#include using namespace std; using uint = unsigned int; using ll = long long; #define CIN( LL , A ) LL A; cin >> A #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr ) #define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( ll VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ ) #define RETURN( ANSWER ) cout << ( ANSWER ) << "\n"; return 0 // 以下、自分のライブラリ（https://github.com/p-adic/cpp）よりソースコードをコピーして編集している。 template INT Residue( const INT& M , const INT& n ) noexcept; template INT Residue( const INT& M , const INT& n ) noexcept { if( M == 0 ){ return 0; } const INT M_abs = ( M > 0 ? M : -M ); if( n < 0 ){ const INT n_abs = -n; const INT res = n_abs % M_abs; return res == 0 ? res : M_abs - res; } return n % M_abs; } template using VLArray = list; using INT_TYPE_FOR_MOD = long long int; // ここをtempate などにしてしまうとoperator+などを呼び出す際に型推論に失敗する。整数型を変えたい時はINT_TYPE_FOR_MODの型エイリアスを変更する。 template class Mod { protected: INT_TYPE_FOR_MOD m_n; INT_TYPE_FOR_MOD m_inv; public: inline Mod() noexcept; inline Mod( const INT_TYPE_FOR_MOD& n ) noexcept; inline Mod( const Mod& n ) noexcept; inline Mod& operator=( const INT_TYPE_FOR_MOD& n ) noexcept; Mod& operator=( const Mod& n ) noexcept; Mod& operator+=( const INT_TYPE_FOR_MOD& n ) noexcept; inline Mod& operator+=( const Mod& n ) noexcept; inline Mod& operator-=( const INT_TYPE_FOR_MOD& n ) noexcept; inline Mod& operator-=( const Mod& n ) noexcept; Mod& operator*=( const INT_TYPE_FOR_MOD& n ) noexcept; Mod& operator*=( const Mod& n ) noexcept; // INT_TYPE_FOR_MODでの割り算ではないことに注意 virtual Mod& operator/=( const INT_TYPE_FOR_MOD& n ); virtual Mod& operator/=( const Mod& n ); Mod& operator%=( const INT_TYPE_FOR_MOD& n ); inline Mod& operator%=( const Mod& n ); inline Mod operator-() const noexcept; // 前置++/--を使うつもりがないので後置++/--と同じものとして定義する inline Mod& operator++() noexcept; inline Mod& operator++( int ) noexcept; inline Mod& operator--() noexcept; inline Mod& operator--( int ) noexcept; inline const INT_TYPE_FOR_MOD& Represent() const noexcept; void Invert() noexcept; bool CheckInvertible() noexcept; bool IsSmallerThan( const INT_TYPE_FOR_MOD& n ) const noexcept; bool IsBiggerThan( const INT_TYPE_FOR_MOD& n ) const noexcept; }; template inline bool operator==( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator==( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator==( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator==( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator!=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator!=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator!=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator!=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator<( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator<( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator<( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator<=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator<=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator<=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator<=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator>( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator>( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator>=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator>=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>=( const Mod& n0 , const Mod& n1 ) noexcept; template Mod operator+( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template Mod operator+( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template Mod operator+( const Mod& n0 , const Mod& n1 ) noexcept; template inline Mod operator-( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template Mod operator-( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template Mod operator-( const Mod& n0 , const Mod& n1 ) noexcept; template Mod operator*( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template Mod operator*( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template Mod operator*( const Mod& n0 , const Mod& n1 ) noexcept; template Mod operator/( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ); template Mod operator/( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ); template Mod operator/( const Mod& n0 , const Mod& n1 ); template Mod