/**
 * code generated by JHelper
 * More info: https://github.com/AlexeyDmitriev/JHelper
 * @author aajisaka
 */


#include<bits/stdc++.h>

using namespace std;

void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
  cerr << " " << to_string(H);
  debug_out(T...);
}
#ifdef LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif

#define SPEED ios_base::sync_with_stdio(false);cin.tie(nullptr)
#define rep(i,n) for(int i=0; i<(int)(n); i++)
#define all(v) v.begin(), v.end()
template<class T> inline bool chmax(T& a, T b) { if (a < b) { a = b; return true; } return false; }
template<class T> inline bool chmin(T& a, T b) { if (a > b) { a = b; return true; } return false; }

using ll = long long;
using ull = unsigned long long;
using P = pair<ll, ll>;

constexpr long double PI = 3.14159265358979323846264338327950288L;
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());


#include <utility>

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 moduler 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`
        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;
      for (long long a : {2, 7, 61}) {
        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);
}

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

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;

}





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

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>;
}

#include <algorithm>
#include <array>


#include <vector>


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

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 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))];
        }
      }
    }
}

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;
}


using mint = modint998244353;
auto half = 1/mint(2);

class FHeightsAndPairs {
public:

    vector<mint> get_vec(int n, vector<mint>& ifact) {
      vector<mint> ret;
      ret.emplace_back(1);
      while(n >= 2) {
        ret.push_back(ret.back());
        ret.back() *= n*(n-1)/2;
        n -= 2;
      }
      for(int i=0; i<ret.size(); i++) {
        ret[i] *= ifact[i];
      }
      return ret;
    }

    void solve(istream& cin, ostream& cout) {
      SPEED;
      int n; cin >> n;
      vector<int> a(2*n);
      vector<int> posa(100000);

      vector<mint> fact(n+1), ifact(n+1);
      fact[0] = 1;
      rep(i, n) {
        fact[i+1] = fact[i]*(i+1);
      }
      ifact[n] = fact[n].inv();
      for(int i=n-1; i>=0; i--) {
        ifact[i] = ifact[i+1]*(i+1);
      }

      rep(i, 2*n) {
        cin >> a[i];
        a[i]--;
        posa[a[i]]++;
      }
      vector<int> v;
      rep(i, 100000) {
        if (posa[i] >= 1) {
          v.push_back(posa[i]);
        }
        if (posa[i] > n) {
          cout << 0 << endl;
          return;
        }
      }

      vector<mint> dp[v.size()];
      priority_queue<P, vector<P>, greater<>> que;
      rep(i, v.size()) {
        dp[i] = get_vec(v[i], ifact);
        que.emplace(dp[i].size(), i);
      }

      while(que.size() >= 2) {
        auto p = que.top(); que.pop();
        auto q = que.top(); que.pop();
        int newPos = q.second;
        dp[newPos] = convolution(dp[p.second], dp[q.second]);
        que.emplace(dp[newPos].size(), newPos);
      }

      mint ans = 0;
      auto r = que.top();
      auto& ret = dp[r.second];
      vector<mint> pairs(n+1);
      pairs[1] = 1;
      for(int i=2; i<=n; i++) {
        pairs[i] = pairs[i-1] * (i*2-1) * (i*2) * half;
      }
      for(int i=2; i<=n; i++) {
        pairs[i] *= ifact[i];
      }

      for(int i=0; i<ret.size(); i++) {
        mint now = ret[i];
        now *= pairs[n-i];
        debug(i, now.val());
        if (i%2!=0) {
          now *= -1;
        }
        ans += now;
      }
      cout << ans.val() << endl;
    }
};

signed main() {
  FHeightsAndPairs solver;
  std::istream& in(std::cin);
  std::ostream& out(std::cout);
  solver.solve(in, out);
  return 0;
}