#ifdef DEBUG
  #define _GLIBCXX_DEBUG
  #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr ); signal( SIGABRT , &AlertAbort )
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , DEBUG_VALUE )
  #define CERR( MESSAGE ) cerr << MESSAGE << endl;
  #define COUT( ANSWER ) cout << ANSWER << endl
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " << ( MIN ) << ( ( MIN ) <= A ? "<=" : ">" ) << A << ( A <= ( MAX ) ? "<=" : ">" ) << ( MAX ) ); assert( ( MIN ) <= A && A <= ( MAX ) )
#define LIBRARY_SEARCH bool searched_library = false; LibrarySearch( searched_library ); if( searched_library ){ QUIT; };
  #define START_WATCH( PROCESS_NAME ) StartWatch( PROCESS_NAME )
  #define STOP_WATCH( HOW_MANY_TIMES ) StopWatch( HOW_MANY_TIMES )
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr )
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , VALUE )
  #define CERR( MESSAGE ) 
  #define COUT( ANSWER ) cout << ANSWER << "\n"
  #define ASSERT( A , MIN , MAX ) assert( ( MIN ) <= A && A <= ( MAX ) )
  #define LIBRARY_SEARCH
  #define START_WATCH( PROCESS_NAME )
  #define STOP_WATCH( HOW_MANY_TIMES )
#endif
// #define RANDOM_TEST
#include <bits/stdc++.h>
using namespace std;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define TYPE_OF( VAR ) decay_t<decltype( VAR )>
#define CEXPR( LL , BOUND , VALUE ) constexpr LL BOUND = VALUE
#define CIN( LL , A ) LL A; cin >> A
#define CIN_ASSERT( A , MIN , MAX ) TYPE_OF( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define GETLINE( A ) string A; getline( cin , A )
#define GETLINE_SEPARATE( A , SEPARATOR ) string A; getline( cin , A , SEPARATOR )
#define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( TYPE_OF( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( TYPE_OF( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( TYPE_OF( INITIAL ) VAR = INITIAL ; VAR >= FINAL ; VAR -- )
#define AUTO_ITR( ARRAY ) auto itr_ ## ARRAY = ARRAY .begin() , end_ ## ARRAY = ARRAY .end()
#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 QUIT return 0
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS_ )

#ifdef DEBUG
  inline void 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。" ); }
  void StartWatch( const string& process_name = "nothing" );
  void StopWatch( const int& how_many_times = 1 );
#endif
#if defined( DEBUG ) && defined( RANDOM_TEST )
  inline CEXPR( int , bound_random_test_num , 1000 );
  #define START_MAIN FOR( random_test_num , 0 , bound_random_test_num ){ CERR( "(" << random_test_num << ")" );
  ll GetRand( const ll& Rand_min , const ll& Rand_max );
  #define SET_ASSERT( A , MIN , MAX ) CERR( #A << " = " << ( A = GetRand( MIN , MAX ) ) )
  #define RETURN( ANSWER ) if( ( ANSWER ) == guchoku ){ CERR( ( ANSWER ) << " == " << guchoku ); continue; } else { CERR( ( ANSWER ) << " != " << guchoku ); QUIT; }
  #define FINISH_MAIN CERR( "" ); }
#else
  #define START_MAIN 
  #define SET_ASSERT( A , MIN , MAX ) cin >> A; ASSERT( A , MIN , MAX )
  #define RETURN( ANSWER ) COUT( ( ANSWER ) ); QUIT
  #define FINISH_MAIN 
#endif

template <typename T> inline T Absolute( const T& a ){ return a > 0 ? a : -a; }
template <typename T> inline T Residue( const T& a , const T& p ){ return a >= 0 ? a % p : p - 1 - ( ( - ( a + 1 ) ) % p ); }

#define POWER( ANSWER , ARGUMENT , EXPONENT )				\
  static_assert( ! is_same<TYPE_OF( ARGUMENT ),int>::value && ! is_same<TYPE_OF( ARGUMENT ),uint>::value ); \
  TYPE_OF( ARGUMENT ) ANSWER{ 1 };					\
  {									\
    TYPE_OF( ARGUMENT ) ARGUMENT_FOR_SQUARE_FOR_POWER = ( ARGUMENT );	\
    TYPE_OF( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT );	\
    while( 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 = ( ( MODULO ) + ( ( ARGUMENT ) % ( MODULO ) ) ) % ( MODULO ); \
    TYPE_OF( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT );	\
    while( 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 , CONSTEXPR_LENGTH , MODULO ) \
  static ll ANSWER[CONSTEXPR_LENGTH];					\
  static ll ANSWER_INV[CONSTEXPR_LENGTH];				\
  static ll INVERSE[CONSTEXPR_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 ); \
    }									\
  }									\

// 二分探索テンプレート
// EXPRESSIONがANSWERの広義単調関数の時、EXPRESSION >= TARGETの整数解を格納。
#define BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , DESIRED_INEQUALITY , TARGET , INEQUALITY_FOR_CHECK , UPDATE_U , UPDATE_L , UPDATE_ANSWER ) \
  static_assert( ! is_same<TYPE_OF( TARGET ),uint>::value && ! is_same<TYPE_OF( TARGET ),ull>::value ); \
  ll ANSWER = MINIMUM;							\
  if( MINIMUM <= MAXIMUM ){						\
    ll VARIABLE_FOR_BINARY_SEARCH_L = MINIMUM;				\
    ll VARIABLE_FOR_BINARY_SEARCH_U = MAXIMUM;				\
    ANSWER = ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2; \
    ll VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH;			\
    while( VARIABLE_FOR_BINARY_SEARCH_L != VARIABLE_FOR_BINARY_SEARCH_U ){ \
      VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH = ( EXPRESSION ) - ( TARGET ); \
      CERR( "二分探索中： " << VARIABLE_FOR_BINARY_SEARCH_L << "<=" << ANSWER << "<=" << VARIABLE_FOR_BINARY_SEARCH_U << ":" << EXPRESSION << "-" << TARGET << "=" << VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH ); \
      if( VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH INEQUALITY_FOR_CHECK 0 ){	\
	VARIABLE_FOR_BINARY_SEARCH_U = UPDATE_U;			\
      } else {								\
	VARIABLE_FOR_BINARY_SEARCH_L = UPDATE_L;			\
      }									\
      ANSWER = UPDATE_ANSWER;						\
    }									\
    CERR( "二分探索終了： " << VARIABLE_FOR_BINARY_SEARCH_L << "<=" << ANSWER << "<=" << VARIABLE_FOR_BINARY_SEARCH_U << ":" << EXPRESSION << ( EXPRESSION > TARGET ? ">" : EXPRESSION < TARGET ? "<" : "=" ) << TARGET ); \
    CERR( ( EXPRESSION DESIRED_INEQUALITY TARGET ? "二分探索成功" : "二分探索失敗" ) ); \
    assert( EXPRESSION DESIRED_INEQUALITY TARGET );			\
  } else {								\
    CERR( "二分探索失敗： " << MINIMUM << ">" << MAXIMUM );		\
    assert( MINIMUM <= MAXIMUM );					\
  }									\

