]) {
      return {0, 1};
    } else {
      return {1, 0};
    }
  }
  if (n < THRESHOLD_NAIVE) {
    return sa_naive(s);
  }
  if (n < THRESHOLD_DOUBLING) {
    return sa_doubling(s);
  }

  std::vector<int> sa(n);
  std::vector<bool> ls(n);
  for (int i = n - 2; i >= 0; i--) {
    ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);
  }
  std::vector<int> sum_l(upper + 1), sum_s(upper + 1);
  for (int i = 0; i < n; i++) {
    if (!ls[i]) {
      sum_s[s[i]]++;
    } else {
      sum_l[s[i] + 1]++;
    }
  }
  for (int i = 0; i <= upper; i++) {
    sum_s[i] += sum_l[i];
    if (i < upper)
      sum_l[i + 1] += sum_s[i];
  }

  auto induce = [&](const std::vector<int> &lms) {
    std::fill(sa.begin(), sa.end(), -1);
    std::vector<int> buf(upper + 1);
    std::copy(sum_s.begin(), sum_s.end(), buf.begin());
    for (auto d : lms) {
      if (d == n)
        continue;
      sa[buf[s[d]]++] = d;
    }
    std::copy(sum_l.begin(), sum_l.end(), buf.begin());
    sa[buf[s[n - 1]]++] = n - 1;
    for (int i = 0; i < n; i++) {
      int v = sa[i];
      if (v >= 1 && !ls[v - 1]) {
        sa[buf[s[v - 1]]++] = v - 1;
      }
    }
    std::copy(sum_l.begin(), sum_l.end(), buf.begin());
    for (int i = n - 1; i >= 0; i--) {
      int v = sa[i];
      if (v >= 1 && ls[v - 1]) {
        sa[--buf[s[v - 1] + 1]] = v - 1;
      }
    }
  };

  std::vector<int> lms_map(n + 1, -1);
  int m = 0;
  for (int i = 1; i < n; i++) {
    if (!ls[i - 1] && ls[i]) {
      lms_map[i] = m++;
    }
  }
  std::vector<int> lms;
  lms.reserve(m);
  for (int i = 1; i < n; i++) {
    if (!ls[i - 1] && ls[i]) {
      lms.push_back(i);
    }
  }

  induce(lms);

  if (m) {
    std::vector<int> sorted_lms;
    sorted_lms.reserve(m);
    for (int v : sa) {
      if (lms_map[v] != -1)
        sorted_lms.push_back(v);
    }
    std::vector<int> rec_s(m);
    int rec_upper = 0;
    rec_s[lms_map[sorted_lms[0]]] = 0;
    for (int i = 1; i < m; i++) {
      int l = sorted_lms[i - 1], r = sorted_lms[i];
      int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;
      int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;
      bool same = true;
      if (end_l - l != end_r - r) {
        same = false;
      } else {
        while (l < end_l) {
          if (s[l] != s[r]) {
            break;
          }
          l++;
          r++;
        }
        if (l == n || s[l] != s[r])
          same = false;
      }
      if (!same)
        rec_upper++;
      rec_s[lms_map[sorted_lms[i]]] = rec_upper;
    }

    auto rec_sa = sa_is<THRESHOLD_NAIVE, THRESHOLD_DOUBLING>(rec_s, rec_upper);

    for (int i = 0; i < m; i++) {
      sorted_lms[i] = lms[rec_sa[i]];
    }
    induce(sorted_lms);
  }
  return sa;
}

} // namespace internal

std::vector<int> suffix_array(const std::vector<int> &s, int upper) {
  assert(0 <= upper);
  for (int d : s) {
    assert(0 <= d && d <= upper);
  }
  auto sa = internal::sa_is(s, upper);
  return sa;
}

template <class T> std::vector<int> suffix_array(const std::vector<T> &s) {
  int n = int(s.size());
  std::vector<int> idx(n);
  iota(idx.begin(), idx.end(), 0);
  sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });
  std::vector<int> s2(n);
  int now = 0;
  for (int i = 0; i < n; i++) {
    if (i && s[idx[i - 1]] != s[idx[i]])
      now++;
    s2[idx[i]] = now;
  }
  return internal::sa_is(s2, now);
}

std::vector<int> suffix_array(const std::string &s) {
  int n = int(s.size());
  std::vector<int> s2(n);
  for (int i = 0; i < n; i++) {
    s2[i] = s[i];
  }
  return internal::sa_is(s2, 255);
}

// Reference:
// T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park,
// Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its
// Applications
template <class T>
std::vector<int> lcp_array(const std::vector<T> &s,
                           const std::vector<int> &sa) {
  int n = int(s.size());
  assert(n >= 1);
  std::vector<int> rnk(n);
  for (int i = 0; i < n; i++) {
    rnk[sa[i]] = i;
  }
  std::vector<int> lcp(n - 1);
  int h = 0;
  for (int i = 0; i < n; i++) {
    if (h > 0)
      h--;
    if (rnk[i] == 0)
      continue;
    int j = sa[rnk[i] - 1];
    for (; j + h < n && i + h < n; h++) {
      if (s[j + h] != s[i + h])
        break;
    }
    lcp[rnk[i] - 1] = h;
  }
  return lcp;
}

std::vector<int> lcp_array(const std::string &s, const std::vector<int> &sa) {
  int n = int(s.size());
  std::vector<int> s2(n);
  for (int i = 0; i < n; i++) {
    s2[i] = s[i];
  }
  return lcp_array(s2, sa);
}

// Reference:
// D. Gusfield,
// Algorithms on Strings, Trees, and Sequences: Computer Science and
// Computational Biology
template <class T> std::vector<int> z_algorithm(const std::vector<T> &s) {
  int n = int(s.size());
  if (n == 0)
    return {};
  std::vector<int> z(n);
  z[0] = 0;
  for (int i = 1, j = 0; i < n; i++) {
    int &k = z[i];
    k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);
    while (i + k < n && s[k] == s[i + k])
      k++;
    if (j + z[j] < i + z[i])
      j = i;
  }
  z[0] = n;
  return z;
}

std::vector<int> z_algorithm(const std::string &s) {
  int n = int(s.size());
  std::vector<int> s2(n);
  for (int i = 0; i < n; i++) {
    s2[i] = s[i];
  }
  return z_algorithm(s2);
}

} // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

// Reference:
// B. Aspvall, M. Plass, and R. Tarjan,
// A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean
// Formulas
struct two_sat {
public:
  two_sat() : _n(0), scc(0) {}
  two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}

  void add_clause(int i, bool f, int j, bool g) {
    assert(0 <= i && i < _n);
    assert(0 <= j && j < _n);
    scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
    scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
  }
  bool satisfiable() {
    auto id = scc.scc_ids().second;
    for (int i = 0; i < _n; i++) {
      if (id[2 * i] == id[2 * i + 1])
        return false;
      _answer[i] = id[2 * i] < id[2 * i + 1];
    }
    return true;
  }
  std::vector<bool> answer() { return _answer; }

private:
  int _n;
  std::vector<bool> _answer;
  internal::scc_graph scc;
};

} // namespace atcoder

using namespace std;
using namespace atcoder;
typedef long long ll;
/*
#define i_7 (ll)(1E9+7)
//#define i_7 998244353
#define i_5 i_7-2
ll mod(ll a){
    ll c=a%i_7;
    if(c>=0)return c;
    return c+i_7;
}
void Add(ll &pos,ll val){pos=mod(pos+val);}
void mod_print(ll k){
    ll P=50000;
    rep(y,1,P){
        ll x=mod(y*k);

        if(abs(x)<=P||x+P>=i_7){
            if(x+P>=i_7){
                x-=i_7;
            }
            cout<<x<<"/"<<y<<endl;
            //cout<<setprecision(5)<<(dd)x/(dd)y;
            return;
        }
    }
    cout<<"nun"<<endl;
}
*/

