#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  // #define USE_GETLINE
#endif

#ifdef INCLUDE_MAIN

IN VO Solve()
{
  CEXPR( uint , P , 998244353 );
  using MP = Mod<P>;
  CEXPR( int , bound_N , 5000 );
  CIN_ASSERT( N , 1 , bound_N);
  SecondStirlingNumberCalculator<MP,bound_N+1> ssn{};
  MP overlapping[N+1][N+1] = {};
  overlapping[0][1] = MP::one();
  overlapping[1][1] = MP::Derepresent( 2 );
  FOREQ( j , 2 , N ){
    overlapping[j][1] = overlapping[j-1][1] * overlapping[1][1];
  }
  FOREQ( j , 0 , N ){
    auto& overlapping_j = overlapping[j];
    overlapping_j[0] = MP::one();
    overlapping_j[1] -= MP::Derepresent( j );
    FOREQ( i , 2 , N ){
      overlapping_j[i] = overlapping_j[i-1] * overlapping_j[1];
    }
  }
  MP answer{};
  FOREQ( i , 0 , N ){
    FOREQ( j , 0 , i ){
      answer += ssn.CountDisjointCover( N , i , j ) * overlapping[j][N-i];
    }
  }
  RETURN( answer );
}
REPEAT_MAIN(1);

#else // INCLUDE_MAIN

#ifdef INCLUDE_SUB

// COMPAREに使用。圧縮時は削除する。
ll Naive( int N , int M , int K )
{
  ll answer = N + M + K;
  return answer;
}

// COMPAREに使用。圧縮時は削除する。
ll Answer( ll N , ll M , ll K )
{
  // START_WATCH;
  ll answer = N + M + K;

  // // TLに準じる乱択や全探索。デフォルトの猶予は100.0[ms]。
  // CEXPR( double , TL , 2000.0 );
  // while( CHECK_WATCH( TL ) ){

  // }
  return answer;
}

// 圧縮時は中身だけ削除する。
IN VO Experiment()
{
  // CEXPR( int , bound , 10 );
  // FOREQ( N , 0 , bound ){
  //   FOREQ( M , 0 , bound ){
  //     FOREQ( K , 0 , bound ){
  //   	COUT( N , M , K , ":" , Naive( N , M , K ) );
  //     }
  //   }
  //   // cout << Naive( N ) << ",\n"[N==bound];
  // }
}

// 圧縮時は中身だけ削除する。
IN VO SmallTest()
{
  // CEXPR( int , bound , 10 );
  // FOREQ( N , 0 , bound ){
  //   FOREQ( M , 0 , bound ){
  //     FOREQ( K , 0 , bound ){
  //   	COMPARE( N , M , K );
  //     }
  //   }
  //   // COMPARE( N );
  // }
}

#define INCLUDE_MAIN
#include __FILE__

#else // INCLUDE_SUB

#ifdef INCLUDE_LIBRARY

/*

C-x 3 C-x o C-x C-fによるファイル操作用

BFS (5KB)
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/BreadthFirstSearch/compress.txt

CoordinateCompress (3KB)
c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/CoordinateCompress/compress.txt

DFSOnTree (11KB)
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/DepthFirstSearch/Tree/a.hpp

Divisor (4KB)
c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Prime/Divisor/compress.txt

IntervalAddBIT (9KB)
c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/AffineSpace/BIT/IntervalAdd/compress.txt

Polynomial (21KB)
c:/Users/user/Documents/Programming/Mathematics/Polynomial/compress.txt

UnionFind (3KB)
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/UnionFindForest/compress.txt

*/

// VVV 常設でないライブラリは以下に挿入する。

template <uint M , typename INT> inline constexpr INT Residue( INT n ) noexcept { return move( n < 0 ? ( ( ( ( ++n ) *= -1 ) %= M ) *= -1 ) += M - 1 : n %= M ); }
template <typename INT1 , typename INT2> inline constexpr INT1 Residue( INT1 n , const INT2& M ) noexcept { return move( n < 0 ? ( ( ( ( ++n ) *= -1 ) %= M ) *= -1 ) += M - 1 : n %= M ); }

template <typename INT> inline constexpr INT& Residue998244353( INT& n ) noexcept { constexpr const uint trunc = ( 1 << 23 ) - 1; INT n_u = n >> 23; n &= trunc; INT n_uq = ( n_u / 7 ) / 17; n_u -= n_uq * 119; n += n_u << 23; return n < n_uq ? n += 998244353 - n_uq : n -= n_uq; }

using INT_TYPE_FOR_MOD = uint;

template <INT_TYPE_FOR_MOD M> class Mod;

template <INT_TYPE_FOR_MOD M>
class ConstantsForMod
{

  friend class Mod<M>;
  
private:
  ConstantsForMod() = delete;
  static constexpr const INT_TYPE_FOR_MOD g_memory_bound = 1000000;
  static constexpr const INT_TYPE_FOR_MOD g_memory_length = M < g_memory_bound ? M : g_memory_bound;

  static constexpr INT_TYPE_FOR_MOD g_M_minus = M - 1;
  static constexpr INT_TYPE_FOR_MOD g_M_minus_2 = M - 2;
  static constexpr INT_TYPE_FOR_MOD g_M_minus_2_neg = 2 - M;

};

#define DECLARATION_OF_COMPARISON_FOR_MOD( FUNC )			\
  inline constexpr bool operator FUNC( const Mod<M>& n ) const noexcept	\

#define DECLARATION_OF_ARITHMETIC_FOR_MOD( FUNC )			\
  inline constexpr Mod<M> operator FUNC( const Mod<M>& n ) const noexcept; \

#define DEFINITION_OF_COMPARISON_FOR_MOD( FUNC )			\
  template <INT_TYPE_FOR_MOD M> inline constexpr bool Mod<M>::operator FUNC( const Mod<M>& n ) const noexcept { return m_n FUNC n.m_n; } \