// 単調増加の時にEXPRESSION >= TARGETの最小解を格納。
#define BS1( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , TARGET , >= , ANSWER , ANSWER + 1 , ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \

// 単調増加の時にEXPRESSION <= TARGETの最大解を格納。
#define BS2( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , TARGET , > , ANSWER - 1 , ANSWER , ( VARIABLE_FOR_BINARY_SEARCH_L + 1 + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \

// 単調減少の時にEXPRESSION >= TARGETの最大解を格納。
#define BS3( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , TARGET , < , ANSWER - 1 , ANSWER , ( VARIABLE_FOR_BINARY_SEARCH_L + 1 + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \
    
// 単調減少の時にEXPRESSION <= TARGETの最小解を格納。
#define BS4( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , TARGET , <= , ANSWER , ANSWER + 1 , ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \

// 圧縮用
#define TE template
#define TY typename
#define US using
#define ST static
#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 MO move
#define TH this
#define CRI CO int&
#define CRUI CO uint&
#define CRL CO ll&

// 配列の各要素がint型の範疇でも総和がそうでない場合はTをint型にすると正しく動作しないことに注意。
// InitialSegmentSumで負の入力を扱うためにuintではなくintをテンプレート引数にする。

// 使用演算：
// T& T::operator=( const T& )
// T& T::operator+=( const T& )
// T operator-( const T& , const T& )（ただしIntervalSumを用いない場合は不要）
// T operator<( const T& , const T& )（ただしBinarySearchを用いない場合は不要）
template <typename T , int N>
class BIT
{
private:
  T m_fenwick[N + 1];

public:
  inline BIT();
  BIT( const T ( & a )[N] );

  // const参照でないことに注意。
  inline T Get( const int& i ) const;
  inline void Set( const int& i , const T& n );

  inline BIT<T,N>& operator+=( const T ( & a )[N] );
  void Add( const int& i , const T& n );

  T InitialSegmentSum( const int& i_final ) const;
  inline T IntervalSum( const int& i_start , const int& i_final ) const;
  
  // operator+=の単位元T()より小さくない要素のみを成分に持つ場合のみサポート。
  // InitialSegmentSum( i )がn以上となるiが存在する場合にその最小値を2進法で探索。
  int BinarySearch( const T& n ) const;
  // IntervalSum( i_start , i )がt以上となるi_start以上のiが存在する場合にその最小値を2進法で探索。
  inline int BinarySearch( const int& i_start , const T& n ) const;
  
};

template <typename T , int N> inline BIT<T,N>::BIT() : m_fenwick() {}
template <typename T , int N>
BIT<T,N>::BIT( const T ( & a )[N] ) : m_fenwick()
{

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

    T& fenwick_j = m_fenwick[j];
    int i = j - 1;
    fenwick_j = a[i];
    int i_lim = j - ( j & -j );

    while( i != i_lim ){

      fenwick_j += m_fenwick[i];
      i -= ( i & -i );

    }

  }

}

template <typename T , int N> inline T BIT<T,N>::Get( const int& i ) const { return IntervalSum( i , i ); }
template <typename T , int N> inline void BIT<T,N>::Set( const int& i , const T& n ) { Add( i , n - IntervalSum( i , i ) ); }

template <typename T , int N> inline BIT<T,N>& BIT<T,N>::operator+=( const T ( & a )[N] ) { for( int i = 0 ; i < N ; i++ ){ Add( i , a[i] ); } return *this; }

template <typename T , int N>
void BIT<T,N>::Add( const int& i , const T& n )
{
  
  int j = i + 1;

  while( j <= N ){

    m_fenwick[j] += n;
    j += ( j & -j );

  }

  return;
  
}

template <typename T , int N> 
T BIT<T,N>::InitialSegmentSum( const int& i_final ) const
{

  T sum = 0;
  int j = ( i_final < N ? i_final : N - 1 ) + 1;

  while( j > 0 ){

    sum += m_fenwick[j];
    j -= j & -j;
    
  }

  return sum;
  
}

template <typename T , int N> inline T BIT<T,N>::IntervalSum( const int& i_start , const int& i_final ) const { return InitialSegmentSum( i_final ) - InitialSegmentSum( i_start - 1 ); }

// inline CEXPR( int , bound_N , 10 );
inline DEXPR( int , bound_N , 2000 , 10 );
// inline CEXPR( int , bound_N , 1000000000 ); // 0が9個
// inline CEXPR( ll , bound_N , 1000000000000000000 ); // 0が18個

inline CEXPR( TYPE_OF( bound_N ) , bound_M , bound_N );
// inline CEXPR( int , bound_M , 10 );
// inline DEXPR( int , bound_M , 100000 , 100 ); // 0が5個
// inline CEXPR( int , bound_M , 1000000000 ); // 0が9個
// inline CEXPR( ll , bound_M , 1000000000000000000 ); // 0が18個
inline CEXPR( TYPE_OF( bound_N ) , bound_K , bound_N );

// inline DEXPR( int , bound_H , 1000 , 10 );
// // inline DEXPR( int , bound_H , 100000 , 10 ); // 0が5個
// // inline CEXPR( int , bound_H , 1000000000 ); // 0が9個
// inline CEXPR( int , bound_W , bound_H );
// #if bound_H < ( 1 << 16 )
//   inline CEXPR( ll , bound_HW , ll( bound_H ) * bound_W );
// #else
//   inline CEXPR( int , bound_HW , bound_H * bound_W );
// #endif
// // CEXPR( int , bound_HW , 100000 ); // 0が5個
// // CEXPR( int , bound_HW , 1000000000 ); // 0が5個

// list<int> e[bound_N];
// list<int> E( const int& i )
// {
//   list<int> answer = e[i];
//   // 入力によらない処理
//   return answer;
// }

// inline CEXPR( ll , P , 998244353 );
// inline CEXPR( ll , P , 1000000007 );

US ull = unsigned long long;IN CEXPR(uint,P,998244353);TE <uint M,TY INT> IN CE INT& RS(INT& n)NE{RE n < 0?((((++n)*= -1)%= M)*= -1)+= M - 1:n %= M;}TE <uint M> IN CE uint& RS(uint& n)NE{RE n %= M;}TE <uint M> IN CE ull& RS(ull& n)NE{RE n %= M;}TE <TY INT> IN CE INT& RSP(INT& n)NE{CE CO 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;RE n < n_uq?n += P - n_uq:n -= n_uq;}TE <> IN CE ull& RS<P,ull>(ull& n)NE{CE CO ull Pull = P;CE CO ull Pull2 = (Pull - 1)* (Pull - 1);RE RSP(n > Pull2?n -= Pull2:n);}TE <uint M,TY INT> IN CE INT RS(INT&& n)NE{RE MO(RS<M>(n));}TE <uint M,TY INT> IN CE INT RS(CO INT& n)NE{RE RS<M>(INT(n));}