operator%( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ); template inline Mod operator%( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ); template inline Mod operator%( const Mod& n0 , const Mod& n1 ); template Mod Inverse( const Mod& n ); template Mod Power( const Mod& n , const INT_TYPE_FOR_MOD& p , const string& method = "normal" ); template <> inline Mod<2> Power( const Mod<2>& n , const INT_TYPE_FOR_MOD& p , const string& method ); // M乗が1になるよう定義されていることに注意 template inline Mod Power( const Mod& n , const Mod& p , const string& method = "normal" ); template <> inline Mod<2> Power( const Mod<2>& n , const Mod<2>& p , const string& method ); // ../Power/a_Body.hppにて定義 template inline T Square( const T& t ); template <> inline Mod<2> Square >( const Mod<2>& t ); template inline string to_string( const Mod& n ) noexcept; void LazyEvaluationOfModularInverse( const INT_TYPE_FOR_MOD& M , const INT_TYPE_FOR_MOD& n , INT_TYPE_FOR_MOD& m ); template inline Mod::Mod() noexcept : m_n( 0 ) , m_inv( M ){} template inline Mod::Mod( const INT_TYPE_FOR_MOD& n ) noexcept : m_n( Residue( M , n ) ) , m_inv( 0 ){} template inline Mod::Mod( const Mod& n ) noexcept : m_n( n.m_n ) , m_inv( 0 ){} template inline Mod& Mod::operator=( const INT_TYPE_FOR_MOD& n ) noexcept { return operator=( Mod( n ) ); } template Mod& Mod::operator=( const Mod& n ) noexcept { m_n = n.m_n; m_inv = n.m_inv; return *this; } template Mod& Mod::operator+=( const INT_TYPE_FOR_MOD& n ) noexcept { m_n = Residue( M , m_n + n ); m_inv = 0; return *this; } template inline Mod& Mod::operator+=( const Mod& n ) noexcept { return operator+=( n.m_n ); }; template inline Mod& Mod::operator-=( const INT_TYPE_FOR_MOD& n ) noexcept { return operator+=( -n ); } template inline Mod& Mod::operator-=( const Mod& n ) noexcept { return operator-=( n.m_n ); } template Mod& Mod::operator*=( const INT_TYPE_FOR_MOD& n ) noexcept { m_n = Residue( M , m_n * n ); m_inv = 0; return *this; } template Mod& Mod::operator*=( const Mod& n ) noexcept { m_n = Residue( M , m_n * n.m_n ); if( m_inv == 0 || n.m_inv == 0 ){ m_inv = 0; } else if( m_inv == M || n.m_inv == M ){ m_inv = M; } else { Residue( M , m_inv * n.m_inv ); } return *this; } // 仮想関数なのでinline指定しない。 template Mod& Mod::operator/=( const INT_TYPE_FOR_MOD& n ) { return operator/=( Mod( n ) ); } template Mod& Mod::operator/=( const Mod& n ) { return operator*=( Inverse( n ) ); } template Mod& Mod::operator%=( const INT_TYPE_FOR_MOD& n ) { m_n %= Residue( M , n ); m_inv = 0; return *this; } template inline Mod& Mod::operator%=( const Mod& n ) { return operator%=( n.m_n ); } template inline Mod Mod::operator-() const noexcept { return Mod( 0 ).operator-=( *this ); } template inline Mod& Mod::operator++() noexcept { return operator+=( 1 ); } template inline Mod& Mod::operator++( int ) noexcept { return operator++(); } template inline Mod& Mod::operator--() noexcept { return operator-=( 1 ); } template inline Mod& Mod::operator--( int ) noexcept { return operator-=(); } template inline const INT_TYPE_FOR_MOD& Mod::Represent() const noexcept { return m_n; } template void Mod::Invert() noexcept { if( CheckInvertible() ){ INT_TYPE_FOR_MOD i = m_inv; m_inv = m_n; m_n = i; } else { // m_nがMになるのはここの処理に限るのでRepresent()の値を参照することで例外処理可能 m_n = M; m_inv = M; } return; } template bool Mod::CheckInvertible() noexcept { if( m_inv == 0 ){ LazyEvaluationOfModularInverse( M , m_n , m_inv ); } return m_inv != M; } template inline bool Mod::IsSmallerThan( const INT_TYPE_FOR_MOD& n ) const noexcept { return m_n < Residue( M , n ); } template inline bool Mod::IsBiggerThan( const INT_TYPE_FOR_MOD& n ) const noexcept { return m_n > Residue( M , n ); } template inline bool operator==( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0 == Mod( n1 ); } template inline bool operator==( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return Mod( n0 ) == n0; } template inline bool operator==( const Mod& n0 , const Mod& n1 ) noexcept { return n0.Represent() == n1.Represent(); } template inline bool operator!=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n0 == n1 ); } template inline bool operator!=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return !( n0 == n1 ); } template inline bool operator!=( const Mod& n0 , const Mod& n1 ) noexcept { return !