#define DEFINITION_OF_ARITHMETIC_FOR_MOD( FUNC )			\
  template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Mod<M>::operator FUNC( const Mod<M>& n ) const noexcept { return move( Mod<M>( *this ) FUNC ## = n ); } \
  template <INT_TYPE_FOR_MOD M , typename T> inline constexpr Mod<M> operator FUNC( T n0 , const Mod<M>& n1 ) noexcept { return move( Mod<M>( move( n0 ) ) FUNC ## = n1 ); } \

template <INT_TYPE_FOR_MOD M>
class Mod
{

protected:
  INT_TYPE_FOR_MOD m_n;

public:
  inline constexpr Mod() noexcept;
  inline constexpr Mod( const Mod<M>& n ) noexcept;
  inline constexpr Mod( Mod<M>&& n ) noexcept;
  template <typename T> inline constexpr Mod( T n ) noexcept;

  inline constexpr Mod<M>& operator=( Mod<M> n ) noexcept;
  inline constexpr Mod<M>& operator+=( const Mod<M>& n ) noexcept;
  inline constexpr Mod<M>& operator-=( const Mod<M>& n ) noexcept;
  inline constexpr Mod<M>& operator*=( const Mod<M>& n ) noexcept;
  inline Mod<M>& operator/=( const Mod<M>& n );
  // n>=0である場合のみサポート。計算量O(log n)で2^n倍する。
  inline constexpr Mod<M>& operator<<=( int n ) noexcept;
  // n>=0かつMが奇数である場合のみサポート。計算量O(n)で2^{-n}倍する。
  inline constexpr Mod<M>& operator>>=( int n ) noexcept;

  inline constexpr Mod<M>& operator++() noexcept;
  inline constexpr Mod<M> operator++( int ) noexcept;
  inline constexpr Mod<M>& operator--() noexcept;
  inline constexpr Mod<M> operator--( int ) noexcept;

  DECLARATION_OF_COMPARISON_FOR_MOD( == );
  DECLARATION_OF_COMPARISON_FOR_MOD( != );
  DECLARATION_OF_COMPARISON_FOR_MOD( < );
  DECLARATION_OF_COMPARISON_FOR_MOD( <= );
  DECLARATION_OF_COMPARISON_FOR_MOD( > );
  DECLARATION_OF_COMPARISON_FOR_MOD( >= );

  DECLARATION_OF_ARITHMETIC_FOR_MOD( + );
  DECLARATION_OF_ARITHMETIC_FOR_MOD( - );
  DECLARATION_OF_ARITHMETIC_FOR_MOD( * );
  DECLARATION_OF_ARITHMETIC_FOR_MOD( / );
  // n>=0である場合のみサポート。計算量O(log n)で2^n倍を返す。
  inline constexpr Mod<M> operator<<( int n ) const noexcept;
  // n>=0かつMが奇数である場合のみサポート。計算量O(n)で2^{-n}倍を返す。
  inline constexpr Mod<M> operator>>( int n ) const noexcept;

  inline constexpr Mod<M> operator-() const noexcept;
  // -1倍する。
  inline constexpr Mod<M>& SignInvert() noexcept;
  // Mが素数である場合のみサポート。-1乗する。
  inline Mod<M>& Invert();
  // Mが素数であるかexponent>=0である場合にのみサポート。exponent乗する。
  template <typename INT> inline constexpr Mod<M>& Power( INT exponent );
  // グローバルスコープでswapを定義するためのもの。
  inline constexpr void swap( Mod<M>& n ) noexcept;

  inline constexpr const INT_TYPE_FOR_MOD& Represent() const noexcept;
  // 0 <= n < Mの場合のみサポート。
  static inline constexpr Mod<M> Derepresent( const INT_TYPE_FOR_MOD& n ) noexcept;
  
  // Mが素数かつn < g_memory_lengthである場合のみサポート。
  static inline const Mod<M>& Inverse( const INT_TYPE_FOR_MOD& n ) noexcept;
  // n < g_memory_lengthである場合のみサポート。
  static inline const Mod<M>& Factorial( const INT_TYPE_FOR_MOD& n ) noexcept;
  // Mが素数かつn < g_memory_lengthである場合のみサポート。
  static inline const Mod<M>& FactorialInverse( const INT_TYPE_FOR_MOD& n ) noexcept;
  // Mが素数かつn < g_memory_lengthである場合のみサポート。
  static inline Mod<M> Combination( const INT_TYPE_FOR_MOD& n , const INT_TYPE_FOR_MOD& i ) noexcept;

  static inline const Mod<M>& zero() noexcept;
  static inline const Mod<M>& one() noexcept;

private:
  template <typename INT> inline constexpr Mod<M>& PositivePower( INT exponent ) noexcept;
  template <typename INT> inline constexpr Mod<M>& NonNegativePower( INT exponent ) noexcept;
  template <typename T> inline constexpr Mod<M>& Ref( T&& n ) noexcept;

  // 0 <= n < Mの場合のみサポート。
  static inline constexpr INT_TYPE_FOR_MOD& Normalise( INT_TYPE_FOR_MOD& n ) noexcept;

};

// Mが素数でありnが0でない場合にのみサポート。
template <INT_TYPE_FOR_MOD M> inline Mod<M> Inverse( const Mod<M>& n );
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Inverse_constexpr( Mod<M> n );

// Mが素数であるかexponent>=0である場合にのみサポート。
template <INT_TYPE_FOR_MOD M , typename T> inline constexpr Mod<M> Power( Mod<M> n , T exponent );

template <INT_TYPE_FOR_MOD M> inline constexpr void swap( Mod<M>& n0 , Mod<M>& n1 ) noexcept;

template <INT_TYPE_FOR_MOD M> inline string to_string( const Mod<M>& n ) noexcept;

template <INT_TYPE_FOR_MOD M , class Traits> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , Mod<M>& n );
template <INT_TYPE_FOR_MOD M , class Traits> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const Mod<M>& n );

template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>::Mod() noexcept : m_n() {}
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>::Mod( const Mod<M>& n ) noexcept : m_n( n.m_n ) {}
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>::Mod( Mod<M>&& n ) noexcept : m_n( move( n.m_n ) ) {}
template <INT_TYPE_FOR_MOD M> template <typename T> inline constexpr Mod<M>::Mod( T n ) noexcept : m_n( Residue<M>( move( n ) ) ) { static_assert( is_constructible_v<INT_TYPE_FOR_MOD,decay_t<T> > ); }

template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator=( Mod<M> n ) noexcept { return Ref( m_n = move( n.m_n ) ); }