#define SFINAE_FOR_MOD(DEFAULT)TY T,enable_if_t<is_constructible<uint,decay_t<T> >::value>* DEFAULT
#define DC_OF_CM_FOR_MOD(FUNC)IN bool OP FUNC(CO Mod<M>& n)CO NE
#define DC_OF_AR_FOR_MOD(FUNC)IN Mod<M> OP FUNC(CO Mod<M>& n)CO NE;TE <SFINAE_FOR_MOD(= nullptr)> IN Mod<M> OP FUNC(T&& n)CO NE;
#define DF_OF_CM_FOR_MOD(FUNC)TE <uint M> IN bool Mod<M>::OP FUNC(CO Mod<M>& n)CO NE{RE m_n FUNC n.m_n;}
#define DF_OF_AR_FOR_MOD(FUNC,FORMULA)TE <uint M> IN Mod<M> Mod<M>::OP FUNC(CO Mod<M>& n)CO NE{RE MO(Mod<M>(*TH)FUNC ## = n);}TE <uint M> TE <SFINAE_FOR_MOD()> IN Mod<M> Mod<M>::OP FUNC(T&& n)CO NE{RE FORMULA;}TE <uint M,SFINAE_FOR_MOD(= nullptr)> IN Mod<M> OP FUNC(T&& n0,CO Mod<M>& n1)NE{RE MO(Mod<M>(forward<T>(n0))FUNC ## = n1);}

TE <uint M>CL Mod{PU:uint m_n;IN CE Mod()NE;IN CE Mod(CO Mod<M>& n)NE;IN CE Mod(Mod<M>& n)NE;IN CE Mod(Mod<M>&& n)NE;TE <SFINAE_FOR_MOD(= nullptr)> IN CE Mod(CO T& n)NE;TE <SFINAE_FOR_MOD(= nullptr)> IN CE Mod(T& n)NE;TE <SFINAE_FOR_MOD(= nullptr)> IN CE Mod(T&& n)NE;IN CE Mod<M>& OP=(CO Mod<M>& n)NE;IN CE Mod<M>& OP=(Mod<M>&& n)NE;IN CE Mod<M>& OP+=(CO Mod<M>& n)NE;IN CE Mod<M>& OP-=(CO Mod<M>& n)NE;IN CE Mod<M>& OP*=(CO Mod<M>& n)NE;IN Mod<M>& OP/=(CO Mod<M>& n);IN CE Mod<M>& OP<<=(int n)NE;IN CE Mod<M>& OP>>=(int n)NE;IN CE Mod<M>& OP++()NE;IN CE Mod<M> OP++(int)NE;IN CE Mod<M>& OP--()NE;IN CE Mod<M> OP--(int)NE;DC_OF_CM_FOR_MOD(==);DC_OF_CM_FOR_MOD(!=);DC_OF_CM_FOR_MOD(<);DC_OF_CM_FOR_MOD(<=);DC_OF_CM_FOR_MOD(>);DC_OF_CM_FOR_MOD(>=);DC_OF_AR_FOR_MOD(+);DC_OF_AR_FOR_MOD(-);DC_OF_AR_FOR_MOD(*);DC_OF_AR_FOR_MOD(/);IN CE Mod<M> OP<<(int n)CO NE;IN CE Mod<M> OP>>(int n)CO NE;IN CE Mod<M> OP-()CO NE;IN CE Mod<M>& SignInvert()NE;IN CE Mod<M>& Double()NE;IN CE Mod<M>& Halve()NE;IN Mod<M>& Invert();TE <TY T> IN CE Mod<M>& PositivePW(T&& EX)NE;TE <TY T> IN CE Mod<M>& NonNegativePW(T&& EX)NE;TE <TY T> IN CE Mod<M>& PW(T&& EX);IN CE VO swap(Mod<M>& n)NE;IN CE CO uint& RP()CO NE;ST IN CE Mod<M> DeRP(CO uint& n)NE;ST IN CE uint& Normalise(uint& n)NE;ST IN CO Mod<M>& Inverse(CO uint& n)NE;ST IN CO Mod<M>& Factorial(CO uint& n)NE;ST IN CO Mod<M>& FactorialInverse(CO uint& n)NE;ST IN CO Mod<M>& zero()NE;ST IN CO Mod<M>& one()NE;TE <TY T> IN CE Mod<M>& Ref(T&& n)NE;};

#define SFINAE_FOR_MN(DEFAULT)TY T,enable_if_t<is_constructible<Mod<M>,decay_t<T> >::value>* DEFAULT
#define DC_OF_AR_FOR_MN(FUNC)IN MN<M> OP FUNC(CO MN<M>& n)CO NE;TE <SFINAE_FOR_MOD(= nullptr)> IN MN<M> OP FUNC(T&& n)CO NE;
#define DF_OF_CM_FOR_MN(FUNC)TE <uint M> IN bool MN<M>::OP FUNC(CO MN<M>& n)CO NE{RE m_n FUNC n.m_n;}
#define DF_OF_AR_FOR_MN(FUNC,FORMULA)TE <uint M> IN MN<M> MN<M>::OP FUNC(CO MN<M>& n)CO NE{RE MO(MN<M>(*TH)FUNC ## = n);}TE <uint M> TE <SFINAE_FOR_MOD()> IN MN<M> MN<M>::OP FUNC(T&& n)CO NE{RE FORMULA;}TE <uint M,SFINAE_FOR_MOD(= nullptr)> IN MN<M> OP FUNC(T&& n0,CO MN<M>& n1)NE{RE MO(MN<M>(forward<T>(n0))FUNC ## = n1);}