( n0 == n1 ); } template inline bool operator<( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0.IsSmallerThan( n1 ); } template inline bool operator<( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1.IsBiggerThan( n0 ); } template inline bool operator<( const Mod& n0 , const Mod& n1 ) noexcept { return n0.Represent() < n1.Represent(); } template inline bool operator<=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n1 < n0 ); } template inline bool operator<=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return !( n1 < n0 ); } template inline bool operator<=( const Mod& n0 , const Mod& n1 ) noexcept { return !( n1 < n0 ); } template inline bool operator>( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n1 < n0; } template inline bool operator>( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1 < n0; } template inline bool operator>( const Mod& n0 , const Mod& n1 ) noexcept { return n1 < n0; } template inline bool operator>=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n0 < n1 ); } template inline bool operator>=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return !( n0 < n1 ); } template inline bool operator>=( const Mod& n0 , const Mod& n1 ) noexcept { return !( n0 < n1 ); } template Mod operator+( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { auto n = n0; n += n1; return n; } template inline Mod operator+( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1 + n0; } template inline Mod operator+( const Mod& n0 , const Mod& n1 ) noexcept { return n0 + n1.Represent(); } template inline Mod operator-( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0 + ( -n1 ); } template inline Mod operator-( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return Mod( n0 - n1.Represent() ); } template inline Mod operator-( const Mod& n0 , const Mod& n1 ) noexcept { return n0 - n1.Represent(); } template Mod operator*( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { auto n = n0; n *= n1; return n; } template inline Mod operator*( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1 * n0; } template Mod operator*( const Mod& n0 , const Mod& n1 ) noexcept { auto n = n0; n *= n1; return n; } template inline Mod operator/( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) { return n0 / Mod( n1 ); } template inline Mod operator/( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) { return Mod( n0 ) / n1; } template Mod operator/( const Mod& n0 , const Mod& n1 ) { auto n = n0; n /= n1; return n; } template Mod operator%( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) { auto n = n0; n %= n1; return n; } template inline Mod operator%( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) { return Mod( n0 ) % n1.Represent(); } template inline Mod operator%( const Mod& n0 , const Mod& n1 ) { return n0 % n1.Represent(); } template Mod Inverse( const Mod& n ) { auto n_copy = n; n_copy.Invert(); return n_copy; } template Mod Power( const Mod& n , const INT_TYPE_FOR_MOD& p , const string& method ) { if( p >= 0 ){ return Power,INT_TYPE_FOR_MOD>( n , p , 1 , true , true , method ); } return Inverse( Power( n , -p , method ) ); } template <> inline Mod<2> Power( const Mod<2>& n , const INT_TYPE_FOR_MOD& p , const string& method ) { return p == 0 ? 1 : n; } template inline Mod Power( const Mod& n , const Mod& p , const string& method ) { return Power,INT_TYPE_FOR_MOD>( n , p.Represent() , method ); } template <> inline Mod<2> Power( const Mod<2>& n , const Mod<2>& p , const string& method ) { return p == 0 ? 1 : n; } template <> inline Mod<2> Square >( const Mod<2>& t ) { return t; } template inline string to_string( const Mod& n ) noexcept { return to_string( n.Represent() ) + " + MZ"; } void LazyEvaluationOfModularInverse( const INT_TYPE_FOR_MOD& M , const INT_TYPE_FOR_MOD& n , INT_TYPE_FOR_MOD& m ) { static VLArray memory_M{}; // vectorでなくVLArrayだと引数が小さい順に呼び出した時の計算量がO(1)からO(n)に跳ね上がってしまう。 