template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator+=( const Mod<M>& n ) noexcept { return Ref( Normalise( m_n += n.m_n ) ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator-=( const Mod<M>& n ) noexcept { return Ref( m_n < n.m_n ? ( m_n += M ) -= n.m_n : m_n -= n.m_n ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator*=( const Mod<M>& n ) noexcept { return Ref( m_n = Residue<M>( ull( m_n ) * n.m_n ) ); }
template <> inline constexpr Mod<998244353>& Mod<998244353>::operator*=( const Mod<998244353>& n ) noexcept { ull m_n_copy = m_n; return Ref( m_n = move( ( m_n_copy *= n.m_n ) < 998244353 ? m_n_copy : Residue998244353( m_n_copy ) ) ); }

template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator/=( const Mod<M>& n ) { return operator*=( Mod<M>( n ).Invert() ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator<<=( int n ) noexcept { assert( n >= 0 ); return *this *= Derepresent( 2 ).NonNegativePower( move( n ) ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator>>=( int n ) noexcept { assert( n >=0 ); while( n-- > 0 ){ ( ( m_n & 1 ) == 0 ? m_n : m_n += M ) >>= 1; } return *this; }

template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator++() noexcept { return Ref( m_n < ConstantsForMod<M>::g_M_minus ? ++m_n : m_n = 0 ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Mod<M>::operator++( int ) noexcept { Mod<M> n{ *this }; operator++(); return n; }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::operator--() noexcept { return Ref( m_n == 0 ? m_n = ConstantsForMod<M>::g_M_minus : --m_n ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Mod<M>::operator--( int ) noexcept { Mod<M> n{ *this }; operator--(); return n; }

DEFINITION_OF_COMPARISON_FOR_MOD( == );
DEFINITION_OF_COMPARISON_FOR_MOD( != );
DEFINITION_OF_COMPARISON_FOR_MOD( > );
DEFINITION_OF_COMPARISON_FOR_MOD( >= );
DEFINITION_OF_COMPARISON_FOR_MOD( < );
DEFINITION_OF_COMPARISON_FOR_MOD( <= );

DEFINITION_OF_ARITHMETIC_FOR_MOD( + );
DEFINITION_OF_ARITHMETIC_FOR_MOD( - );
DEFINITION_OF_ARITHMETIC_FOR_MOD( * );
DEFINITION_OF_ARITHMETIC_FOR_MOD( / );
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Mod<M>::operator<<( int n ) const noexcept { return move( Mod<M>( *this ) <<= n ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Mod<M>::operator>>( int n ) const noexcept { return move( Mod<M>( *this ) >>= n ); }

template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Mod<M>::operator-() const noexcept { return move( Mod<M>( *this ).SignInvert() ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M>& Mod<M>::SignInvert() noexcept { return Ref( m_n > 0 ? m_n = M - m_n : m_n ); }

template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::Invert() { assert( m_n != 0 ); INT_TYPE_FOR_MOD m_n_neg; return m_n < ConstantsForMod<M>::g_memory_length ? Ref( m_n = Inverse( m_n ).m_n ) : ( ( m_n_neg = M - m_n ) < ConstantsForMod<M>::g_memory_length ) ? Ref( m_n = M - Inverse( m_n_neg ).m_n ) : PositivePower( INT_TYPE_FOR_MOD( ConstantsForMod<M>::g_M_minus_2 ) ); }

template <INT_TYPE_FOR_MOD M> template <typename INT> inline constexpr Mod<M>& Mod<M>::PositivePower( INT exponent ) noexcept { Mod<M> power{ *this }; exponent--; while( exponent != 0 ){ ( exponent & 1 ) == 1 ? operator*=( power ) : *this; exponent >>= 1; power *= power; } return *this; }
template <INT_TYPE_FOR_MOD M> template <typename INT> inline constexpr Mod<M>& Mod<M>::NonNegativePower( INT exponent ) noexcept { return exponent == 0 ? Ref( m_n = 1 ) : Ref( PositivePower( move( exponent ) ) ); }
template <INT_TYPE_FOR_MOD M> template <typename INT> inline constexpr Mod<M>& Mod<M>::Power( INT exponent ) { bool neg = exponent < 0; assert( !( neg && m_n == 0 ) ); return neg ? PositivePower( move( exponent *= ConstantsForMod<M>::g_M_minus_2_neg ) ) : NonNegativePower( move( exponent ) ); }

template <INT_TYPE_FOR_MOD M> inline constexpr void Mod<M>::swap( Mod<M>& n ) noexcept { std::swap( m_n , n.m_n ); }

template <INT_TYPE_FOR_MOD M> inline const Mod<M>& Mod<M>::Inverse( const INT_TYPE_FOR_MOD& n ) noexcept { static Mod<M> memory[ConstantsForMod<M>::g_memory_length] = { zero() , one() }; static INT_TYPE_FOR_MOD length_curr = 2; while( length_curr <= n ){ memory[length_curr].m_n = M - memory[M % length_curr].m_n * ull( M / length_curr ) % M; length_curr++; } return memory[n]; }
template <INT_TYPE_FOR_MOD M> inline const Mod<M>& Mod<M>::Factorial( const INT_TYPE_FOR_MOD& n ) noexcept { static Mod<M> memory[ConstantsForMod<M>::g_memory_length] = { one() , one() }; static INT_TYPE_FOR_MOD length_curr = 2; while( length_curr <= n ){ ( memory[length_curr] = memory[length_curr - 1] ) *= length_curr; length_curr++; } return memory[n]; }
template <INT_TYPE_FOR_MOD M> inline const Mod<M>& Mod<M>::FactorialInverse( const INT_TYPE_FOR_MOD& n ) noexcept { static Mod<M> memory[ConstantsForMod<M>::g_memory_length] = { one() , one() }; static INT_TYPE_FOR_MOD length_curr = 2; while( length_curr <= n ){ ( memory[length_curr] = memory[length_curr - 1] ) *= Inverse( length_curr ); length_curr++; } return memory[n]; }
template <INT_TYPE_FOR_MOD M> inline Mod<M> Mod<M>::Combination( const INT_TYPE_FOR_MOD& n , const INT_TYPE_FOR_MOD& i ) noexcept { return Factorial( n ) * FactorialInverse( i ) * FactorialInverse( n - i ); }

template <INT_TYPE_FOR_MOD M> inline constexpr const INT_TYPE_FOR_MOD& Mod<M>::Represent() const noexcept { return m_n; }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Mod<M>::Derepresent( const INT_TYPE_FOR_MOD& n ) noexcept { Mod<M> n_copy{}; n_copy.m_n = n; return n_copy; }