TE <uint M>CL MN :PU Mod<M>{PU:IN CE MN()NE;IN CE MN(CO MN<M>& n)NE;IN CE MN(MN<M>& n)NE;IN CE MN(MN<M>&& n)NE;TE <SFINAE_FOR_MN(= nullptr)> IN CE MN(CO T& n)NE;TE <SFINAE_FOR_MN(= nullptr)> IN CE MN(T&& n)NE;IN CE MN<M>& OP=(CO MN<M>& n)NE;IN CE MN<M>& OP=(MN<M>&& n)NE;IN CE MN<M>& OP+=(CO MN<M>& n)NE;IN CE MN<M>& OP-=(CO MN<M>& n)NE;IN CE MN<M>& OP*=(CO MN<M>& n)NE;IN MN<M>& OP/=(CO MN<M>& n);IN CE MN<M>& OP<<=(int n)NE;IN CE MN<M>& OP>>=(int n)NE;IN CE MN<M>& OP++()NE;IN CE MN<M> OP++(int)NE;IN CE MN<M>& OP--()NE;IN CE MN<M> OP--(int)NE;DC_OF_AR_FOR_MN(+);DC_OF_AR_FOR_MN(-);DC_OF_AR_FOR_MN(*);DC_OF_AR_FOR_MN(/);IN CE MN<M> OP<<(int n)CO NE;IN CE MN<M> OP>>(int n)CO NE;IN CE MN<M> OP-()CO NE;IN CE MN<M>& SignInvert()NE;IN CE MN<M>& Double()NE;IN CE MN<M>& Halve()NE;IN CE MN<M>& Invert();TE <TY T> IN CE MN<M>& PositivePW(T&& EX)NE;TE <TY T> IN CE MN<M>& NonNegativePW(T&& EX)NE;TE <TY T> IN CE MN<M>& PW(T&& EX);IN CE uint RP()CO NE;IN CE Mod<M> Reduce()CO NE;ST IN CE MN<M> DeRP(CO uint& n)NE;ST IN CO MN<M>& Formise(CO uint& n)NE;ST IN CO MN<M>& Inverse(CO uint& n)NE;ST IN CO MN<M>& Factorial(CO uint& n)NE;ST IN CO MN<M>& FactorialInverse(CO uint& n)NE;ST IN CO MN<M>& zero()NE;ST IN CO MN<M>& one()NE;ST IN CE uint Form(CO uint& n)NE;ST IN CE ull& Reduction(ull& n)NE;ST IN CE ull& ReducedMU(ull& n,CO uint& m)NE;ST IN CE uint MU(CO uint& n0,CO uint& n1)NE;ST IN CE uint BaseSquareTruncation(uint& n)NE;TE <TY T> IN CE MN<M>& Ref(T&& n)NE;};TE <uint M> IN CE MN<M> Twice(CO MN<M>& n)NE;TE <uint M> IN CE MN<M> Half(CO MN<M>& n)NE;TE <uint M> IN CE MN<M> Inverse(CO MN<M>& n);TE <uint M,TY T> IN CE MN<M> PW(CO MN<M>& n,CO T& EX);TE <TY T> IN CE MN<2> PW(CO MN<2>& n,CO T& p);TE <TY T> IN CE T Square(CO T& t);TE <> IN CE MN<2> Square<MN<2> >(CO MN<2>& t);TE <uint M> IN CE VO swap(MN<M>& n0,MN<M>& n1)NE;TE <uint M> IN string to_string(CO MN<M>& n)NE;TE<uint M,CL Traits> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO MN<M>& n);

TE <uint M>CL COantsForMod{PU:COantsForMod()= delete;ST CE CO bool g_even = ((M & 1)== 0);ST CE CO uint g_memory_bound = 1000000;ST CE CO uint g_memory_LE = M < g_memory_bound?M:g_memory_bound;ST IN CE ull MNBasePW(ull&& EX)NE;ST CE uint g_M_minus = M - 1;ST CE uint g_M_minus_2 = M - 2;ST CE uint g_M_minus_2_neg = 2 - M;ST CE CO int g_MN_digit = 32;ST CE CO ull g_MN_base = ull(1)<< g_MN_digit;ST CE CO uint g_MN_base_minus = uint(g_MN_base - 1);ST CE CO uint g_MN_digit_half = (g_MN_digit + 1)>> 1;ST CE CO uint g_MN_base_sqrt_minus = (1 << g_MN_digit_half)- 1;ST CE CO uint g_MN_M_neg_inverse = uint((g_MN_base - MNBasePW((ull(1)<< (g_MN_digit - 1))- 1))& g_MN_base_minus);ST CE CO uint g_MN_base_mod = uint(g_MN_base % M);ST CE CO uint g_MN_base_square_mod = uint(((g_MN_base % M)* (g_MN_base % M))% M);};TE <uint M> IN CE ull COantsForMod<M>::MNBasePW(ull&& EX)NE{ull prod = 1;ull PW = M;WH(EX != 0){(EX & 1)== 1?(prod *= PW)&= g_MN_base_minus:prod;EX >>= 1;(PW *= PW)&= g_MN_base_minus;}RE prod;}

#include <immintrin.h>
#define SET_VE_32_128_FOR_SIMD(UINT,VE_NAME,SCALAR0,SCALAR1,SCALAR2,SCALAR3)CE CO UINT VE_NAME ## _copy[4] ={SCALAR0,SCALAR1,SCALAR2,SCALAR3};ST CO __m128i v_ ## VE_NAME = _mm_load_si128((__m128i*)VE_NAME ##_copy)
#define SET_VE_64_128_FOR_SIMD(UINT,VE_NAME,SCALAR0,SCALAR1)CE CO UINT VE_NAME ## _copy[2] ={SCALAR0,SCALAR1};ST CO __m128i v_ ## VE_NAME = _mm_load_si128((__m128i*)VE_NAME ##_copy)
#define SET_VE_64_256_FOR_SIMD(ULL,VE_NAME,SCALAR0,SCALAR1,SCALAR2,SCALAR3)CE CO ULL VE_NAME ## _copy[4] ={SCALAR0,SCALAR1,SCALAR2,SCALAR3};ST CO __m256i v_ ## VE_NAME = _mm256_load_si256((__m256i*)VE_NAME ##_copy)
#define SET_CO_VE_32_128_FOR_SIMD(UINT,VE_NAME,SCALAR)SET_VE_32_128_FOR_SIMD(UINT,VE_NAME,SCALAR,SCALAR,SCALAR,SCALAR)
#define SET_CO_VE_64_128_FOR_SIMD(ULL,VE_NAME,SCALAR)SET_VE_64_128_FOR_SIMD(ULL,VE_NAME,SCALAR,SCALAR)
#define SET_CO_VE_64_256_FOR_SIMD(ULL,VE_NAME,SCALAR)SET_VE_64_256_FOR_SIMD(ULL,VE_NAME,SCALAR,SCALAR,SCALAR,SCALAR)