#include "boost/multiprecision/cpp_dec_float.hpp"
#include "boost/multiprecision/cpp_int.hpp"
namespace multi = boost::multiprecision;
typedef multi::cpp_int LL;
typedef multi::number<multi::cpp_dec_float<1024>>
    DD; // 仮数部が1024ビットの浮動小数点数型(TLEしたら小さくする)

typedef long double dd;
typedef pair<ll, ll> l_l;
typedef pair<dd, dd> d_d;
ll inf = (ll)1E16;
#define rep(i, l, r) for (ll i = l; i <= r; i++)
#define rrep(i, r, l) for (ll i = r; i >= l; i--)
#define pb push_back
ll max(ll a, ll b) {
  if (a < b)
    return b;
  else
    return a;
}
ll min(ll a, ll b) {
  if (a > b)
    return b;
  else
    return a;
}
void Max(ll &pos, ll val) { pos = max(pos, val); } // Max(dp[n],dp[n-1]);
void Min(ll &pos, ll val) { pos = min(pos, val); }
dd EPS = 1E-12;
#define fastio                                                                 \\
  ios::sync_with_stdio(false);                                                 \\
  cin.tie(0);                                                                  \\
  cout.tie(0);
#define fi first
#define se second

#define SORT(v) sort(v.begin(), v.end())
#define ERASE(v) v.erase(unique(v.begin(), v.end()), v.end())
#define POSL(v, x) (lower_bound(v.begin(), v.end(), x) - v.begin())
#define POSU(v, x) (upper_bound(v.begin(), v.end(), x) - v.begin())
template <class T, class S> inline bool chmax(T &a, S b) {
  if (a < b) {
    a = (T)b;
    return true;
  }
  return false;
}
template <class T, class S> inline bool chmin(T &a, S b) {
  if (a > b) {
    a = (T)b;
    return true;
  }
  return false;
}
#define endl                                                                   \\
  "\\n" // インタラクティブで消す！！！！！！！！！！！！！！！！！！！！！
#define all(c) c.begin(), c.end()

using mint = modint998244353;
// using mint = modint1000000007;
// using mint=modint;
// using mint=static_modint<100>;

typedef vector<ll> vl;
typedef vector<mint> vi;
dd PI = acos((dd)-1);
//////////////////////////

#define N 200004
mint kai[N];
mint kai2[N];
void calc() {
  kai[0] = 1;
  kai2[0] = 1;
  rep(i, 1, N - 1) { kai[i] = kai[i - 1] * i; }
  kai2[N - 1] = 1 / kai[N - 1];
  for (ll i = N - 2; i >= 0; i--) {
    kai2[i] = kai2[i + 1] * (i + 1);
  }
}
mint comb(ll n, ll k) {
  if (n < k)
    return 0;
  if (n == 0)
    return 1;
  return kai[n] * kai2[n - k] * kai2[k];
}
mint C2[N];

int main() {
  fastio calc();
  rep(i, 2, N - 1) { C2[i] = comb(i, 2); }
  ll n;
  cin >> n;
  vl h(2 * n);
  rep(i, 0, 2 * n - 1) cin >> h[i];
  SORT(h);
  vector<ll> a;
  rep(i, 0, 2 * n - 1) {
    ll c = 1;
    while (i + 1 <= 2 * n - 1 && h[i] == h[i + 1]) {
      c++;
      i++;
    }
    a.pb(c);
  }
  sort(all(a), [](ll x, ll y) { return x > y; });
  if (a[0] > n) {
    cout << 0 << endl;
    return 0;
  }
  //  for(auto x:a)cout<<x<<" ";cout<<endl;
  vector<vector<mint>> V;
  for (auto x : a) {
    vi v;
    mint mulc = 1;
    v.pb(1);
    for (ll i = 2; i <= x; i += 2) {
      mulc *= C2[x + 2 - i] / (i / 2);
      v.pb(mulc);
    }
    V.pb(v);
  }
  vi t;
  t.pb(1);
  while (V.size() >= 2) {
    vector<vi> nx;
    ll vs = V.size();
    for (ll i = 0; i <= vs - 1; i += 2) {
      if (i + 1 <= vs - 1)
        nx.pb(convolution(V[i], V[i + 1]));
      else
        nx.pb(V[i]);
    }
    V = nx;
    if (V.size() == 1)
      t = V[0];
  }

  // for(auto v:V)t=convolution(t,v);

  //  for(auto x:t)cout<<x.val()<<" ";cout<<endl;
  mint A[n + 1];
  A[n] = 1;
  rrep(i, n - 1, 0) A[i] = A[i + 1] * C2[2 * n - 2 * i];
  ll ts = t.size();
  rep(i, 0, ts - 1) {
    t[i] *= A[i];
    t[i] /= kai[n - i];
  }
  // for(auto x:t)cout<<x.val()<<" ";cout<<endl;
  mint ans = 0;
  rep(i, 0, ts - 1) {
    if (i % 2 == 0) {
      ans += t[i];
    } else {
      ans -= t[i];
    }
  }
  cout << ans.val() << endl;

  return 0;
}


Error: TLE

Fixed submission:


#include <bits/stdc++.h>
// #include "bits/stdc++.h"

#include <algorithm>
#include <array>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
  int x = 0;
  while ((1U << x) < (unsigned int)(n))
    x++;
  return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
  unsigned long index;
  _BitScanForward(&index, n);
  return index;
#else
  return __builtin_ctz(n);
#endif
}

} // namespace internal

} // namespace atcoder

#include <utility>

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
  x %= m;
  if (x < 0)
    x += m;
  return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
  unsigned int _m;
  unsigned long long im;

  // @param m `1 <= m < 2^31`
  barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

  // @return m
  unsigned int umod() const { return _m; }

  // @param a `0 <= a < m`
  // @param b `0 <= b < m`
  // @return `a * b % m`
  unsigned int mul(unsigned int a, unsigned int b) const {
    // [1] m = 1
    // a = b = im = 0, so okay

    // [2] m >= 2
    // im = ceil(2^64 / m)
    // -> im * m = 2^64 + r (0 <= r < m)
    // let z = a*b = c*m + d (0 <= c, d < m)
    // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
    // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) <
    // 2^64 * 2
    // ((ab * im) >> 64) == c or c + 1
    unsigned long long z = a;
    z *= b;
#ifdef _MSC_VER
    unsigned long long x;
    _umul128(z, im, &x);
#else
    unsigned long long x =
        (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
    unsigned int v = (unsigned int)(z - x * _m);
    if (_m <= v)
      v += _m;
    return v;
  }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
  if (m == 1)
    return 0;
  unsigned int _m = (unsigned int)(m);
  unsigned long long r = 1;
  unsigned long long y = safe_mod(x, m);
  while (n) {
    if (n & 1)
      r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  long long d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  constexpr long long bases[3] = {2, 7, 61};
  for (long long a : bases) {
    long long t = d;
    long long y = pow_mod_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1) {
      y = y * y % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0) {
      return false;
    }
  }
  return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
  a = safe_mod(a, b);
  if (a == 0)
    return {b, 0};

  // Contracts:
  // [1] s - m0 * a = 0 (mod b)
  // [2] t - m1 * a = 0 (mod b)
  // [3] s * |m1| + t * |m0| <= b
  long long s = b, t = a;
  long long m0 = 0, m1 = 1;

  while (t) {
    long long u = s / t;
    s -= t * u;
    m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b

    // [3]:
    // (s - t * u) * |m1| + t * |m0 - m1 * u|
    // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
    // = s * |m1| + t * |m0| <= b

    auto tmp = s;
    s = t;
    t = tmp;
    tmp = m0;
    m0 = m1;
    m1 = tmp;
  }
  // by [3]: |m0| <= b/g
  // by g != b: |m0| < b/g
  if (m0 < 0)
    m0 += b / s;
  return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
  if (m == 2)
    return 1;
  if (m == 167772161)
    return 3;
  if (m == 469762049)
    return 3;
  if (m == 754974721)
    return 11;
  if (m == 998244353)
    return 3;
  int divs[20] = {};
  divs[0] = 2;
  int cnt = 1;
  int x = (m - 1) / 2;
  while (x % 2 == 0)
    x /= 2;
  for (int i = 3; (long long)(i)*i <= x; i += 2) {
    if (x % i == 0) {
      divs[cnt++] = i;
      while (x % i == 0) {
        x /= i;
      }
    }
  }
  if (x > 1) {
    divs[cnt++] = x;
  }
  for (int g = 2;; g++) {
    bool ok = true;
    for (int i = 0; i < cnt; i++) {
      if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
        ok = false;
        break;
      }
    }
    if (ok)
      return g;
  }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral =
    typename std::conditional<std::is_integral<T>::value ||
                                  is_signed_int128<T>::value ||
                                  is_unsigned_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_signed_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_signed<T>::value) ||
                                  is_signed_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value, make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using to_unsigned =
    typename std::conditional<is_signed_int<T>::value, std::make_unsigned<T>,
                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