template <INT_TYPE_FOR_MOD M> inline const Mod<M>& Mod<M>::zero() noexcept { static constexpr const Mod<M> z{}; return z; }
template <INT_TYPE_FOR_MOD M> inline const Mod<M>& Mod<M>::one() noexcept { static constexpr const Mod<M> o{ Derepresent( 1 ) }; return o; }

template <INT_TYPE_FOR_MOD M> template <typename T> inline constexpr Mod<M>& Mod<M>::Ref( T&& n ) noexcept { return *this; }
template <INT_TYPE_FOR_MOD M> inline constexpr INT_TYPE_FOR_MOD& Mod<M>::Normalise( INT_TYPE_FOR_MOD& n ) noexcept { return n < M ? n : n -= M; }

template <INT_TYPE_FOR_MOD M> inline Mod<M> Inverse( const Mod<M>& n ) { return move( Mod<M>( n ).Invert() ); }
template <INT_TYPE_FOR_MOD M> inline constexpr Mod<M> Inverse_constrexpr( Mod<M> n ) noexcept { return move( n.NonNegativePower( M - 2 ) ); }

template <INT_TYPE_FOR_MOD M , typename INT> inline constexpr Mod<M> Power( Mod<M> n , INT exponent ) { return move( n.Power( move( exponent ) ) ); }

template <INT_TYPE_FOR_MOD M> inline constexpr void swap( Mod<M>& n0 , Mod<M>& n1 ) noexcept { n0.swap( n1 ); }

template <INT_TYPE_FOR_MOD M> inline string to_string( const Mod<M>& n ) noexcept { return to_string( n.Represent() ) + " + " + to_string( M ) + "Z"; }

template <INT_TYPE_FOR_MOD M , class Traits> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , Mod<M>& n ) { ll m; is >> m; n = m; return is; }
template <INT_TYPE_FOR_MOD M , class Traits> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const Mod<M>& n ) { return os << n.Represent(); }

template <typename T , int length>
class SecondStirlingNumberCalculator
{

private:
  // N元集合の非交叉非空部分集合i個による被覆の個数をm_val[N][i]に格納する。
  T m_val[length][length];

public:
  // （コンパイル時に）計算量O(length^2)で構築する。
  constexpr inline SecondStirlingNumberCalculator();

  constexpr inline const T ( &operator[]( const int& i ) const )[length];

  // 以下N<lengthの場合のみサポート。（i<lengthでなくてもよい）

  // N元集合の非交叉非空部分集合i個による被覆の個数を返す。（O(1)）
  constexpr inline T CountDisjointCover( const int& N , const int& i ) const;
  // N元集合の非交叉非空部分集合i個の個数を返す。（O(1)）
  constexpr inline T CountDisjointSubset( const int& N , const int& i ) const;

  // 以下Tが正整数Mに対するMod<M>と表せる場合のみサポート。

  // N元集合の長さiの非交叉非空部分集合列による被覆の個数を返す。（O(log min{N,i})）
  // （i彩色の個数はこれらを足し合わせればよい）
  inline T CountDisjointCoverSequence( const int& N , const int& i ) const;
  // N元集合の長さiの非交叉非空部分集合列の個数を返す。（O(log min{N,i})）
  inline T CountDisjointSubsetSequence( const int& N , const int& i ) const;

  // 以下Mがlength以上の素数である場合のみサポート。

  // N元集合の要素数nの部分集合の非交叉非空部分集合i個による被覆の個数を返す。（O(log N)）
  inline T CountDisjointCover( const int& N , const int& n , const int& i ) const;
  // N元集合の要素数nの部分集合の長さiの非交叉非空部分集合列による被覆の個数を返す。（O(log N)）
  inline T CountDisjointCoverSequence( const int& N , const int& n , const int& i ) const;
  
};

template <typename T , int length> constexpr inline SecondStirlingNumberCalculator<T,length>::SecondStirlingNumberCalculator() : m_val() 
{
  
  m_val[0][0] = 1;

  for( int i = 1 ; i < length ; i++ ){

    auto& m_val_i = m_val[i];
    const auto& m_val_i_minus = m_val[i - 1];

    for( int j = 1 ; j < i ; j++ ){

      ( ( m_val_i[j] = m_val_i_minus[j] ) *= j ) += m_val_i_minus[j - 1];

    }

    m_val_i[i] = 1;

  }

}

template <typename T , int length> constexpr inline const T ( &SecondStirlingNumberCalculator<T,length>::operator[]( const int& i ) const )[length] { assert( i < length ); return m_val[i]; }

template <typename T , int length> constexpr inline T SecondStirlingNumberCalculator<T,length>::CountDisjointCover( const int& N , const int& i ) const { assert( N < length ); return i <= N ? m_val[N][i] : T(); }
template <typename T , int length> constexpr inline T SecondStirlingNumberCalculator<T,length>::CountDisjointSubset( const int& N , const int& i ) const { assert( N < length ); return i < N ? m_val[N][i] + m_val[N][i+1] : i == N ? m_val[N][i] : T(); }

template <typename T , int length> inline T SecondStirlingNumberCalculator<T,length>::CountDisjointCoverSequence( const int& N , const int& i ) const { return CountDisjointCover( N , i ) * T::Factorial( i ); }
template <typename T , int length> inline T SecondStirlingNumberCalculator<T,length>::CountDisjointSubsetSequence( const int& N , const int& i ) const { return CountDisjointSubset( N , i ) * T::Factorial( i ); }

template <typename T , int length> inline T SecondStirlingNumberCalculator<T,length>::CountDisjointCover( const int& N , const int& n , const int& i ) const { return CountDisjointCover( n , i ) * T::Combination( N , n ); }
template <typename T , int length> inline T SecondStirlingNumberCalculator<T,length>::CountDisjointCoverSequence( const int& N , const int& n , const int& i ) const { return CountDisjointCoverSequence( n , i ) * T::Combination( N , n ); }