static VLArray > memory_inverse{}; auto itr_M = memory_M.begin() , end_M = memory_M.end(); auto itr_inverse = memory_inverse.begin(); vector* p_inverse = nullptr; while( itr_M != end_M && p_inverse == nullptr ){ if( *itr_M == M ){ p_inverse = &( *itr_inverse ); } itr_M++; itr_inverse++; } if( p_inverse == nullptr ){ memory_M.push_front( M ); memory_inverse.push_front( vector() ); p_inverse = &( memory_inverse.front() ); p_inverse->push_back( M ); } const INT_TYPE_FOR_MOD size = p_inverse->size(); for( INT_TYPE_FOR_MOD i = size ; i <= n ; i++ ){ p_inverse->push_back( 0 ); } INT_TYPE_FOR_MOD& n_inv = ( *p_inverse )[n]; if( n_inv != 0 ){ m = n_inv; return; } const INT_TYPE_FOR_MOD M_abs = M >= 0 ? M : -M; const INT_TYPE_FOR_MOD n_sub = M_abs % n; INT_TYPE_FOR_MOD n_sub_inv = ( *p_inverse )[n_sub]; if( n_sub_inv == 0 ){ LazyEvaluationOfModularInverse( M , n_sub , n_sub_inv ); } if( n_sub_inv != M ){ n_inv = M_abs - ( ( n_sub_inv * ( M_abs / n ) ) % M_abs ); m = n_inv; return; } for( INT_TYPE_FOR_MOD i = 1 ; i < M_abs ; i++ ){ if( ( n * i ) % M_abs == 1 ){ n_inv = i; m = n_inv; return; } } n_inv = M; m = n_inv; return; } template inline constexpr const uint LimitOfPowerForFFT; template inline const T ( &PrimitiveRootOfTwoForFFT() noexcept )[LimitOfPowerForFFT]; template inline const T ( &InversePrimitiveRootOfTwoForFFT() noexcept )[LimitOfPowerForFFT]; template vector FFT( const vector& f , const uint& N ); template vector FFT( const vector& f , const uint& two_power , const uint& exponent ); template vector IFFT( const vector& f , const uint& N ); template vector IFFT( const vector& f , const uint& two_power , const T& two_power_inv , const uint& exponent ); template <> inline constexpr const uint LimitOfPowerForFFT > = 24; template <> inline const Mod<998244353> ( &PrimitiveRootOfTwoForFFT() noexcept )[LimitOfPowerForFFT >] { static const Mod<998244353> PRT[ LimitOfPowerForFFT > ] = { Mod<998244353>( 1 ) , Mod<998244353>( 998244352 ) , Mod<998244353>( 911660635 ) , Mod<998244353>( 625715529 ) , Mod<998244353>( 373294451 ) , Mod<998244353>( 827987769 ) , Mod<998244353>( 280333251 ) , Mod<998244353>( 581015842 ) , Mod<998244353>( 628092333 ) , Mod<998244353>( 300892551 ) , Mod<998244353>( 586046298 ) , Mod<998244353>( 615001099 ) , Mod<998244353>( 318017948 ) , Mod<998244353>( 64341522 ) , Mod<998244353>( 106061068 ) , Mod<998244353>( 304605202 ) , Mod<998244353>( 631920086 ) , Mod<998244353>( 857779016 ) , Mod<998244353>( 841431251 ) , Mod<998244353>( 805775211 ) , Mod<998244353>( 390359979 ) , Mod<998244353>( 923521 ) , Mod<998244353>( 961 ) , Mod<998244353>( 31 ) }; return PRT; } template <> inline const Mod<998244353> ( &InversePrimitiveRootOfTwoForFFT() noexcept )[LimitOfPowerForFFT >] { static const Mod<998244353> PRT[ LimitOfPowerForFFT > ] = { Mod<998244353>( 1 ) , Mod<998244353>( 998244352 ) , Mod<998244353>( 86583718 ) , Mod<998244353>( 488723995 ) , Mod<998244353>( 369330050 ) , Mod<998244353>( 543653592 ) , Mod<998244353>( 382946991 ) , Mod<998244353>( 844956623 ) , Mod<998244353>( 91420391 ) , Mod<998244353>( 433414612 ) , Mod<998244353>( 288894979 ) , Mod<998244353>( 260490556 ) , Mod<998244353>( 857007890 ) , Mod<998244353>( 736054570 ) , Mod<998244353>( 474649464 ) , Mod<998244353>( 948509906 ) , Mod<998244353>( 114942468 ) , Mod<998244353>( 962405921 ) , Mod<998244353>( 667573957 ) , Mod<998244353>( 46809892 ) , Mod<998244353>( 304321983 ) , Mod<998244353>( 30429817 ) , Mod<998244353>( 293967900 ) , Mod<998244353>( 128805723 ) }; return PRT; } template static vector FFT_Body( const vector& f , const uint& two_power , const uint& exponent , const uint& start , const uint& depth , const T ( &PRT )[LimitOfPowerForFFT] ); template vector FFT( const vector& f , const uint& N ) { uint two_power = 1; uint exponent = 0; while( N > two_power ){ two_power *= 2; exponent++; } return FFT( f , two_power , exponent ); } template vector FFT( const vector& f , const uint& two_power , const uint& exponent ) { const uint size = f.size(); if( size == two_power ){ return FFT_Body( f , two_power , exponent , 0 , 1 , PrimitiveRootOfTwoForFFT() ); } vector f_copy = f; const T zero{}; for( uint i = size ; i < two_power ; i++ ){ f_copy.push_back( zero ); } return FFT_Body( f_copy , two_power , exponent , 0 , 1 , PrimitiveRootOfTwoForFFT() ); } template vector IFFT( const vector& f , const uint& N ) { uint two_power = 1; T two_power_inv{ 1 }; uint exponent = 0; while( N > two_power ){ two_power *= 2; two_power_inv /= 2; exponent++; } return IFFT( f , two_power , two_power_inv , exponent ); } template vector IFFT( const vector& f , const uint& two_power , const T& two_power_inv , const uint& exponent ) { vector