} // namespace internal

template <int m, std::enable_if_t<(1 <= m)> * = nullptr>
struct static_modint : internal::static_modint_base {
  using mint = static_modint;

public:
  static constexpr int mod() { return m; }
  static mint raw(int v) {
    mint x;
    x._v = v;
    return x;
  }

  static_modint() : _v(0) {}
  template <class T, internal::is_signed_int_t<T> * = nullptr>
  static_modint(T v) {
    long long x = (long long)(v % (long long)(umod()));
    if (x < 0)
      x += umod();
    _v = (unsigned int)(x);
  }
  template <class T, internal::is_unsigned_int_t<T> * = nullptr>
  static_modint(T v) {
    _v = (unsigned int)(v % umod());
  }
  static_modint(bool v) { _v = ((unsigned int)(v) % umod()); }

  unsigned int val() const { return _v; }

  mint &operator++() {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint &operator--() {
    if (_v == 0)
      _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int) {
    mint result = *this;
    ++*this;
    return result;
  }
  mint operator--(int) {
    mint result = *this;
    --*this;
    return result;
  }

  mint &operator+=(const mint &rhs) {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint &operator-=(const mint &rhs) {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint &operator*=(const mint &rhs) {
    unsigned long long z = _v;
    z *= rhs._v;
    _v = (unsigned int)(z % umod());
    return *this;
  }
  mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }

  mint pow(long long n) const {
    assert(0 <= n);
    mint x = *this, r = 1;
    while (n) {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const {
    if (prime) {
      assert(_v);
      return pow(umod() - 2);
    } else {
      auto eg = internal::inv_gcd(_v, m);
      assert(eg.first == 1);
      return eg.second;
    }
  }

  friend mint operator+(const mint &lhs, const mint &rhs) {
    return mint(lhs) += rhs;
  }
  friend mint operator-(const mint &lhs, const mint &rhs) {
    return mint(lhs) -= rhs;
  }
  friend mint operator*(const mint &lhs, const mint &rhs) {
    return mint(lhs) *= rhs;
  }
  friend mint operator/(const mint &lhs, const mint &rhs) {
    return mint(lhs) /= rhs;
  }
  friend bool operator==(const mint &lhs, const mint &rhs) {
    return lhs._v == rhs._v;
  }
  friend bool operator!=(const mint &lhs, const mint &rhs) {
    return lhs._v != rhs._v;
  }

private:
  unsigned int _v;
  static constexpr unsigned int umod() { return m; }
  static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
  using mint = dynamic_modint;

public:
  static int mod() { return (int)(bt.umod()); }
  static void set_mod(int m) {
    assert(1 <= m);
    bt = internal::barrett(m);
  }
  static mint raw(int v) {
    mint x;
    x._v = v;
    return x;
  }

  dynamic_modint() : _v(0) {}
  template <class T, internal::is_signed_int_t<T> * = nullptr>
  dynamic_modint(T v) {
    long long x = (long long)(v % (long long)(mod()));
    if (x < 0)
      x += mod();
    _v = (unsigned int)(x);
  }
  template <class T, internal::is_unsigned_int_t<T> * = nullptr>
  dynamic_modint(T v) {
    _v = (unsigned int)(v % mod());
  }
  dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); }

  unsigned int val() const { return _v; }

  mint &operator++() {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint &operator--() {
    if (_v == 0)
      _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int) {
    mint result = *this;
    ++*this;
    return result;
  }
  mint operator--(int) {
    mint result = *this;
    --*this;
    return result;
  }

  mint &operator+=(const mint &rhs) {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint &operator-=(const mint &rhs) {
    _v += mod() - rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint &operator*=(const mint &rhs) {
    _v = bt.mul(_v, rhs._v);
    return *this;
  }
  mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }

  mint pow(long long n) const {
    assert(0 <= n);
    mint x = *this, r = 1;
    while (n) {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const {
    auto eg = internal::inv_gcd(_v, mod());
    assert(eg.first == 1);
    return eg.second;
  }

  friend mint operator+(const mint &lhs, const mint &rhs) {
    return mint(lhs) += rhs;
  }
  friend mint operator-(const mint &lhs, const mint &rhs) {
    return mint(lhs) -= rhs;
  }
  friend mint operator*(const mint &lhs, const mint &rhs) {
    return mint(lhs) *= rhs;
  }
  friend mint operator/(const mint &lhs, const mint &rhs) {
    return mint(lhs) /= rhs;
  }
  friend bool operator==(const mint &lhs, const mint &rhs) {
    return lhs._v == rhs._v;
  }
  friend bool operator!=(const mint &lhs, const mint &rhs) {
    return lhs._v != rhs._v;
  }

private:
  unsigned int _v;
  static internal::barrett bt;
  static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt = 998244353;

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {

namespace internal {

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly(std::vector<mint> &a) {
  static constexpr int g = internal::primitive_root<mint::mod()>;
  int n = int(a.size());
  int h = internal::ceil_pow2(n);

  static bool first = true;
  static mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
  if (first) {
    first = false;
    mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
    int cnt2 = bsf(mint::mod() - 1);
    mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
    for (int i = cnt2; i >= 2; i--) {
      // e^(2^i) == 1
      es[i - 2] = e;
      ies[i - 2] = ie;
      e *= e;
      ie *= ie;
    }
    mint now = 1;
    for (int i = 0; i <= cnt2 - 2; i++) {
      sum_e[i] = es[i] * now;
      now *= ies[i];
    }
  }
  for (int ph = 1; ph <= h; ph++) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    mint now = 1;
    for (int s = 0; s < w; s++) {
      int offset = s << (h - ph + 1);
      for (int i = 0; i < p; i++) {
        auto l = a[i + offset];
        auto r = a[i + offset + p] * now;
        a[i + offset] = l + r;
        a[i + offset + p] = l - r;
      }
      now *= sum_e[bsf(~(unsigned int)(s))];
    }
  }
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly_inv(std::vector<mint> &a) {
  static constexpr int g = internal::primitive_root<mint::mod()>;
  int n = int(a.size());
  int h = internal::ceil_pow2(n);

  static bool first = true;
  static mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
  if (first) {
    first = false;
    mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
    int cnt2 = bsf(mint::mod() - 1);
    mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
    for (int i = cnt2; i >= 2; i--) {
      // e^(2^i) == 1
      es[i - 2] = e;
      ies[i - 2] = ie;
      e *= e;
      ie *= ie;
    }
    mint now = 1;
    for (int i = 0; i <= cnt2 - 2; i++) {
      sum_ie[i] = ies[i] * now;
      now *= es[i];
    }
  }

  for (int ph = h; ph >= 1; ph--) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    mint inow = 1;
    for (int s = 0; s < w; s++) {
      int offset = s << (h - ph + 1);
      for (int i = 0; i < p; i++) {
        auto l = a[i + offset];
        auto r = a[i + offset + p];
        a[i + offset] = l + r;
        a[i + offset + p] =
            (unsigned long long)(mint::mod() + l.val() - r.val()) * inow.val();
      }
      inow *= sum_ie[bsf(~(unsigned int)(s))];
    }
  }
}