// AAA 常設でないライブラリは以上に挿入する。

#define INCLUDE_SUB
#include __FILE__

#else // INCLUDE_LIBRARY

#ifdef DEBUG
  #define _GLIBCXX_DEBUG
  #define REPEAT_MAIN( BOUND ) START_MAIN; signal( SIGABRT , &AlertAbort ); AutoCheck( exec_mode , use_getline ); if( exec_mode == sample_debug_mode || exec_mode == submission_debug_mode || exec_mode == library_search_mode ){ RE 0; } else if( exec_mode == experiment_mode ){ Experiment(); RE 0; } else if( exec_mode == small_test_mode ){ SmallTest(); RE 0; }; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if( exec_mode == solve_mode ){ if CE( bound_test_case_num > 1 ){ CERR( "テストケースの個数を入力してください。" ); SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } } else if( exec_mode == random_test_mode ){ CERR( "ランダムテストを行う回数を指定してください。" ); SET_LL( test_case_num ); } FINISH_MAIN
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) if( exec_mode == solve_mode ){ SET_LL( A ); ASSERT( A , MIN , MAX ); } else if( exec_mode == random_test_mode ){ CERR( #A , " = " , ( A = GetRand( MIN , MAX ) ) ); } else { AS( false ); }
  #define SOLVE_ONLY ST_AS( __FUNCTION__[0] == 'S' )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define COUT( ... ) VariadicCout( cout << "出力： " , __VA_ARGS__ ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << endl
  #define COUT_A( A , N ) cout << "出力： "; OUTPUT_ARRAY( cout , A , N ) << endl
  #define CERR_ITR( A ) OUTPUT_ITR( cerr , A ) << endl
  #define COUT_ITR( A ) cout << "出力： "; OUTPUT_ITR( cout , A ) << endl
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if CE( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } FINISH_MAIN
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) SET_LL( A ); ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #define CERR( ... ) 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define CERR_A( A , N ) 
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout , A , N ) << ENDL
  #define CERR_ITR( A ) 
  #define COUT_ITR( A ) OUTPUT_ITR( cout , A ) << ENDL
#endif
#ifdef REACTIVE
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#ifdef USE_GETLINE
  #define SET_LL( A ) { GETLINE( A ## _str ); A = stoll( A ## _str ); }
  #define GETLINE_SEPARATE( SEPARATOR , ... ) SOLVE_ONLY; string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) SOLVE_ONLY; GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
#else
  #define SET_LL( A ) cin >> A
  #define CIN( LL , ... ) SOLVE_ONLY; LL __VA_ARGS__; VariadicCin( cin , __VA_ARGS__ )
  #define SET_A( A , N ) SOLVE_ONLY; FOR( VARIABLE_FOR_SET_A , 0 , N ){ cin >> A[VARIABLE_FOR_SET_A]; }
  #define CIN_A( LL , A , N ) VE<LL> A( N ); SET_A( A , N );
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define FINISH_MAIN REPEAT( test_case_num ){ if CE( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } Solve(); CERR( "" ); } }
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now()
#define CURRENT_TIME static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME < TL_MS - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( A , N , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A] , MIN , MAX ); }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( A , N , MIN , MAX ) vector<decldecay_t( MAX )> A( N ); SET_A_ASSERT( A , N , MIN , MAX )
#define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( decldecay_t( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( decldecay_t( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( decldecay_t( INITIAL ) VAR = INITIAL ; VAR + 1 > FINAL ; VAR -- )
#define AUTO_ITR( ARRAY ) auto itr_ ## ARRAY = ARRAY .BE() , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( AUTO_ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT_ ## HOW_MANY_TIMES , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS )
#define OUTPUT_ARRAY( OS , A , N ) FOR( VARIABLE_FOR_OUTPUT_ARRAY , 0 , N ){ OS << A[VARIABLE_FOR_OUTPUT_ARRAY] << (VARIABLE_FOR_OUTPUT_ARRAY==N-1?"":" "); } OS
#define OUTPUT_ITR( OS , A ) { auto ITERATOR_FOR_OUTPUT_ITR = A.BE() , EN_FOR_OUTPUT_ITR = A.EN(); bool VARIABLE_FOR_OUTPUT_ITR = ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR; WH( VARIABLE_FOR_OUTPUT_ITR ){ OS << *ITERATOR_FOR_COUT_ITR; ( VARIABLE_FOR_OUTPUT_ITR = ++ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR ) ? OS : OS << " "; } } OS
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ ); auto answer = Answer( __VA_ARGS__ ); bool match = naive == answer; COUT( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive == " , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ RE; }

// 圧縮用
#define TE template
#define TY typename
#define US using
#define ST static
#define AS assert
#define IN inline
#define CL class
#define PU public
#define OP operator
#define CE constexpr
#define CO const
#define NE noexcept
#define RE return 
#define WH while
#define VO void
#define VE vector
#define LI list
#define BE begin
#define EN end
#define SZ size
#define LE length
#define PW Power
#define MO move
#define TH this
#define CRI CO int&
#define CRUI CO uint&
#define CRL CO ll&
#define VI virtual 
#define ST_AS static_assert
#define reMO_CO remove_const
#define is_COructible_v is_constructible_v
#define rBE rbegin
#define reSZ resize

// 型のエイリアス
#define decldecay_t( VAR ) decay_t<decltype( VAR )>
TE <TY F , TY...Args> US ret_t = decltype( declval<F>()( declval<Args>()... ) );
TE <TY T> US inner_t = TY T::type;
US uint = unsigned int;
US ll = long long;
US ull = unsigned long long;
US ld = long double;
US lld = __float128;
TE <TY INT> US T2 = pair<INT,INT>;
TE <TY INT> US T3 = tuple<INT,INT,INT>;
TE <TY INT> US T4 = tuple<INT,INT,INT,INT>;
US path = pair<int,ll>;

// 入出力用
TE <CL Traits> IN basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is ) { RE is; }
TE <CL Traits , TY Arg , TY... ARGS> IN basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is , Arg& arg , ARGS&... args ) { RE VariadicCin( is >> arg , args... ); }
TE <CL Traits> IN basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , CO char& separator ) { RE is; }
TE <CL Traits , TY Arg , TY... ARGS> IN basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , CO char& separator , Arg& arg , ARGS&... args ) { RE VariadicGetline( getline( is , arg , separator ) , separator , args... ); }
TE <CL Traits , TY Arg> IN basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , CO VE<Arg>& arg ) { auto BE = arg.BE() , EN = arg.EN(); auto itr = BE; WH( itr != EN ){ ( itr == BE ? os : os << " " ) << *itr; itr++; } RE os; }
TE <CL Traits , TY Arg1 , TY Arg2> IN basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , CO pair<Arg1,Arg2>& arg ) { RE os << arg.first << " " << arg.second; }
TE <CL Traits , TY Arg> IN basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , CO Arg& arg ) { RE os << arg; }
TE <CL Traits , TY Arg1 , TY Arg2 , TY... ARGS> IN basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , CO Arg1& arg1 , CO Arg2& arg2 , CO ARGS&... args ) { RE VariadicCout( os << arg1 << " " , arg2 , args... ); }