TE <uint M>CL COantsForSIMDForMod{PU:COantsForSIMDForMod()= delete;ST IN CO __m128i& v_M()NE;ST IN CO __m128i& v_Mull()NE;ST IN CO __m128i& v_M_minus()NE;ST IN CO __m128i& v_M_neg_inverse()NE;ST IN CO __m128i& v_digitull()NE;};TE <uint M> IN CO __m128i& COantsForSIMDForMod<M>::v_M()NE{SET_CO_VE_32_128_FOR_SIMD(uint,M,M);RE v_M;}TE <uint M> IN CO __m128i& COantsForSIMDForMod<M>::v_Mull()NE{SET_CO_VE_64_128_FOR_SIMD(ull,Mull,M);RE v_Mull;}TE <uint M> IN CO __m128i& COantsForSIMDForMod<M>::v_M_minus()NE{SET_CO_VE_32_128_FOR_SIMD(uint,M_minus,M - 1);RE v_M_minus;}TE <uint M> IN CO __m128i& COantsForSIMDForMod<M>::v_M_neg_inverse()NE{SET_CO_VE_32_128_FOR_SIMD(uint,M_neg_inverse,COantsForMod<M>::g_MN_M_neg_inverse);RE v_M_neg_inverse;}TE <uint M> IN CO __m128i& COantsForSIMDForMod<M>::v_digitull()NE{SET_CO_VE_64_128_FOR_SIMD(ull,digitull,COantsForMod<M>::g_MN_digit);RE v_digitull;}TE <uint M> IN __m128i& SIMD_RS_32_128(__m128i& v)NE{CO __m128i& v_M = COantsForSIMDForMod<M>::v_M();RE v -= v_M * _mm_cmpgt_epi32(v,v_M);}TE <uint M> IN __m128i& SIMD_RS_64_128(__m128i& v)NE{ull v_copy[2];_mm_store_si128((__m128i*)v_copy,v);for(uint i = 0;i < 2;i++){ull& v_copy_i = v_copy[i];v_copy_i = (v_copy_i < M?0:M);}RE v -= _mm_load_si128((__m128i*)v_copy);}TE <uint M> IN __m256i& SIMD_RS_64_256(__m256i& v)NE{ull v_copy[4];_mm256_store_si256((__m256i*)v_copy,v);for(uint i = 0;i < 4;i++){ull& v_copy_i = v_copy[i];v_copy_i = (v_copy_i < M?0:M);}RE v -= _mm256_load_si256((__m256i*)v_copy);}IN CE int SIMD_Shuffle(CRI a0,CRI a1,CRI a2,CRI a3)NE{RE (a0 << (0 << 1))+ (a1 << (1 << 1))+ (a2 << (2 << 1))+ (a3 << (3 << 1));}TE <uint M> IN VO SIMD_Addition_32_64(CO Mod<M>& a0,CO Mod<M>& a1,CO Mod<M>& b0,CO Mod<M>& b1,Mod<M>& c0,Mod<M>& c1)NE{uint a_copy[4] ={a0.m_n,a1.m_n,0,0};uint b_copy[4] ={b0.m_n,b1.m_n,0,0};__m128i v_a = _mm_load_si128((__m128i*)a_copy);v_a += _mm_load_si128((__m128i*)b_copy);ST CO __m128i& v_M_minus = COantsForSIMDForMod<M>::v_M_minus();ST CO __m128i& v_M = COantsForSIMDForMod<M>::v_M();v_a += _mm_cmpgt_epi32(v_a,v_M_minus)& v_M;_mm_store_si128((__m128i*)a_copy,v_a);c0.m_n = MO(a_copy[0]);c1.m_n = MO(a_copy[1]);RE;}TE <uint M> IN VO SIMD_Addition_32_128(CO Mod<M>& a0,CO Mod<M>& a1,CO Mod<M>& a2,CO Mod<M>& a3,CO Mod<M>& b0,CO Mod<M>& b1,CO Mod<M>& b2,CO Mod<M>& b3,Mod<M>& c0,Mod<M>& c1,Mod<M>& c2,Mod<M>& c3)NE{uint a_copy[4] ={a0.m_n,a1.m_n,a2.m_n,a3.m_n};uint b_copy[4] ={b0.m_n,b1.m_n,b2.m_n,b3.m_n};__m128i v_a = _mm_load_si128((__m128i*)a_copy)+ _mm_load_si128((__m128i*)b_copy);_mm_store_si128((__m128i*)a_copy,v_a);for(uint i = 0;i < 4;i++){b_copy[i] = a_copy[i] < M?0:M;}v_a -= _mm_load_si128((__m128i*)b_copy);_mm_store_si128((__m128i*)a_copy,v_a);c0.m_n = MO(a_copy[0]);c1.m_n = MO(a_copy[1]);c2.m_n = MO(a_copy[2]);c3.m_n = MO(a_copy[3]);RE;}TE <uint M> IN VO SIMD_Substracition_32_64(CO Mod<M>& a0,CO Mod<M>& a1,CO Mod<M>& b0,CO Mod<M>& b1,Mod<M>& c0,Mod<M>& c1)NE{uint a_copy[4] ={a0.m_n,a1.m_n,0,0};uint b_copy[4] ={b0.m_n,b1.m_n,0,0};__m128i v_a = _mm_load_si128((__m128i*)a_copy);__m128i v_b = _mm_load_si128((__m128i*)b_copy);_mm_store_si128((__m128i*)a_copy,v_a);for(uint i = 0;i < 2;i++){b_copy[i] = a_copy[i] < b_copy[i]?M:0;}(v_a += _mm_load_si128((__m128i*)b_copy))-= v_b;_mm_store_si128((__m128i*)a_copy,v_a);c0.m_n = MO(a_copy[0]);c1.m_n = MO(a_copy[1]);RE;}TE <uint M> IN VO SIMD_Subtraction_32_128(CO Mod<M>& a0,CO Mod<M>& a1,CO Mod<M>& a2,CO Mod<M>& a3,CO Mod<M>& b0,CO Mod<M>& b1,CO Mod<M>& b2,CO Mod<M>& b3,Mod<M>& c0,Mod<M>& c1,Mod<M>& c2,Mod<M>& c3)NE{uint a_copy[4] ={a0.m_n,a1.m_n,a2.m_n,a3.m_n};uint b_copy[4] ={b0.m_n,b1.m_n,b2.m_n,b3.m_n};__m128i v_a = _mm_load_si128((__m128i*)a_copy);__m128i v_b = _mm_load_si128((__m128i*)b_copy);_mm_store_si128((__m128i*)a_copy,v_a);for(uint i = 0;i < 4;i++){b_copy[i] = a_copy[i] < b_copy[i]?M:0;}(v_a += _mm_load_si128((__m128i*)b_copy))-= v_b;_mm_store_si128((__m128i*)a_copy,v_a);c0.m_n = MO(a_copy[0]);c1.m_n = MO(a_copy[1]);c2.m_n = MO(a_copy[2]);c3.m_n = MO(a_copy[3]);RE;}