} // namespace internal

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
  int n = int(a.size()), m = int(b.size());
  if (!n || !m)
    return {};
  if (std::min(n, m) <= 60) {
    if (n < m) {
      std::swap(n, m);
      std::swap(a, b);
    }
    std::vector<mint> ans(n + m - 1);
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        ans[i + j] += a[i] * b[j];
      }
    }
    return ans;
  }
  int z = 1 << internal::ceil_pow2(n + m - 1);
  a.resize(z);
  internal::butterfly(a);
  b.resize(z);
  internal::butterfly(b);
  for (int i = 0; i < z; i++) {
    a[i] *= b[i];
  }
  internal::butterfly_inv(a);
  a.resize(n + m - 1);
  mint iz = mint(z).inv();
  for (int i = 0; i < n + m - 1; i++)
    a[i] *= iz;
  return a;
}

template <unsigned int mod = 998244353, class T,
          std::enable_if_t<internal::is_integral<T>::value> * = nullptr>
std::vector<T> convolution(const std::vector<T> &a, const std::vector<T> &b) {
  int n = int(a.size()), m = int(b.size());
  if (!n || !m)
    return {};

  using mint = static_modint<mod>;
  std::vector<mint> a2(n), b2(m);
  for (int i = 0; i < n; i++) {
    a2[i] = mint(a[i]);
  }
  for (int i = 0; i < m; i++) {
    b2[i] = mint(b[i]);
  }
  auto c2 = convolution(move(a2), move(b2));
  std::vector<T> c(n + m - 1);
  for (int i = 0; i < n + m - 1; i++) {
    c[i] = c2[i].val();
  }
  return c;
}

std::vector<long long> convolution_ll(const std::vector<long long> &a,
                                      const std::vector<long long> &b) {
  int n = int(a.size()), m = int(b.size());
  if (!n || !m)
    return {};

  static constexpr unsigned long long MOD1 = 754974721; // 2^24
  static constexpr unsigned long long MOD2 = 167772161; // 2^25
  static constexpr unsigned long long MOD3 = 469762049; // 2^26
  static constexpr unsigned long long M2M3 = MOD2 * MOD3;
  static constexpr unsigned long long M1M3 = MOD1 * MOD3;
  static constexpr unsigned long long M1M2 = MOD1 * MOD2;
  static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

  static constexpr unsigned long long i1 =
      internal::inv_gcd(MOD2 * MOD3, MOD1).second;
  static constexpr unsigned long long i2 =
      internal::inv_gcd(MOD1 * MOD3, MOD2).second;
  static constexpr unsigned long long i3 =
      internal::inv_gcd(MOD1 * MOD2, MOD3).second;

  auto c1 = convolution<MOD1>(a, b);
  auto c2 = convolution<MOD2>(a, b);
  auto c3 = convolution<MOD3>(a, b);

  std::vector<long long> c(n + m - 1);
  for (int i = 0; i < n + m - 1; i++) {
    unsigned long long x = 0;
    x += (c1[i] * i1) % MOD1 * M2M3;
    x += (c2[i] * i2) % MOD2 * M1M3;
    x += (c3[i] * i3) % MOD3 * M1M2;
    // B = 2^63, -B <= x, r(real value) < B
    // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
    // r = c1[i] (mod MOD1)
    // focus on MOD1
    // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
    // r = x,
    //     x - M' + (0 or 2B),
    //     x - 2M' + (0, 2B or 4B),
    //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
    // (r - x) = 0, (0)
    //           - M' + (0 or 2B), (1)
    //           -2M' + (0 or 2B or 4B), (2)
    //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
    // we checked that
    //   ((1) mod MOD1) mod 5 = 2
    //   ((2) mod MOD1) mod 5 = 3
    //   ((3) mod MOD1) mod 5 = 4
    long long diff =
        c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
    if (diff < 0)
      diff += MOD1;
    static constexpr unsigned long long offset[5] = {0, 0, M1M2M3, 2 * M1M2M3,
                                                     3 * M1M2M3};
    x -= offset[diff % 5];
    c[i] = x;
  }

  return c;
}

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

// Implement (union by size) + (path compression)
// Reference:
// Zvi Galil and Giuseppe F. Italiano,
// Data structures and algorithms for disjoint set union problems
struct dsu {
public:
  dsu() : _n(0) {}
  dsu(int n) : _n(n), parent_or_size(n, -1) {}

  int merge(int a, int b) {
    assert(0 <= a && a < _n);
    assert(0 <= b && b < _n);
    int x = leader(a), y = leader(b);
    if (x == y)
      return x;
    if (-parent_or_size[x] < -parent_or_size[y])
      std::swap(x, y);
    parent_or_size[x] += parent_or_size[y];
    parent_or_size[y] = x;
    return x;
  }

  bool same(int a, int b) {
    assert(0 <= a && a < _n);
    assert(0 <= b && b < _n);
    return leader(a) == leader(b);
  }

  int leader(int a) {
    assert(0 <= a && a < _n);
    if (parent_or_size[a] < 0)
      return a;
    return parent_or_size[a] = leader(parent_or_size[a]);
  }

  int size(int a) {
    assert(0 <= a && a < _n);
    return -parent_or_size[leader(a)];
  }

  std::vector<std::vector<int>> groups() {
    std::vector<int> leader_buf(_n), group_size(_n);
    for (int i = 0; i < _n; i++) {
      leader_buf[i] = leader(i);
      group_size[leader_buf[i]]++;
    }
    std::vector<std::vector<int>> result(_n);
    for (int i = 0; i < _n; i++) {
      result[i].reserve(group_size[i]);
    }
    for (int i = 0; i < _n; i++) {
      result[leader_buf[i]].push_back(i);
    }
    result.erase(
        std::remove_if(result.begin(), result.end(),
                       [&](const std::vector<int> &v) { return v.empty(); }),
        result.end());
    return result;
  }

private:
  int _n;
  // root node: -1 * component size
  // otherwise: parent
  std::vector<int> parent_or_size;
};

} // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

// Reference: https://en.wikipedia.org/wiki/Fenwick_tree
template <class T> struct fenwick_tree {
  using U = internal::to_unsigned_t<T>;

public:
  fenwick_tree() : _n(0) {}
  fenwick_tree(int n) : _n(n), data(n) {}

  void add(int p, T x) {
    assert(0 <= p && p < _n);
    p++;
    while (p <= _n) {
      data[p - 1] += U(x);
      p += p & -p;
    }
  }

  T sum(int l, int r) {
    assert(0 <= l && l <= r && r <= _n);
    return sum(r) - sum(l);
  }

private:
  int _n;
  std::vector<U> data;

  U sum(int r) {
    U s = 0;
    while (r > 0) {
      s += data[r - 1];
      r -= r & -r;
    }
    return s;
  }
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
namespace atcoder {

template <class S, S (*op)(S, S), S (*e)(), class F, S (*mapping)(F, S),
          F (*composition)(F, F), F (*id)()>
struct lazy_segtree {
public:
  lazy_segtree() : lazy_segtree(0) {}
  lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
  lazy_segtree(const std::vector<S> &v) : _n(int(v.size())) {
    log = internal::ceil_pow2(_n);
    size = 1 << log;
    d = std::vector<S>(2 * size, e());
    lz = std::vector<F>(size, id());
    for (int i = 0; i < _n; i++)
      d[size + i] = v[i];
    for (int i = size - 1; i >= 1; i--) {
      update(i);
    }
  }

  void set(int p, S x) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--)
      push(p >> i);
    d[p] = x;
    for (int i = 1; i <= log; i++)
      update(p >> i);
  }

  S get(int p) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--)
      push(p >> i);
    return d[p];
  }

  S prod(int l, int r) {
    assert(0 <= l && l <= r && r <= _n);
    if (l == r)
      return e();

    l += size;
    r += size;

    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l)
        push(l >> i);
      if (((r >> i) << i) != r)
        push(r >> i);
    }

    S sml = e(), smr = e();
    while (l < r) {
      if (l & 1)
        sml = op(sml, d[l++]);
      if (r & 1)
        smr = op(d[--r], smr);
      l >>= 1;
      r >>= 1;
    }

    return op(sml, smr);
  }

  S all_prod() { return d[1]; }

  void apply(int p, F f) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--)
      push(p >> i);
    d[p] = mapping(f, d[p]);
    for (int i = 1; i <= log; i++)
      update(p >> i);
  }
  void apply(int l, int r, F f) {
    assert(0 <= l && l <= r && r <= _n);
    if (l == r)
      return;

    l += size;
    r += size;

    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l)
        push(l >> i);
      if (((r >> i) << i) != r)
        push((r - 1) >> i);
    }