// 算術用
TE <TY T> CE T PositiveBaseResidue( CO T& a , CO T& p ){ RE a >= 0 ? a % p : p - 1 - ( ( - ( a + 1 ) ) % p ); }
TE <TY T> CE T Residue( CO T& a , CO T& p ){ RE PositiveBaseResidue( a , p < 0 ? -p : p ); }
TE <TY T> CE T PositiveBaseQuotient( CO T& a , CO T& p ){ RE ( a - PositiveBaseResidue( a , p ) ) / p; }
TE <TY T> CE T Quotient( CO T& a , CO T& p ){ RE p < 0 ? PositiveBaseQuotient( -a , -p ) : PositiveBaseQuotient( a , p ); }

#define POWER( ANSWER , ARGUMENT , EXPONENT )				\
  ST_AS( ! is_same<decldecay_t( ARGUMENT ),int>::value && ! is_same<decldecay_t( ARGUMENT ),uint>::value ); \
  decldecay_t( ARGUMENT ) ANSWER{ 1 };					\
  {									\
    decldecay_t( ARGUMENT ) ARGUMENT_FOR_SQUARE_FOR_POWER = ( ARGUMENT ); \
    decldecay_t( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT ); \
    WH( EXPONENT_FOR_SQUARE_FOR_POWER != 0 ){				\
      if( EXPONENT_FOR_SQUARE_FOR_POWER % 2 == 1 ){			\
	ANSWER *= ARGUMENT_FOR_SQUARE_FOR_POWER;			\
      }									\
      ARGUMENT_FOR_SQUARE_FOR_POWER *= ARGUMENT_FOR_SQUARE_FOR_POWER;	\
      EXPONENT_FOR_SQUARE_FOR_POWER /= 2;				\
    }									\
  }									\

#define POWER_MOD( ANSWER , ARGUMENT , EXPONENT , MODULO )		\
  ll ANSWER{ 1 };							\
  {									\
    ll ARGUMENT_FOR_SQUARE_FOR_POWER = ( ( ARGUMENT ) % ( MODULO ) ) % ( MODULO ); \
    ARGUMENT_FOR_SQUARE_FOR_POWER < 0 ? ARGUMENT_FOR_SQUARE_FOR_POWER += ( MODULO ) : ARGUMENT_FOR_SQUARE_FOR_POWER; \
    decldecay_t( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT ); \
    WH( EXPONENT_FOR_SQUARE_FOR_POWER != 0 ){				\
      if( EXPONENT_FOR_SQUARE_FOR_POWER % 2 == 1 ){			\
	ANSWER = ( ANSWER * ARGUMENT_FOR_SQUARE_FOR_POWER ) % ( MODULO ); \
      }									\
      ARGUMENT_FOR_SQUARE_FOR_POWER = ( ARGUMENT_FOR_SQUARE_FOR_POWER * ARGUMENT_FOR_SQUARE_FOR_POWER ) % ( MODULO ); \
      EXPONENT_FOR_SQUARE_FOR_POWER /= 2;				\
    }									\
  }									\

#define FACTORIAL_MOD( ANSWER , ANSWER_INV , INVERSE , MAX_INDEX , CE_LENGTH , MODULO ) \
  ll ANSWER[CE_LENGTH];							\
  ll ANSWER_INV[CE_LENGTH];						\
  ll INVERSE[CE_LENGTH];						\
  {									\
    ll VARIABLE_FOR_PRODUCT_FOR_FACTORIAL = 1;				\
    ANSWER[0] = VARIABLE_FOR_PRODUCT_FOR_FACTORIAL;			\
    FOREQ( i , 1 , MAX_INDEX ){						\
      ANSWER[i] = ( VARIABLE_FOR_PRODUCT_FOR_FACTORIAL *= i ) %= ( MODULO ); \
    }									\
    ANSWER_INV[0] = ANSWER_INV[1] = INVERSE[1] = VARIABLE_FOR_PRODUCT_FOR_FACTORIAL = 1; \
    FOREQ( i , 2 , MAX_INDEX ){						\
      ANSWER_INV[i] = ( VARIABLE_FOR_PRODUCT_FOR_FACTORIAL *= INVERSE[i] = ( MODULO ) - ( ( ( ( MODULO ) / i ) * INVERSE[ ( MODULO ) % i ] ) % ( MODULO ) ) ) %= ( MODULO ); \
    }									\
  }									\