US MP = Mod<P>;US MNP = MN<P>;TE <uint M> IN CE uint MN<M>::Form(CO uint& n)NE{ull n_copy = n;RE uint(MO(Reduction(n_copy *= COantsForMod<M>::g_MN_base_square_mod)));}TE <uint M> IN CE ull& MN<M>::Reduction(ull& n)NE{ull n_sub = n & COantsForMod<M>::g_MN_base_minus;RE ((n += ((n_sub *= COantsForMod<M>::g_MN_M_neg_inverse)&= COantsForMod<M>::g_MN_base_minus)*= M)>>= COantsForMod<M>::g_MN_digit)< M?n:n -= M;}TE <uint M> IN CE ull& MN<M>::ReducedMU(ull& n,CO uint& m)NE{RE Reduction(n *= m);}TE <uint M> IN CE uint MN<M>::MU(CO uint& n0,CO uint& n1)NE{ull n0_copy = n0;RE uint(MO(ReducedMU(ReducedMU(n0_copy,n1),COantsForMod<M>::g_MN_base_square_mod)));}TE <uint M> IN CE uint MN<M>::BaseSquareTruncation(uint& n)NE{CO uint n_u = n >> COantsForMod<M>::g_MN_digit_half;n &= COantsForMod<M>::g_MN_base_sqrt_minus;RE n_u;}TE <uint M> IN CE MN<M>::MN()NE:Mod<M>(){static_assert(! COantsForMod<M>::g_even);}TE <uint M> IN CE MN<M>::MN(CO MN<M>& n)NE:Mod<M>(n){}TE <uint M> IN CE MN<M>::MN(MN<M>& n)NE:Mod<M>(n){}TE <uint M> IN CE MN<M>::MN(MN<M>&& n)NE:Mod<M>(MO(n)){}TE <uint M> TE <SFINAE_FOR_MN()> IN CE MN<M>::MN(CO T& n)NE:Mod<M>(n){static_assert(! COantsForMod<M>::g_even);Mod<M>::m_n = Form(Mod<M>::m_n);}TE <uint M> TE <SFINAE_FOR_MN()> IN CE MN<M>::MN(T&& n)NE:Mod<M>(forward<T>(n)){static_assert(! COantsForMod<M>::g_even);Mod<M>::m_n = Form(Mod<M>::m_n);}TE <uint M> IN CE MN<M>& MN<M>::OP=(CO MN<M>& n)NE{RE Ref(Mod<M>::OP=(n));}TE <uint M> IN CE MN<M>& MN<M>::OP=(MN<M>&& n)NE{RE Ref(Mod<M>::OP=(MO(n)));}TE <uint M> IN CE MN<M>& MN<M>::OP+=(CO MN<M>& n)NE{RE Ref(Mod<M>::OP+=(n));}TE <uint M> IN CE MN<M>& MN<M>::OP-=(CO MN<M>& n)NE{RE Ref(Mod<M>::OP-=(n));}TE <uint M> IN CE MN<M>& MN<M>::OP*=(CO MN<M>& n)NE{ull m_n_copy = Mod<M>::m_n;RE Ref(Mod<M>::m_n = MO(ReducedMU(m_n_copy,n.m_n)));}TE <uint M> IN MN<M>& MN<M>::OP/=(CO MN<M>& n){RE OP*=(MN<M>(n).Invert());}TE <uint M> IN CE MN<M>& MN<M>::OP<<=(int n)NE{RE Ref(Mod<M>::OP<<=(n));}TE <uint M> IN CE MN<M>& MN<M>::OP>>=(int n)NE{RE Ref(Mod<M>::OP>>=(n));}TE <uint M> IN CE MN<M>& MN<M>::OP++()NE{RE Ref(Mod<M>::Normalise(Mod<M>::m_n += COantsForMod<M>::g_MN_base_mod));}TE <uint M> IN CE MN<M> MN<M>::OP++(int)NE{MN<M> n{*TH};OP++();RE n;}TE <uint M> IN CE MN<M>& MN<M>::OP--()NE{RE Ref(Mod<M>::m_n < COantsForMod<M>::g_MN_base_mod?((Mod<M>::m_n += M)-= COantsForMod<M>::g_MN_base_mod):Mod<M>::m_n -= COantsForMod<M>::g_MN_base_mod);}TE <uint M> IN CE MN<M> MN<M>::OP--(int)NE{MN<M> n{*TH};OP--();RE n;}DF_OF_AR_FOR_MN(+,MN<M>(forward<T>(n))+= *TH);DF_OF_AR_FOR_MN(-,MN<M>(forward<T>(n)).SignInvert()+= *TH);DF_OF_AR_FOR_MN(*,MN<M>(forward<T>(n))*= *TH);DF_OF_AR_FOR_MN(/,MN<M>(forward<T>(n)).Invert()*= *TH);TE <uint M> IN CE MN<M> MN<M>::OP<<(int n)CO NE{RE MO(MN<M>(*TH)<<= n);}TE <uint M> IN CE MN<M> MN<M>::OP>>(int n)CO NE{RE MO(MN<M>(*TH)>>= n);}TE <uint M> IN CE MN<M> MN<M>::OP-()CO NE{RE MO(MN<M>(*TH).SignInvert());}TE <uint M> IN CE MN<M>& MN<M>::SignInvert()NE{RE Ref(Mod<M>::m_n > 0?Mod<M>::m_n = M - Mod<M>::m_n:Mod<M>::m_n);}TE <uint M> IN CE MN<M>& MN<M>::Double()NE{RE Ref(Mod<M>::Double());}TE <uint M> IN CE MN<M>& MN<M>::Halve()NE{RE Ref(Mod<M>::Halve());}TE <uint M> IN CE MN<M>& MN<M>::Invert(){assert(Mod<M>::m_n > 0);RE PositivePW(uint(COantsForMod<M>::g_M_minus_2));}TE <uint M> TE <TY T> IN CE MN<M>& MN<M>::PositivePW(T&& EX)NE{MN<M> PW{*TH};(--EX)%= COantsForMod<M>::g_M_minus_2;WH(EX != 0){(EX & 1)== 1?OP*=(PW):*TH;EX >>= 1;PW *= PW;}RE *TH;}TE <uint M> TE <TY T> IN CE MN<M>& MN<M>::NonNegativePW(T&& EX)NE{RE EX == 0?Ref(Mod<M>::m_n = 1):PositivePW(forward<T>(EX));}TE <uint M> TE <TY T> IN CE MN<M>& MN<M>::PW(T&& EX){bool neg = EX < 0;assert(!(neg && Mod<M>::m_n == 0));RE neg?PositivePW(forward<T>(EX *= COantsForMod<M>::g_M_minus_2_neg)):NonNegativePW(forward<T>(EX));}TE <uint M> IN CE uint MN<M>::RP()CO NE{ull m_n_copy = Mod<M>::m_n;RE MO(Reduction(m_n_copy));}TE <uint M> IN CE Mod<M> MN<M>::Reduce()CO NE{ull m_n_copy = Mod<M>::m_n;RE Mod<M>::DeRP(MO(Reduction(m_n_copy)));}TE <uint M> IN CE MN<M> MN<M>::DeRP(CO uint& n)NE{RE MN<M>(Mod<M>::DeRP(n));}TE <uint M> IN CO MN<M>& MN<M>::Formise(CO uint& n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={zero(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = DeRP(LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN CO MN<M>& MN<M>::Inverse(CO uint& n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={zero(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = MN<M>(Mod<M>::Inverse(LE_curr));LE_curr++;}RE memory[n];}TE <uint M> IN CO MN<M>& MN<M>::Factorial(CO uint& n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;ST MN<M> val_curr{one()};MN<M> val_last{one()};WH(LE_curr <= n){memory[LE_curr++] = val_curr *= ++val_last;}RE memory[n];}TE <uint M> IN CO MN<M>& MN<M>::FactorialInverse(CO uint& n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;ST MN<M> val_curr{one()};MN<M> val_last{one()};WH(LE_curr <= n){memory[LE_curr] = val_curr *= Inverse(LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN CO MN<M>& MN<M>::zero()NE{ST CE CO MN<M> z{};RE z;}TE <uint M> IN CO MN<M>& MN<M>::one()NE{ST CE CO MN<M> o{DeRP(1)};RE o;}TE <uint M> TE <TY T> IN CE MN<M>& MN<M>::Ref(T&& n)NE{RE *TH;}TE <uint M> IN CE MN<M> Twice(CO MN<M>& n)NE{RE MO(MN<M>(n).Double());}TE <uint M> IN CE MN<M> Half(CO MN<M>& n)NE{RE MO(MN<M>(n).Halve());}TE <uint M> IN CE MN<M> Inverse(CO MN<M>& n){RE MO(MN<M>(n).Invert());}TE <uint M,TY T> IN CE MN<M> PW(CO MN<M>& n,CO T& EX){RE MO(MN<M>(n).PW(T(EX)));}TE <uint M> IN CE VO swap(MN<M>& n0,MN<M>& n1)NE{n0.swap(n1);}TE <uint M> IN string to_string(CO MN<M>& n)NE{RE to_string(n.RP())+ " + MZ";}TE<uint M,CL Traits> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO MN<M>& n){RE os << n.RP();}