    {
      int l2 = l, r2 = r;
      while (l < r) {
        if (l & 1)
          all_apply(l++, f);
        if (r & 1)
          all_apply(--r, f);
        l >>= 1;
        r >>= 1;
      }
      l = l2;
      r = r2;
    }

    for (int i = 1; i <= log; i++) {
      if (((l >> i) << i) != l)
        update(l >> i);
      if (((r >> i) << i) != r)
        update((r - 1) >> i);
    }
  }

  template <bool (*g)(S)> int max_right(int l) {
    return max_right(l, [](S x) { return g(x); });
  }
  template <class G> int max_right(int l, G g) {
    assert(0 <= l && l <= _n);
    assert(g(e()));
    if (l == _n)
      return _n;
    l += size;
    for (int i = log; i >= 1; i--)
      push(l >> i);
    S sm = e();
    do {
      while (l % 2 == 0)
        l >>= 1;
      if (!g(op(sm, d[l]))) {
        while (l < size) {
          push(l);
          l = (2 * l);
          if (g(op(sm, d[l]))) {
            sm = op(sm, d[l]);
            l++;
          }
        }
        return l - size;
      }
      sm = op(sm, d[l]);
      l++;
    } while ((l & -l) != l);
    return _n;
  }

  template <bool (*g)(S)> int min_left(int r) {
    return min_left(r, [](S x) { return g(x); });
  }
  template <class G> int min_left(int r, G g) {
    assert(0 <= r && r <= _n);
    assert(g(e()));
    if (r == 0)
      return 0;
    r += size;
    for (int i = log; i >= 1; i--)
      push((r - 1) >> i);
    S sm = e();
    do {
      r--;
      while (r > 1 && (r % 2))
        r >>= 1;
      if (!g(op(d[r], sm))) {
        while (r < size) {
          push(r);
          r = (2 * r + 1);
          if (g(op(d[r], sm))) {
            sm = op(d[r], sm);
            r--;
          }
        }
        return r + 1 - size;
      }
      sm = op(d[r], sm);
    } while ((r & -r) != r);
    return 0;
  }

private:
  int _n, size, log;
  std::vector<S> d;
  std::vector<F> lz;

  void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
  void all_apply(int k, F f) {
    d[k] = mapping(f, d[k]);
    if (k < size)
      lz[k] = composition(f, lz[k]);
  }
  void push(int k) {
    all_apply(2 * k, lz[k]);
    all_apply(2 * k + 1, lz[k]);
    lz[k] = id();
  }
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>

namespace atcoder {

long long pow_mod(long long x, long long n, int m) {
  assert(0 <= n && 1 <= m);
  if (m == 1)
    return 0;
  internal::barrett bt((unsigned int)(m));
  unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
  while (n) {
    if (n & 1)
      r = bt.mul(r, y);
    y = bt.mul(y, y);
    n >>= 1;
  }
  return r;
}

long long inv_mod(long long x, long long m) {
  assert(1 <= m);
  auto z = internal::inv_gcd(x, m);
  assert(z.first == 1);
  return z.second;
}

// (rem, mod)
std::pair<long long, long long> crt(const std::vector<long long> &r,
                                    const std::vector<long long> &m) {
  assert(r.size() == m.size());
  int n = int(r.size());
  // Contracts: 0 <= r0 < m0
  long long r0 = 0, m0 = 1;
  for (int i = 0; i < n; i++) {
    assert(1 <= m[i]);
    long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
    if (m0 < m1) {
      std::swap(r0, r1);
      std::swap(m0, m1);
    }
    if (m0 % m1 == 0) {
      if (r0 % m1 != r1)
        return {0, 0};
      continue;
    }
    // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)

    // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
    // r2 % m0 = r0
    // r2 % m1 = r1
    // -> (r0 + x*m0) % m1 = r1
    // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
    // -> x = (r1 - r0) / g * inv(u0) (mod u1)

    // im = inv(u0) (mod u1) (0 <= im < u1)
    long long g, im;
    std::tie(g, im) = internal::inv_gcd(m0, m1);

    long long u1 = (m1 / g);
    // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
    if ((r1 - r0) % g)
      return {0, 0};

    // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
    long long x = (r1 - r0) / g % u1 * im % u1;

    // |r0| + |m0 * x|
    // < m0 + m0 * (u1 - 1)
    // = m0 + m0 * m1 / g - m0
    // = lcm(m0, m1)
    r0 += x * m0;
    m0 *= u1; // -> lcm(m0, m1)
    if (r0 < 0)
      r0 += m0;
  }
  return {r0, m0};
}

long long floor_sum(long long n, long long m, long long a, long long b) {
  long long ans = 0;
  if (a >= m) {
    ans += (n - 1) * n * (a / m) / 2;
    a %= m;
  }
  if (b >= m) {
    ans += n * (b / m);
    b %= m;
  }

  long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
  if (y_max == 0)
    return ans;
  ans += (n - (x_max + a - 1) / a) * y_max;
  ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
  return ans;
}

} // namespace atcoder

#include <algorithm>

#include <vector>

namespace atcoder {

namespace internal {

template <class T> struct simple_queue {
  std::vector<T> payload;
  int pos = 0;
  void reserve(int n) { payload.reserve(n); }
  int size() const { return int(payload.size()) - pos; }
  bool empty() const { return pos == int(payload.size()); }
  void push(const T &t) { payload.push_back(t); }
  T &front() { return payload[pos]; }
  void clear() {
    payload.clear();
    pos = 0;
  }
  void pop() { pos++; }
};

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

template <class Cap> struct mf_graph {
public:
  mf_graph() : _n(0) {}
  mf_graph(int n) : _n(n), g(n) {}

  int add_edge(int from, int to, Cap cap) {
    assert(0 <= from && from < _n);
    assert(0 <= to && to < _n);
    assert(0 <= cap);
    int m = int(pos.size());
    pos.push_back({from, int(g[from].size())});
    int from_id = int(g[from].size());
    int to_id = int(g[to].size());
    if (from == to)
      to_id++;
    g[from].push_back(_edge{to, to_id, cap});
    g[to].push_back(_edge{from, from_id, 0});
    return m;
  }

  struct edge {
    int from, to;
    Cap cap, flow;
  };

  edge get_edge(int i) {
    int m = int(pos.size());
    assert(0 <= i && i < m);
    auto _e = g[pos[i].first][pos[i].second];
    auto _re = g[_e.to][_e.rev];
    return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
  }
  std::vector<edge> edges() {
    int m = int(pos.size());
    std::vector<edge> result;
    for (int i = 0; i < m; i++) {
      result.push_back(get_edge(i));
    }
    return result;
  }
  void change_edge(int i, Cap new_cap, Cap new_flow) {
    int m = int(pos.size());
    assert(0 <= i && i < m);
    assert(0 <= new_flow && new_flow <= new_cap);
    auto &_e = g[pos[i].first][pos[i].second];
    auto &_re = g[_e.to][_e.rev];
    _e.cap = new_cap - new_flow;
    _re.cap = new_flow;
  }

  Cap flow(int s, int t) { return flow(s, t, std::numeric_limits<Cap>::max()); }
  Cap flow(int s, int t, Cap flow_limit) {
    assert(0 <= s && s < _n);
    assert(0 <= t && t < _n);
    assert(s != t);

    std::vector<int> level(_n), iter(_n);
    internal::simple_queue<int> que;

    auto bfs = [&]() {
      std::fill(level.begin(), level.end(), -1);
      level[s] = 0;
      que.clear();
      que.push(s);
      while (!que.empty()) {
        int v = que.front();
        que.pop();
        for (auto e : g[v]) {
          if (e.cap == 0 || level[e.to] >= 0)
            continue;
          level[e.to] = level[v] + 1;
          if (e.to == t)
            return;
          que.push(e.to);
        }
      }
    };
    auto dfs = [&](auto self, int v, Cap up) {
      if (v == s)
        return up;
      Cap res = 0;
      int level_v = level[v];
      for (int &i = iter[v]; i < int(g[v].size()); i++) {
        _edge &e = g[v][i];
        if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0)
          continue;
        Cap d = self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
        if (d <= 0)
          continue;
        g[v][i].cap += d;
        g[e.to][e.rev].cap -= d;
        res += d;
        if (res == up)
          break;
      }
      return res;
    };