// 二分探索用
// EXPRESSIONがANSWERの広義単調関数の時、EXPRESSION >= CO_TARGETの整数解を格納。
#define BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , DESIRED_INEQUALITY , CO_TARGET , INEQUALITY_FOR_CHECK , UPDATE_U , UPDATE_L , UPDATE_ANSWER ) \
  ST_AS( ! is_same<decldecay_t( CO_TARGET ),uint>::value && ! is_same<decldecay_t( CO_TARGET ),ull>::value ); \
  ll ANSWER = MINIMUM;							\
  {									\
    ll L_BS = MINIMUM;							\
    ll U_BS = MAXIMUM;							\
    ANSWER = UPDATE_ANSWER;						\
    ll EXPRESSION_BS;							\
    CO ll CO_TARGET_BS = ( CO_TARGET );			\
    ll DIFFERENCE_BS;							\
    WH( L_BS < U_BS ){						\
      DIFFERENCE_BS = ( EXPRESSION_BS = ( EXPRESSION ) ) - CO_TARGET_BS; \
      CERR( "二分探索中：" , "L_BS =" , L_BS , "<=" , #ANSWER , "=" , ANSWER , "<=" , U_BS , "= U_BS :" , #EXPRESSION , "=" , EXPRESSION_BS , DIFFERENCE_BS > 0 ? ">" : DIFFERENCE_BS < 0 ? "<" : "=" , CO_TARGET_BS , "=" , #CO_TARGET ); \
      if( DIFFERENCE_BS INEQUALITY_FOR_CHECK 0 ){			\
	U_BS = UPDATE_U;						\
      } else {								\
	L_BS = UPDATE_L;						\
      }									\
      ANSWER = UPDATE_ANSWER;						\
    }									\
    if( L_BS > U_BS ){							\
      CERR( "二分探索失敗：" , "L_BS =" , L_BS , ">" , U_BS , "= U_BS :" , #ANSWER , ":=" , #MAXIMUM , "+ 1 =" , MAXIMUM + 1  ); \
      CERR( "二分探索マクロにミスがある可能性があります。変更前の版に戻してください。" ); \
      ANSWER = MAXIMUM + 1;						\
    } else {								\
      CERR( "二分探索終了：" , "L_BS =" , L_BS , "<=" , #ANSWER , "=" , ANSWER , "<=" , U_BS , "= U_BS" ); \
      CERR( "二分探索が成功したかを確認するために" , #EXPRESSION , "を計算します。" ); \
      CERR( "成功判定が不要な場合はこの計算を削除しても構いません。" );	\
      EXPRESSION_BS = ( EXPRESSION );					\
      CERR( "二分探索結果：" , #EXPRESSION , "=" , EXPRESSION_BS , ( EXPRESSION_BS > CO_TARGET_BS ? ">" : EXPRESSION_BS < CO_TARGET_BS ? "<" : "=" ) , CO_TARGET_BS ); \
      if( EXPRESSION_BS DESIRED_INEQUALITY CO_TARGET_BS ){		\
	CERR( "二分探索成功：" , #ANSWER , ":=" , ANSWER );		\
      } else {								\
	CERR( "二分探索失敗：" , #ANSWER , ":=" , #MAXIMUM , "+ 1 =" , MAXIMUM + 1 ); \
	CERR( "単調でないか、単調増加性と単調減少性を逆にしてしまったか、探索範囲内に解が存在しません。" ); \
	ANSWER = MAXIMUM + 1;						\
      }									\
    }									\
  }									\

// 単調増加の時にEXPRESSION >= CO_TARGETの最小解を格納。
#define BS1( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CO_TARGET , >= , ANSWER , ANSWER + 1 , ( L_BS + U_BS ) / 2 )
// 単調増加の時にEXPRESSION <= CO_TARGETの最大解を格納。
#define BS2( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CO_TARGET , > , ANSWER - 1 , ANSWER , ( L_BS + 1 + U_BS ) / 2 )
// 単調減少の時にEXPRESSION >= CO_TARGETの最大解を格納。
#define BS3( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CO_TARGET , < , ANSWER - 1 , ANSWER , ( L_BS + 1 + U_BS ) / 2 )
// 単調減少の時にEXPRESSION <= CO_TARGETの最小解を格納。
#define BS4( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CO_TARGET , <= , ANSWER , ANSWER + 1 , ( L_BS + U_BS ) / 2 )
// t以下の値が存在すればその最大値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MaximumLeq( set<T>& S , CO T& t ) { CO auto EN = S.EN(); if( S.empty() ){ RE EN; } auto itr = S.upper_bound( t ); RE itr == EN ? S.find( *( S.rBE() ) ) : itr == S.BE() ? EN : --itr; }
// t未満の値が存在すればその最大値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MaximumLt( set<T>& S , CO T& t ) { CO auto EN = S.EN(); if( S.empty() ){ RE EN; } auto itr = S.lower_bound( t ); RE itr == EN ? S.find( *( S.rBE() ) ) : itr == S.BE() ? EN : --itr; }
// t以上の値が存在すればその最小値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MinimumGeq( set<T>& S , CO T& t ) { RE S.lower_bound( t ); }
// tより大きい値が存在すればその最小値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MinimumGt( set<T>& S , CO T& t ) { RE S.upper_bound( t ); }

// 尺取り法用
// VAR_TPAがINITからUPDATEを繰り返しCONTINUE_CONDITIONを満たす限り、ON_CONDITIONを判定して
// trueならON、falseならOFFとなる。直近のONの区間を[VAR_TPA_L,VAR_TPA_R)で管理する。
#define TPA( VAR_TPA , INIT , UPDATE , CONTINUE_CONDITION , ON_CONDITION , ONON , ONOFF , OFFON , OFFOFF , FINISH ) \
  {									\
    auto VAR_TPA = INIT;						\
    auto VAR_TPA ## _L = VAR_TPA;					\
    auto VAR_TPA ## _R = VAR_TPA;					\
    bool on_TPA = false;						\
    int state_TPA = 3;							\
    WH( CONTINUE_CONDITION ){						\
      bool on_TPA_next = ON_CONDITION;					\
      state_TPA = ( ( on_TPA ? 1 : 0 ) << 1 ) | ( on_TPA_next ? 1 : 0 ); \
      CERR( "尺取り中： [L,R) = [" , VAR_TPA ## _L , "," , VAR_TPA ## _R , ") ," , #VAR_TPA , "=" , VAR_TPA , "," , ( ( state_TPA >> 1 ) & 1 ) == 1 ? "on" : "off" , " ->" , ( state_TPA & 1 ) == 1 ? "on" : "off" ); \
      if( state_TPA == 0 ){						\
	OFFOFF; VAR_TPA ## _L = VAR_TPA ## _R = VAR_TPA; UPDATE;	\
      } else if( state_TPA == 1 ){					\
	OFFON; VAR_TPA ## _L = VAR_TPA; UPDATE; VAR_TPA ## _R = VAR_TPA; \
      } else if( state_TPA == 2 ){					\
	ONOFF; VAR_TPA ## _L = VAR_TPA ## _R = VAR_TPA; UPDATE;		\
      } else {								\
	ONON; UPDATE; VAR_TPA ## _R = VAR_TPA;				\
      }									\
      on_TPA = on_TPA_next;						\
    }									\
    CERR( "尺取り終了： [L,R) = [" , VAR_TPA ## _L , "," , VAR_TPA ## _R , ") ," , #VAR_TPA , "=" , VAR_TPA ); \
    FINISH;								\
  }									\