TE <uint M> IN CE Mod<M>::Mod()NE:m_n(){}TE <uint M> IN CE Mod<M>::Mod(CO Mod<M>& n)NE:m_n(n.m_n){}TE <uint M> IN CE Mod<M>::Mod(Mod<M>& n)NE:m_n(n.m_n){}TE <uint M> IN CE Mod<M>::Mod(Mod<M>&& n)NE:m_n(MO(n.m_n)){}TE <uint M> TE <SFINAE_FOR_MOD()> IN CE Mod<M>::Mod(CO T& n)NE:m_n(RS<M>(n)){}TE <uint M> TE <SFINAE_FOR_MOD()> IN CE Mod<M>::Mod(T& n)NE:m_n(RS<M>(decay_t<T>(n))){}TE <uint M> TE <SFINAE_FOR_MOD()> IN CE Mod<M>::Mod(T&& n)NE:m_n(RS<M>(forward<T>(n))){}TE <uint M> IN CE Mod<M>& Mod<M>::OP=(CO Mod<M>& n)NE{RE Ref(m_n = n.m_n);}TE <uint M> IN CE Mod<M>& Mod<M>::OP=(Mod<M>&& n)NE{RE Ref(m_n = MO(n.m_n));}TE <uint M> IN CE Mod<M>& Mod<M>::OP+=(CO Mod<M>& n)NE{RE Ref(Normalise(m_n += n.m_n));}TE <uint M> IN CE Mod<M>& Mod<M>::OP-=(CO Mod<M>& n)NE{RE Ref(m_n < n.m_n?(m_n += M)-= n.m_n:m_n -= n.m_n);}TE <uint M> IN CE Mod<M>& Mod<M>::OP*=(CO Mod<M>& n)NE{RE Ref(m_n = COantsForMod<M>::g_even?RS<M>(ull(m_n)* n.m_n):MN<M>::MU(m_n,n.m_n));}TE <> IN CE MP& MP::OP*=(CO MP& n)NE{ull m_n_copy = m_n;RE Ref(m_n = MO((m_n_copy *= n.m_n)< P?m_n_copy:RSP(m_n_copy)));}TE <uint M> IN Mod<M>& Mod<M>::OP/=(CO Mod<M>& n){RE OP*=(Mod<M>(n).Invert());}TE <uint M> IN CE Mod<M>& Mod<M>::OP<<=(int n)NE{WH(n-- > 0){Normalise(m_n <<= 1);}RE *TH;}TE <uint M> IN CE Mod<M>& Mod<M>::OP>>=(int n)NE{WH(n-- > 0){((m_n & 1)== 0?m_n:m_n += M)>>= 1;}RE *TH;}TE <uint M> IN CE Mod<M>& Mod<M>::OP++()NE{RE Ref(m_n < COantsForMod<M>::g_M_minus?++m_n:m_n = 0);}TE <uint M> IN CE Mod<M> Mod<M>::OP++(int)NE{Mod<M> n{*TH};OP++();RE n;}TE <uint M> IN CE Mod<M>& Mod<M>::OP--()NE{RE Ref(m_n == 0?m_n = COantsForMod<M>::g_M_minus:--m_n);}TE <uint M> IN CE Mod<M> Mod<M>::OP--(int)NE{Mod<M> n{*TH};OP--();RE n;}DF_OF_CM_FOR_MOD(==);DF_OF_CM_FOR_MOD(!=);DF_OF_CM_FOR_MOD(>);DF_OF_CM_FOR_MOD(>=);DF_OF_CM_FOR_MOD(<);DF_OF_CM_FOR_MOD(<=);DF_OF_AR_FOR_MOD(+,Mod<M>(forward<T>(n))+= *TH);DF_OF_AR_FOR_MOD(-,Mod<M>(forward<T>(n)).SignInvert()+= *TH);DF_OF_AR_FOR_MOD(*,Mod<M>(forward<T>(n))*= *TH);DF_OF_AR_FOR_MOD(/,Mod<M>(forward<T>(n)).Invert()*= *TH);TE <uint M> IN CE Mod<M> Mod<M>::OP<<(int n)CO NE{RE MO(Mod<M>(*TH)<<= n);}TE <uint M> IN CE Mod<M> Mod<M>::OP>>(int n)CO NE{RE MO(Mod<M>(*TH)>>= n);}TE <uint M> IN CE Mod<M> Mod<M>::OP-()CO NE{RE MO(Mod<M>(*TH).SignInvert());}TE <uint M> IN CE Mod<M>& Mod<M>::SignInvert()NE{RE Ref(m_n > 0?m_n = M - m_n:m_n);}TE <uint M> IN CE Mod<M>& Mod<M>::Double()NE{RE Ref(Normalise(m_n <<= 1));}TE <uint M> IN CE Mod<M>& Mod<M>::Halve()NE{RE Ref(((m_n & 1)== 0?m_n:m_n += M)>>= 1);}TE <uint M> IN Mod<M>& Mod<M>::Invert(){assert(m_n > 0);uint m_n_neg;RE m_n < COantsForMod<M>::g_memory_LE?Ref(m_n = Inverse(m_n).m_n):(m_n_neg = M - m_n < COantsForMod<M>::g_memory_LE)?Ref(m_n = M - Inverse(m_n_neg).m_n):PositivePW(uint(COantsForMod<M>::g_M_minus_2));}TE <> IN Mod<2>& Mod<2>::Invert(){assert(m_n > 0);RE *TH;}TE <uint M> TE <TY T> IN CE Mod<M>& Mod<M>::PositivePW(T&& EX)NE{Mod<M> PW{*TH};EX--;WH(EX != 0){(EX & 1)== 1?OP*=(PW):*TH;EX >>= 1;PW *= PW;}RE *TH;}TE <> TE <TY T> IN CE Mod<2>& Mod<2>::PositivePW(T&& EX)NE{RE *TH;}TE <uint M> TE <TY T> IN CE Mod<M>& Mod<M>::NonNegativePW(T&& EX)NE{RE EX == 0?Ref(m_n = 1):Ref(PositivePW(forward<T>(EX)));}TE <uint M> TE <TY T> IN CE Mod<M>& Mod<M>::PW(T&& EX){bool neg = EX < 0;assert(!(neg && m_n == 0));neg?EX *= COantsForMod<M>::g_M_minus_2_neg:EX;RE m_n == 0?*TH:(EX %= COantsForMod<M>::g_M_minus)== 0?Ref(m_n = 1):PositivePW(forward<T>(EX));}TE <uint M> IN CO Mod<M>& Mod<M>::Inverse(CO uint& n)NE{ST Mod<M> memory[COantsForMod<M>::g_memory_LE] ={zero(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr].m_n = M - MN<M>::MU(memory[M % LE_curr].m_n,M / LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::Factorial(CO uint& n)NE{ST Mod<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = MN<M>::Factorial(LE_curr).Reduce();LE_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::FactorialInverse(CO uint& n)NE{ST Mod<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = MN<M>::FactorialInverse(LE_curr).Reduce();LE_curr++;}RE memory[n];}TE <uint M> IN CE VO Mod<M>::swap(Mod<M>& n)NE{std::swap(m_n,n.m_n);}TE <uint M> IN CE CO uint& Mod<M>::RP()CO NE{RE m_n;}TE <uint M> IN CE Mod<M> Mod<M>::DeRP(CO uint& n)NE{Mod<M> n_copy{};n_copy.m_n = n;RE n_copy;}TE <uint M> IN CE uint& Mod<M>::Normalise(uint& n)NE{RE n < M?n:n -= M;}TE <uint M> IN CO Mod<M>& Mod<M>::zero()NE{ST CE CO Mod<M> z{};RE z;}TE <uint M> IN CO Mod<M>& Mod<M>::one()NE{ST CE CO Mod<M> o{DeRP(1)};RE o;}TE <uint M> TE <TY T> IN CE Mod<M>& Mod<M>::Ref(T&& n)NE{RE *TH;}TE <uint M> IN CE Mod<M> Twice(CO Mod<M>& n)NE{RE MO(Mod<M>(n).Double());}TE <uint M> IN CE Mod<M> Half(CO Mod<M>& n)NE{RE MO(Mod<M>(n).Halve());}TE <uint M> IN Mod<M> Inverse(CO Mod<M>& n){RE MO(Mod<M>(n).Invert());}TE <uint M> IN CE Mod<M> Inverse_COrexpr(CO uint& n)NE{RE MO(Mod<M>::DeRP(RS<M>(n)).NonNegativePW(M - 2));}TE <uint M,TY T> IN CE Mod<M> PW(CO Mod<M>& n,CO T& EX){RE MO(Mod<M>(n).PW(T(EX)));}TE <uint M> IN CE VO swap(Mod<M>& n0,Mod<M>& n1)NE{n0.swap(n1);}TE <uint M> IN string to_string(CO Mod<M>& n)NE{RE to_string(n.RP())+ " + MZ";}TE<uint M,CL Traits> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO Mod<M>& n){RE os << n.RP();}