    Cap flow = 0;
    while (flow < flow_limit) {
      bfs();
      if (level[t] == -1)
        break;
      std::fill(iter.begin(), iter.end(), 0);
      while (flow < flow_limit) {
        Cap f = dfs(dfs, t, flow_limit - flow);
        if (!f)
          break;
        flow += f;
      }
    }
    return flow;
  }

  std::vector<bool> min_cut(int s) {
    std::vector<bool> visited(_n);
    internal::simple_queue<int> que;
    que.push(s);
    while (!que.empty()) {
      int p = que.front();
      que.pop();
      visited[p] = true;
      for (auto e : g[p]) {
        if (e.cap && !visited[e.to]) {
          visited[e.to] = true;
          que.push(e.to);
        }
      }
    }
    return visited;
  }

private:
  int _n;
  struct _edge {
    int to, rev;
    Cap cap;
  };
  std::vector<std::pair<int, int>> pos;
  std::vector<std::vector<_edge>> g;
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

template <class Cap, class Cost> struct mcf_graph {
public:
  mcf_graph() {}
  mcf_graph(int n) : _n(n), g(n) {}

  int add_edge(int from, int to, Cap cap, Cost cost) {
    assert(0 <= from && from < _n);
    assert(0 <= to && to < _n);
    int m = int(pos.size());
    pos.push_back({from, int(g[from].size())});
    int from_id = int(g[from].size());
    int to_id = int(g[to].size());
    if (from == to)
      to_id++;
    g[from].push_back(_edge{to, to_id, cap, cost});
    g[to].push_back(_edge{from, from_id, 0, -cost});
    return m;
  }

  struct edge {
    int from, to;
    Cap cap, flow;
    Cost cost;
  };

  edge get_edge(int i) {
    int m = int(pos.size());
    assert(0 <= i && i < m);
    auto _e = g[pos[i].first][pos[i].second];
    auto _re = g[_e.to][_e.rev];
    return edge{
        pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,
    };
  }
  std::vector<edge> edges() {
    int m = int(pos.size());
    std::vector<edge> result(m);
    for (int i = 0; i < m; i++) {
      result[i] = get_edge(i);
    }
    return result;
  }

  std::pair<Cap, Cost> flow(int s, int t) {
    return flow(s, t, std::numeric_limits<Cap>::max());
  }
  std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
    return slope(s, t, flow_limit).back();
  }
  std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
    return slope(s, t, std::numeric_limits<Cap>::max());
  }
  std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
    assert(0 <= s && s < _n);
    assert(0 <= t && t < _n);
    assert(s != t);
    // variants (C = maxcost):
    // -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0
    // reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge
    std::vector<Cost> dual(_n, 0), dist(_n);
    std::vector<int> pv(_n), pe(_n);
    std::vector<bool> vis(_n);
    auto dual_ref = [&]() {
      std::fill(dist.begin(), dist.end(), std::numeric_limits<Cost>::max());
      std::fill(pv.begin(), pv.end(), -1);
      std::fill(pe.begin(), pe.end(), -1);
      std::fill(vis.begin(), vis.end(), false);
      struct Q {
        Cost key;
        int to;
        bool operator<(Q r) const { return key > r.key; }
      };
      std::priority_queue<Q> que;
      dist[s] = 0;
      que.push(Q{0, s});
      while (!que.empty()) {
        int v = que.top().to;
        que.pop();
        if (vis[v])
          continue;
        vis[v] = true;
        if (v == t)
          break;
        // dist[v] = shortest(s, v) + dual[s] - dual[v]
        // dist[v] >= 0 (all reduced cost are positive)
        // dist[v] <= (n-1)C
        for (int i = 0; i < int(g[v].size()); i++) {
          auto e = g[v][i];
          if (vis[e.to] || !e.cap)
            continue;
          // |-dual[e.to] + dual[v]| <= (n-1)C
          // cost <= C - -(n-1)C + 0 = nC
          Cost cost = e.cost - dual[e.to] + dual[v];
          if (dist[e.to] - dist[v] > cost) {
            dist[e.to] = dist[v] + cost;
            pv[e.to] = v;
            pe[e.to] = i;
            que.push(Q{dist[e.to], e.to});
          }
        }
      }
      if (!vis[t]) {
        return false;
      }

      for (int v = 0; v < _n; v++) {
        if (!vis[v])
          continue;
        // dual[v] = dual[v] - dist[t] + dist[v]
        //         = dual[v] - (shortest(s, t) + dual[s] - dual[t]) +
        //         (shortest(s, v) + dual[s] - dual[v]) = - shortest(s, t) +
        //         dual[t] + shortest(s, v) = shortest(s, v) - shortest(s, t) >=
        //         0 - (n-1)C
        dual[v] -= dist[t] - dist[v];
      }
      return true;
    };
    Cap flow = 0;
    Cost cost = 0, prev_cost_per_flow = -1;
    std::vector<std::pair<Cap, Cost>> result;
    result.push_back({flow, cost});
    while (flow < flow_limit) {
      if (!dual_ref())
        break;
      Cap c = flow_limit - flow;
      for (int v = t; v != s; v = pv[v]) {
        c = std::min(c, g[pv[v]][pe[v]].cap);
      }
      for (int v = t; v != s; v = pv[v]) {
        auto &e = g[pv[v]][pe[v]];
        e.cap -= c;
        g[v][e.rev].cap += c;
      }
      Cost d = -dual[s];
      flow += c;
      cost += c * d;
      if (prev_cost_per_flow == d) {
        result.pop_back();
      }
      result.push_back({flow, cost});
      prev_cost_per_flow = d;
    }
    return result;
  }

private:
  int _n;

  struct _edge {
    int to, rev;
    Cap cap;
    Cost cost;
  };

  std::vector<std::pair<int, int>> pos;
  std::vector<std::vector<_edge>> g;
};

} // namespace atcoder

#include <algorithm>

#include <algorithm>
#include <utility>
#include <vector>

namespace atcoder {
namespace internal {

template <class E> struct csr {
  std::vector<int> start;
  std::vector<E> elist;
  csr(int n, const std::vector<std::pair<int, E>> &edges)
      : start(n + 1), elist(edges.size()) {
    for (auto e : edges) {
      start[e.first + 1]++;
    }
    for (int i = 1; i <= n; i++) {
      start[i] += start[i - 1];
    }
    auto counter = start;
    for (auto e : edges) {
      elist[counter[e.first]++] = e.second;
    }
  }
};

// Reference:
// R. Tarjan,
// Depth-First Search and Linear Graph Algorithms
struct scc_graph {
public:
  scc_graph(int n) : _n(n) {}

  int num_vertices() { return _n; }

  void add_edge(int from, int to) { edges.push_back({from, {to}}); }

  // @return pair of (# of scc, scc id)
  std::pair<int, std::vector<int>> scc_ids() {
    auto g = csr<edge>(_n, edges);
    int now_ord = 0, group_num = 0;
    std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
    visited.reserve(_n);
    auto dfs = [&](auto self, int v) -> void {
      low[v] = ord[v] = now_ord++;
      visited.push_back(v);
      for (int i = g.start[v]; i < g.start[v + 1]; i++) {
        auto to = g.elist[i].to;
        if (ord[to] == -1) {
          self(self, to);
          low[v] = std::min(low[v], low[to]);
        } else {
          low[v] = std::min(low[v], ord[to]);
        }
      }
      if (low[v] == ord[v]) {
        while (true) {
          int u = visited.back();
          visited.pop_back();
          ord[u] = _n;
          ids[u] = group_num;
          if (u == v)
            break;
        }
        group_num++;
      }
    };
    for (int i = 0; i < _n; i++) {
      if (ord[i] == -1)
        dfs(dfs, i);
    }
    for (auto &x : ids) {
      x = group_num - 1 - x;
    }
    return {group_num, ids};
  }

  std::vector<std::vector<int>> scc() {
    auto ids = scc_ids();
    int group_num = ids.first;
    std::vector<int> counts(group_num);
    for (auto x : ids.second)
      counts[x]++;
    std::vector<std::vector<int>> groups(ids.first);
    for (int i = 0; i < group_num; i++) {
      groups[i].reserve(counts[i]);
    }
    for (int i = 0; i < _n; i++) {
      groups[ids.second[i]].push_back(i);
    }
    return groups;
  }

private:
  int _n;
  struct edge {
    int to;
  };
  std::vector<std::pair<int, edge>> edges;
};

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

struct scc_graph {
public:
  scc_graph() : internal(0) {}
  scc_graph(int n) : internal(n) {}

  void add_edge(int from, int to) {
    int n = internal.num_vertices();
    assert(0 <= from && from < n);
    assert(0 <= to && to < n);
    internal.add_edge(from, to);
  }

  std::vector<std::vector<int>> scc() { return internal.scc(); }

private:
  internal::scc_graph internal;
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

template <class S, S (*op)(S, S), S (*e)()> struct segtree {
public:
  segtree() : segtree(0) {}
  segtree(int n) : segtree(std::vector<S>(n, e())) {}
  segtree(const std::vector<S> &v) : _n(int(v.size())) {
    log = internal::ceil_pow2(_n);
    size = 1 << log;
    d = std::vector<S>(2 * size, e());
    for (int i = 0; i < _n; i++)
      d[size + i] = v[i];
    for (int i = size - 1; i >= 1; i--) {
      update(i);
    }
  }

  void set(int p, S x) {
    assert(0 <= p && p < _n);
    p += size;
    d[p] = x;
    for (int i = 1; i <= log; i++)
      update(p >> i);
  }