// データ構造用
TE <TY T , TE <TY...> TY V> IN V<T> OP+( CO V<T>& a0 , CO V<T>& a1 ) { if( a0.empty() ){ RE a1; } if( a1.empty() ){ RE a0; } AS( a0.SZ() == a1.SZ() ); V<T> answer{}; for( auto itr0 = a0.BE() , itr1 = a1.BE() , EN0 = a0.EN(); itr0 != EN0 ; itr0++ , itr1++ ){ answer.push_back( *itr0 + *itr1 ); } RE answer; }
TE <TY T , TY U> IN pair<T,U> OP+( CO pair<T,U>& t0 , CO pair<T,U>& t1 ) { RE { t0.first + t1.first , t0.second + t1.second }; }
TE <TY T , TY U , TY V> IN tuple<T,U,V> OP+( CO tuple<T,U,V>& t0 , CO tuple<T,U,V>& t1 ) { RE { get<0>( t0 ) + get<0>( t1 ) , get<1>( t0 ) + get<1>( t1 ) , get<2>( t0 ) + get<2>( t1 ) }; }
TE <TY T , TY U , TY V , TY W> IN tuple<T,U,V,W> OP+( CO tuple<T,U,V,W>& t0 , CO tuple<T,U,V,W>& t1 ) { RE { get<0>( t0 ) + get<0>( t1 ) , get<1>( t0 ) + get<1>( t1 ) , get<2>( t0 ) + get<2>( t1 ) , get<3>( t0 ) + get<3>( t1 ) }; }
TE <TY T> IN T Add( CO T& t0 , CO T& t1 ) { RE t0 + t1; }
TE <TY T> IN T XorAdd( CO T& t0 , CO T& t1 ){ RE t0 ^ t1; }
TE <TY T> IN T Multiply( CO T& t0 , CO T& t1 ) { RE t0 * t1; }
TE <TY T> IN CO T& Zero() { ST CO T z{}; RE z; }
TE <TY T> IN CO T& One() { ST CO T o = 1; RE o; }\
TE <TY T> IN T AddInv( CO T& t ) { RE -t; }
TE <TY T> IN T Id( CO T& v ) { RE v; }
TE <TY T> IN T Min( CO T& a , CO T& b ){ RE a < b ? a : b; }
TE <TY T> IN T Max( CO T& a , CO T& b ){ RE a < b ? b : a; }
TE <TY T , TE <TY...> TY V> IN auto Get( CO V<T>& a ) { return [&]( CRI i = 0 ){ RE a[i]; }; }

// グリッド問題用
int H , W , H_minus , W_minus , HW;
VE<VE<bool>> non_wall;
IN T2<int> EnumHW( CRI v ) { RE { v / W , v % W }; }
IN int EnumHW_inv( CO T2<int>& ij ) { auto& [i,j] = ij; RE i * W + j; }
CO string direction[4] = {"U","R","D","L"};
// (i,j)->(k,h)の方向番号を取得
IN int DirectionNumberOnGrid( CRI i , CRI j , CRI k , CRI h ){RE i<k?2:i>k?0:j<h?1:j>h?3:(AS(false),-1);}
// v->wの方向番号を取得
IN int DirectionNumberOnGrid( CRI v , CRI w ){auto [i,j]=EnumHW(v);auto [k,h]=EnumHW(w);RE DirectionNumberOnGrid(i,j,k,h);}
// 方向番号の反転U<->D、R<->L
IN int ReverseDirectionNumberOnGrid( CRI n ){AS(0<=n&&n<4);RE(n+2)%4;}
IN VO SetEdgeOnGrid( CO string& Si , CRI i , VE<LI<int>>& e , CO char& walkable = '.' ){FOR(j,0,W){if(Si[j]==walkable){int v = EnumHW_inv({i,j});if(i>0){e[EnumHW_inv({i-1,j})].push_back(v);}if(i+1<H){e[EnumHW_inv({i+1,j})].push_back(v);}if(j>0){e[EnumHW_inv({i,j-1})].push_back(v);}if(j+1<W){e[EnumHW_inv({i,j+1})].push_back(v);}}}}
IN VO SetEdgeOnGrid( CO string& Si , CRI i , VE<LI<path>>& e , CO char& walkable = '.' ){FOR(j,0,W){if(Si[j]==walkable){CO int v=EnumHW_inv({i,j});if(i>0){e[EnumHW_inv({i-1,j})].push_back({v,1});}if(i+1<H){e[EnumHW_inv({i+1,j})].push_back({v,1});}if(j>0){e[EnumHW_inv({i,j-1})].push_back({v,1});}if(j+1<W){e[EnumHW_inv({i,j+1})].push_back({v,1});}}}}
IN VO SetWallOnGrid( CO string& Si , CRI i , VE<VE<bool>>& non_wall , CO char& walkable = '.'  , CO char& unwalkable = '#' ){non_wall.push_back(VE<bool>(W));auto& non_wall_i=non_wall[i];FOR(j,0,W){non_wall_i[j]=Si[j]==walkable?true:(assert(Si[j]==unwalkable),false);}}

// デバッグ用
#ifdef DEBUG
  IN VO AlertAbort( int n ) { CERR( "abort関数が呼ばれました。assertマクロのメッセージが出力されていない場合はオーバーフローの有無を確認をしてください。" ); }
  VO AutoCheck( int& exec_mode , CO bool& use_getline );
  IN VO Solve();
  IN VO Experiment();
  IN VO SmallTest();
  IN VO RandomTest();
  ll GetRand( CRL Rand_min , CRL Rand_max );
  IN VO BreakPoint( CRI LINE ) {}
  int exec_mode;
  CEXPR( int , solve_mode , 0 );
  CEXPR( int , sample_debug_mode , 1 );
  CEXPR( int , submission_debug_mode , 2 );
  CEXPR( int , library_search_mode , 3 );
  CEXPR( int , experiment_mode , 4 );
  CEXPR( int , small_test_mode , 5 );
  CEXPR( int , random_test_mode , 6 );
  #ifdef USE_GETLINE
    CEXPR( bool , use_getline , true );
  #else
    CEXPR( bool , use_getline , false );
  #endif
#else
  ll GetRand( CRL Rand_min , CRL Rand_max ) { ll answer = time( NULL ); RE answer * rand() % ( Rand_max + 1 - Rand_min ) + Rand_min; }
#endif

// VVV 常設ライブラリは以下に挿入する。

// AAA 常設ライブラリは以上に挿入する。

#define INCLUDE_LIBRARY
#include __FILE__

#endif // INCLUDE_LIBRARY

#endif // INCLUDE_SUB

#endif // INCLUDE_MAIN