int main()
{
  UNTIE;
  LIBRARY_SEARCH;
  START_MAIN;

  // DEXPR( int , bound_T , 100000 , 100 );
  // CIN_ASSERT( T , 1 , bound_T );
  // REPEAT( T ){

  // }

  CIN_ASSERT( N , 1 , bound_N );
  CIN_ASSERT( M , 1 , bound_M );
  CIN_ASSERT( K , 1 , bound_K );
  int S[bound_N + 1];
  FOREQ( i , 1 , N ){
    CIN_ASSERT( Si, 1 , M );
    S[i] = Si;
  }
  // CIN_ASSERT( H , 1 , bound_H );
  // CIN_ASSERT( W , 1 , bound_W );
  // TYPE_OF( bound_HW ) HW = TYPE_OF( bound_HW )( H ) * W;
  // assert( HW <= bound_HW );
  
  int A[bound_M + 1];
  FOREQ( i , 1 , M ){
    CIN_ASSERT( Ai , 1 , K );
    A[i] = Ai;
  }
  int B[bound_M + 1];
  FOREQ( i , 1 , M ){
    CIN_ASSERT( Bi , 1 , K );
    B[i] = Bi;
  }
  BIT<MP,bound_N+1> bit[bound_K+1] = {};
  bit[0].Set( 0 , 1 );
  int last[bound_K + 1] = {};
  FOREQ( i , 1 , N ){
    int& Si = S[i];
    int& Ai = A[Si];
    int& Bi = B[Si];
    int& last_i = last[Si];
    FOREQ( k , 1 , K ){
      bit[k].Set( i , ( k >= Ai ? bit[k - Ai].IntervalSum( last_i , i - 1 ) : 0 ) + ( k >= Bi ? bit[k - Bi].IntervalSum( last_i , i - 1 ) : 0 ) );
    }
    last_i = i;
  }
  // FOR( i , 0 , M ){
  //   CIN( ui , 1 , N );
  //   CIN( vi , 1 , N );
  //   ui--;
  //   vi--;
  //   e[ui].push_back( vi );
  //   e[vi].push_back( ui );
  // }
  
  // DEXPR( int , bound_Q , 100000 , 100 );
  // CIN_ASSERT( Q , 1 , bound_Q );
  // REPEAT( Q ){
  //   COUT( N );
  // }

  // ll guchoku = Guchoku();
  ll answer = bit[K].IntervalSum( 1 , N ).RP();

  // if( answer == guchoku ){
  //   CERR( answer << " == " << guchoku );
  // } else {
  //   CERR( answer << " != " << guchoku );
  //   QUIT;
  // }

  COUT( ( answer ) );

  FINISH_MAIN;
  QUIT;
}