  S get(int p) {
    assert(0 <= p && p < _n);
    return d[p + size];
  }

  S prod(int l, int r) {
    assert(0 <= l && l <= r && r <= _n);
    S sml = e(), smr = e();
    l += size;
    r += size;

    while (l < r) {
      if (l & 1)
        sml = op(sml, d[l++]);
      if (r & 1)
        smr = op(d[--r], smr);
      l >>= 1;
      r >>= 1;
    }
    return op(sml, smr);
  }

  S all_prod() { return d[1]; }

  template <bool (*f)(S)> int max_right(int l) {
    return max_right(l, [](S x) { return f(x); });
  }
  template <class F> int max_right(int l, F f) {
    assert(0 <= l && l <= _n);
    assert(f(e()));
    if (l == _n)
      return _n;
    l += size;
    S sm = e();
    do {
      while (l % 2 == 0)
        l >>= 1;
      if (!f(op(sm, d[l]))) {
        while (l < size) {
          l = (2 * l);
          if (f(op(sm, d[l]))) {
            sm = op(sm, d[l]);
            l++;
          }
        }
        return l - size;
      }
      sm = op(sm, d[l]);
      l++;
    } while ((l & -l) != l);
    return _n;
  }

  template <bool (*f)(S)> int min_left(int r) {
    return min_left(r, [](S x) { return f(x); });
  }
  template <class F> int min_left(int r, F f) {
    assert(0 <= r && r <= _n);
    assert(f(e()));
    if (r == 0)
      return 0;
    r += size;
    S sm = e();
    do {
      r--;
      while (r > 1 && (r % 2))
        r >>= 1;
      if (!f(op(d[r], sm))) {
        while (r < size) {
          r = (2 * r + 1);
          if (f(op(d[r], sm))) {
            sm = op(d[r], sm);
            r--;
          }
        }
        return r + 1 - size;
      }
      sm = op(d[r], sm);
    } while ((r & -r) != r);
    return 0;
  }

private:
  int _n, size, log;
  std::vector<S> d;

  void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
};

} // namespace atcoder

#include <algorithm>
#include <cassert>
#include <numeric>
#include <string>
#include <vector>

namespace atcoder {

namespace internal {

std::vector<int> sa_naive(const std::vector<int> &s) {
  int n = int(s.size());
  std::vector<int> sa(n);
  std::iota(sa.begin(), sa.end(), 0);
  std::sort(sa.begin(), sa.end(), [&](int l, int r) {
    if (l == r)
      return false;
    while (l < n && r < n) {
      if (s[l] != s[r])
        return s[l] < s[r];
      l++;
      r++;
    }
    return l == n;
  });
  return sa;
}

std::vector<int> sa_doubling(const std::vector<int> &s) {
  int n = int(s.size());
  std::vector<int> sa(n), rnk = s, tmp(n);
  std::iota(sa.begin(), sa.end(), 0);
  for (int k = 1; k < n; k *= 2) {
    auto cmp = [&](int x, int y) {
      if (rnk[x] != rnk[y])
        return rnk[x] < rnk[y];
      int rx = x + k < n ? rnk[x + k] : -1;
      int ry = y + k < n ? rnk[y + k] : -1;
      return rx < ry;
    };
    std::sort(sa.begin(), sa.end(), cmp);
    tmp[sa[0]] = 0;
    for (int i = 1; i < n; i++) {
      tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);
    }
    std::swap(tmp, rnk);
  }
  return sa;
}

// SA-IS, linear-time suffix array construction
// Reference:
// G. Nong, S. Zhang, and W. H. Chan,
// Two Efficient Algorithms for Linear Time Suffix Array Construction
template <int THRESHOLD_NAIVE = 10, int THRESHOLD_DOUBLING = 40>
std::vector<int> sa_is(const std::vector<int> &s, int upper) {
  int n = int(s.size());
  if (n == 0)
    return {};
  if (n == 1)
    return {0};
  if (n == 2) {
    if (s[0] < s[1]) {
      return {0, 1};
    } else {
      return {1, 0};
    }
  }
  if (n < THRESHOLD_NAIVE) {
    return sa_naive(s);
  }
  if (n < THRESHOLD_DOUBLING) {
    return sa_doubling(s);
  }

  std::vector<int> sa(n);
  std::vector<bool> ls(n);
  for (int i = n - 2; i >= 0; i--) {
    ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);
  }
  std::vector<int> sum_l(upper + 1), sum_s(upper + 1);
  for (int i = 0; i < n; i++) {
    if (!ls[i]) {
      sum_s[s[i]]++;
    } else {
      sum_l[s[i] + 1]++;
    }
  }
  for (int i = 0; i <= upper; i++) {
    sum_s[i] += sum_l[i];
    if (i < upper)
      sum_l[i + 1] += sum_s[i];
  }

  auto induce = [&](const std::vector<int> &lms) {
    std::fill(sa.begin(), sa.end(), -1);
    std::vector<int> buf(upper + 1);
    std::copy(sum_s.begin(), sum_s.end(), buf.begin());
    for (auto d : lms) {
      if (d == n)
        continue;
      sa[buf[s[d]]++] = d;
    }
    std::copy(sum_l.begin(), sum_l.end(), buf.begin());
    sa[buf[s[n - 1]]++] = n - 1;
    for (int i = 0; i < n; i++) {
      int v = sa[i];
      if (v >= 1 && !ls[v - 1]) {
        sa[buf[s[v - 1]]++] = v - 1;
      }
    }
    std::copy(sum_l.begin(), sum_l.end(), buf.begin());
    for (int i = n - 1; i >= 0; i--) {
      int v = sa[i];
      if (v >= 1 && ls[v - 1]) {
        sa[--buf[s[v - 1] + 1]] = v - 1;
      }
    }
  };

  std::vector<int> lms_map(n + 1, -1);
  int m = 0;
  for (int i = 1; i < n; i++) {
    if (!ls[i - 1] && ls[i]) {
      lms_map[i] = m++;
    }
  }
  std::vector<int> lms;
  lms.reserve(m);
  for (int i = 1; i < n; i++) {
    if (!ls[i - 1] && ls[i]) {
      lms.push_back(i);
    }
  }

  induce(lms);

  if (m) {
    std::vector<int> sorted_lms;
    sorted_lms.reserve(m);
    for (int v : sa) {
      if (lms_map[v] != -1)
        sorted_lms.push_back(v);
    }
    std::vector<int> rec_s(m);
    int rec_upper = 0;
    rec_s[lms_map[sorted_lms[0]]] = 0;
    for (int i = 1; i < m; i++) {
      int l = sorted_lms[i - 1], r = sorted_lms[i];
      int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;
      int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;
      bool same = true;
      if (end_l - l != end_r - r) {
        same = false;
      } else {
        while (l < end_l) {
          if (s[l] != s[r]) {
            break;
          }
          l++;
          r++;
        }
        if (l == n || s[l] != s[r])
          same = false;
      }
      if (!same)
        rec_upper++;
      rec_s[lms_map[sorted_lms[i]]] = rec_upper;
    }

    auto rec_sa = sa_is<THRESHOLD_NAIVE, THRESHOLD_DOUBLING>(rec_s, rec_upper);

    for (int i = 0; i < m; i++) {
      sorted_lms[i] = lms[rec_sa[i]];
    }
    induce(sorted_lms);
  }
  return sa;
}

} // namespace internal

std::vector<int> suffix_array(const std::vector<int> &s, int upper) {
  assert(0 <= upper);
  for (int d : s) {
    assert(0 <= d && d <= upper);
  }
  auto sa = internal::sa_is(s, upper);
  return sa;
}

template <class T> std::vector<int> suffix_array(const std::vector<T> &s) {
  int n = int(s.size());
  std::vector<int> idx(n);
  iota(idx.begin(), idx.end(), 0);
  sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });
  std::vector<int> s2(n);
  int now = 0;
  for (int i = 0; i < n; i++) {
    if (i && s[idx[i - 1]] != s[idx[i]])
      now++;
    s2[idx[i]] = now;
  }
  return internal::sa_is(s2, now);
}

std::vector<int> suffix_array(const std::string &s) {
  int n = int(s.size());
  std::vector<int> s2(n);
  for (int i = 0; i < n; i++) {
    s2[i] = s[i];
  }
  return internal::sa_is(s2, 255);
}

// Reference:
// T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park,
// Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its
// Applications
template <class T>
std::vector<int> lcp_array(const std::vector<T> &s,
                           const std::vector<int> &sa) {
  int n = int(s.size());
  assert(n >= 1);
  std::vector<int> rnk(n);
  for (int i = 0; i < n; i++) {
    rnk[sa[i]] = i;
  }
  std::vector<int> lcp(n - 1);
  int h = 0;
  for (int i = 0; i < n; i++) {
    if (h > 0)
      h--;
    if (rnk[i] == 0)
      continue;
    int j = sa[rnk[i] - 1];
    for (; j + h < n && i + h < n; h++) {
      if (s[j + h] != s[i + h])
        break;
    }
    lcp[rnk[i] - 1] = h;
  }
  return lcp;
}

std::vector<int> lcp_array(const std::string &s, const std::vector<int> &sa) {
  int n = int(s.size());
  std::vector<int> s2(n);
  for (int i = 0; i < n; i++) {
    s2[i] = s[i];
  }
  return lcp_array(s2, sa);
}

// Reference:
// D. Gusfield,
// Algorithms on Strings, Trees, and Sequences: Computer Science and
// Computational Biology
template <class T> std::vector<int> z_algorithm(const std::vector<T> &s) {
  int n = int(s.size());
  if (n == 0)
    return {};
  std::vector<int> z(n);
  z[0] = 0;
  for (int i = 1, j = 0; i < n; i++) {
    int &k = z[i];
    k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);
    while (i + k < n && s[k] == s[i + k])
      k++;
    if (j + z[j] < i + z[i])
      j = i;
  }
  z[0] = n;
  return z;
}

std::vector<int> z_algorithm(const std::string &s) {
  int n = int(s.size());
  std::vector<int> s2(n);
  for (int i = 0; i < n; i++) {
    s2[i] = s[i];
  }
  return z_algorithm(s2);
}

} // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

// Reference:
// B. Aspvall, M. Plass, and R. Tarjan,
// A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean
// Formulas
struct two_sat {
public:
  two_sat() : _n(0), scc(0) {}
  two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}

  void add_clause(int i, bool f, int j, bool g) {
    assert(0 <= i && i < _n);
    assert(0 <= j && j < _n);
    scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
    scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
  }
  bool satisfiable() {
    auto id = scc.scc_ids().second;
    for (int i = 0; i < _n; i++) {
      if (id[2 * i] == id[2 * i + 1])
        return false;
      _answer[i] = id[2 * i] < id[2 * i + 1];
    }
    return true;
  }
  std::vector<bool> answer() { return _answer; }

private:
  int _n;
  std::vector<bool> _answer;
  internal::scc_graph scc;
};

} // namespace atcoder

using namespace std;
using namespace atcoder;
typedef long long ll;
/*
#define i_7 (ll)(1E9+7)
//#define i_7 998244353
#define i_5 i_7-2
ll mod(ll a){
    ll c=a%i_7;
    if(c>=0)return c;
    return c+i_7;
}
void Add(ll &pos,ll val){pos=mod(pos+val);}
void mod_print(ll k){
    ll P=50000;
    rep(y,1,P){
        ll x=mod(y*k);

        if(abs(x)<=P||x+P>=i_7){
            if(x+P>=i_7){
                x-=i_7;
            }
            cout<<x<<"/"<<y<<endl;
            //cout<<setprecision(5)<<(dd)x/(dd)y;
            return;
        }
    }
    cout<<"nun"<<endl;
}
*/

#include "boost/multiprecision/cpp_dec_float.hpp"
#include "boost/multiprecision/cpp_int.hpp"
namespace multi = boost::multiprecision;
typedef multi::cpp_int LL;
typedef multi::number<multi::cpp_dec_float<1024>>
    DD; // 仮数部が1024ビットの浮動小数点数型(TLEしたら小さくする)

typedef long double dd;
typedef pair<ll, ll> l_l;
typedef pair<dd, dd> d_d;
ll inf = (ll)1E16;
#define rep(i, l, r) for (ll i = l; i <= r; i++)
#define rrep(i, r, l) for (ll i = r; i >= l; i--)
#define pb push_back
ll max(ll a, ll b) {
  if (a < b)
    return b;
  else
    return a;
}
ll min(ll a, ll b) {
  if (a > b)
    return b;
  else
    return a;
}
void Max(ll &pos, ll val) { pos = max(pos, val); } // Max(dp[n],dp[n-1]);
void Min(ll &pos, ll val) { pos = min(pos, val); }
dd EPS = 1E-12;
#define fastio                                                                 \\
  ios::sync_with_stdio(false);                                                 \\
  cin.tie(0);                                                                  \\
  cout.tie(0);
#define fi first
#define se second

#define SORT(v) sort(v.begin(), v.end())
#define ERASE(v) v.erase(unique(v.begin(), v.end()), v.end())
#define POSL(v, x) (lower_bound(v.begin(), v.end(), x) - v.begin())
#define POSU(v, x) (upper_bound(v.begin(), v.end(), x) - v.begin())
template <class T, class S> inline bool chmax(T &a, S b) {
  if (a < b) {
    a = (T)b;
    return true;
  }
  return false;
}
template <class T, class S> inline bool chmin(T &a, S b) {
  if (a > b) {
    a = (T)b;
    return true;
  }
  return false;
}
#define endl                                                                   \\
  "\\n" // インタラクティブで消す！！！！！！！！！！！！！！！！！！！！！
#define all(c) c.begin(), c.end()

using mint = modint998244353;
// using mint = modint1000000007;
// using mint=modint;
// using mint=static_modint<100>;

typedef vector<ll> vl;
typedef vector<mint> vi;
dd PI = acos((dd)-1);
//////////////////////////

#define N 2000004
mint kai[N];
mint kai2[N];
void calc() {
  kai[0] = 1;
  kai2[0] = 1;
  rep(i, 1, N - 1) { kai[i] = kai[i - 1] * i; }
  kai2[N - 1] = 1 / kai[N - 1];
  for (ll i = N - 2; i >= 0; i--) {
    kai2[i] = kai2[i + 1] * (i + 1);
  }
}
mint comb(ll n, ll k) {
  if (n < k)
    return 0;
  if (n == 0)
    return 1;
  return kai[n] * kai2[n - k] * kai2[k];
}

int main() {
  fastio calc();
  ll n;
  cin >> n;
  vl h(2 * n);
  rep(i, 0, 2 * n - 1) cin >> h[i];
  SORT(h);
  vector<ll> a;
  rep(i, 0, 2 * n - 1) {
    ll c = 1;
    while (i + 1 <= 2 * n - 1 && h[i] == h[i + 1]) {
      c++;
      i++;
    }
    a.pb(c);
  }
  //  for(auto x:a)cout<<x<<" ";cout<<endl;
  vi t;
  t.pb(1);
  for (auto x : a) {
    vi v;
    mint mulc = 1;
    v.pb(1);
    for (ll i = 2; i <= x; i += 2) {
      mulc *= comb(x + 2 - i, 2) / (i / 2);
      v.pb(mulc);
    }
    t = convolution(t, v);
  }
  //  for(auto x:t)cout<<x.val()<<" ";cout<<endl;
  mint A[n + 1], kai[n + 1];
  A[n] = 1;
  rrep(i, n - 1, 0) A[i] = A[i + 1] * comb(2 * n - 2 * i, 2);
  kai[0] = 1;
  rep(i, 1, n) kai[i] = kai[i - 1] * i;
  ll ts = t.size();
  rep(i, 0, ts - 1) {
    t[i] *= A[i];
    t[i] /= kai[n - i];
  }
  // for(auto x:t)cout<<x.val()<<" ";cout<<endl;
  mint ans = 0;
  rep(i, 0, ts - 1) {
    if (i % 2 == 0) {
      ans += t[i];
    } else {
      ans -= t[i];
    }
  }
  cout << ans.val() << endl;

  return 0;
}