#pragma GCC optimize ( "O3" ) #pragma GCC target ( "avx" ) #include using namespace std; using uint = unsigned int; using ll = long long; #define TYPE_OF( VAR ) remove_const::type >::type #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr ) #define CEXPR( LL , BOUND , VALUE ) constexpr const LL BOUND = VALUE #define CIN( LL , A ) LL A; cin >> A #define ASSERT( A , MIN , MAX ) assert( MIN <= A && A <= MAX ) #define CIN_ASSERT( A , MIN , MAX ) CIN( TYPE_OF( MAX ) , A ); 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 FOR_ITR( ARRAY , ITR , END ) for( auto ITR = ARRAY .begin() , END = ARRAY .end() ; ITR != END ; ITR ++ ) #define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES ) #define QUIT return 0 #define COUT( ANSWER ) cout << ( ANSWER ) << "\n"; #define RETURN( ANSWER ) COUT( ANSWER ); QUIT #define DOUBLE( PRECISION , ANSWER ) cout << fixed << setprecision( PRECISION ) << ( ANSWER ) << "\n"; QUIT #define POWER( ANSWER , ARGUMENT , EXPONENT ) \ 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 ) \ TYPE_OF( ARGUMENT ) ANSWER{ 1 }; \ { \ TYPE_OF( ARGUMENT ) 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 , MAX_I , LENGTH , MODULO ) \ ll ANSWER[LENGTH]; \ ll ANSWER_INV[LENGTH]; \ { \ ll VARIABLE_FOR_PRODUCT_FOR_FACTORIAL = 1; \ ANSWER[0] = VARIABLE_FOR_PRODUCT_FOR_FACTORIAL; \ FOREQ( i , 1 , MAX_I ){ \ ANSWER[i] = ( VARIABLE_FOR_PRODUCT_FOR_FACTORIAL *= i ) %= MODULO; \ } \ POWER_MOD( FACTORIAL_MAX_INV , ANSWER[MAX_I] , MODULO - 2 , MODULO ); \ ANSWER_INV[MAX_I] = FACTORIAL_MAX_INV; \ FOREQINV( i , MAX_I - 1 , 0 ){ \ ANSWER_INV[i] = ( FACTORIAL_MAX_INV *= i + 1 ) %= MODULO; \ } \ } \ \ // 通常の二分探索 #define BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET ) \ ll ANSWER = MAXIMUM; \ { \ ll VARIABLE_FOR_BINARY_SEARCH_L = MINIMUM; \ ll VARIABLE_FOR_BINARY_SEARCH_U = ANSWER; \ ll VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH = ( TARGET ) - ( EXPRESSION ); \ if( VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH == 0 ){ \ VARIABLE_FOR_BINARY_SEARCH_L = ANSWER; \ } else { \ ANSWER = ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2; \ } \ while( VARIABLE_FOR_BINARY_SEARCH_L != ANSWER ){ \ VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH = ( TARGET ) - ( EXPRESSION ); \ if( VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH == 0 ){ \ VARIABLE_FOR_BINARY_SEARCH_L = ANSWER; \ break; \ } else { \ if( VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH > 0 ){ \ VARIABLE_FOR_BINARY_SEARCH_L = ANSWER; \ } else { \ VARIABLE_FOR_BINARY_SEARCH_U = ANSWER; \ } \ ANSWER = ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2; \ } \ } \ } \ \ // 二進法の二分探索 #define BS2( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET ) \ ll ANSWER = MINIMUM; \ { \ ll VARIABLE_FOR_POWER_FOR_BINARY_SEARCH_2 = 1; \ ll VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH = ( MAXIMUM ) - ANSWER; \ while( VARIABLE_FOR_POWER_FOR_BINARY_SEARCH_2 <= VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH ){ \ VARIABLE_FOR_POWER_FOR_BINARY_SEARCH_2 *= 2; \ } \ VARIABLE_FOR_POWER_FOR_BINARY_SEARCH_2 /= 2; \ ll VARIABLE_FOR_ANSWER_FOR_BINARY_SEARCH_2 = ANSWER; \ while( VARIABLE_FOR_POWER_FOR_BINARY_SEARCH_2 != 0 ){ \ ANSWER = VARIABLE_FOR_ANSWER_FOR_BINARY_SEARCH_2 + VARIABLE_FOR_POWER_FOR_BINARY_SEARCH_2; \ VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH = ( TARGET ) - ( EXPRESSION ); \ if( VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH == 0 ){ \ VARIABLE_FOR_ANSWER_FOR_BINARY_SEARCH_2 = ANSWER; \ break; \ } else if( VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH > 0 ){ \ VARIABLE_FOR_ANSWER_FOR_BINARY_SEARCH_2 = ANSWER; \ } \ VARIABLE_FOR_POWER_FOR_BINARY_SEARCH_2 /= 2; \ } \ ANSWER = VARIABLE_FOR_ANSWER_FOR_BINARY_SEARCH_2; \ } \ \ template inline T Absolute( const T& a ){ return a > 0 ? a : - a; } // template inline T Residue( const T& a , const T& p ){ return a >= 0 ? a % p : p - ( - a - 1 ) % p - 1; } template using VLArray = list; // template // using LineTypeForMatrix = VLArray; // vectorの方が乗法の計算が高速 template using LineTypeForMatrix = vector; template using TableTypeForMatrix = LineTypeForMatrix >; // using SizeTypeForMatrix = uint; using SizeTypeForMatrix = ll; template class Matrix { private: TableTypeForMatrix m_M; public: // argsの長さがXYでなくてもコンパイルエラーとならないがサポート外である。 template Matrix( const Args&... args ) noexcept; inline Matrix( const Matrix& mat ) noexcept; // ( X , Y )行列でないものも引数に取れるがサポート外である。 template inline Matrix( const TableTypeForMatrix& M ) noexcept; Matrix& operator=( const Matrix& mat ) noexcept; Matrix& operator+=( const Matrix& mat ); Matrix& operator-=( const Matrix& mat ); Matrix& operator*=( const T& scalar ) noexcept; // 行や列の長さを変更可能だがサポート外である。 inline TableTypeForMatrix& RefTable() noexcept; inline const TableTypeForMatrix& GetTable() const noexcept; static inline const Matrix& Unit() noexcept; private: static inline void ConstructTable( TableTypeForMatrix& M , LineTypeForMatrix& vec ) noexcept; template static void ConstructTable( TableTypeForMatrix& M , LineTypeForMatrix& vec , const Arg& arg , const Args&... args ) noexcept; static Matrix Unit_Body() noexcept; }; template inline Matrix operator==( const Matrix& mat1 , const Matrix& mat2 ) noexcept; template inline Matrix operator!=( const Matrix& mat1 , const Matrix& mat2 ) noexcept; template Matrix operator+( const Matrix& mat1 , const Matrix& mat2 ); template Matrix operator-( const Matrix& mat1 , const Matrix& mat2 ); template Matrix operator*( const Matrix& mat1 , const Matrix& mat2 ); template Matrix operator*( const T& scalar , const Matrix& mat ); template Matrix Transpose( const Matrix& mat ); template T Trace( const Matrix& mat ); // ../Arithmetic/Power/a_Body.hppにて定義 // template // T PowerBinaryMethod( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication ); // template // Matrix<2,2,T> PowerBinaryMethod( const Matrix<2,2,T>& mat , const UINT& num , const Matrix<2,2,T>& init_dummy , const bool& for_right_multiplication_dummy ); template template Matrix::Matrix( const Args&... args ) noexcept : m_M() { TableTypeForMatrix M{}; LineTypeForMatrix vec{}; ConstructTable( M , vec , args... ); m_M = M; } template inline Matrix::Matrix( const Matrix& mat ) noexcept : m_M( mat.m_M ) {} template template inline Matrix::Matrix( const TableTypeForMatrix& M ) noexcept : m_M( M ) {} template Matrix& Matrix::operator=( const Matrix& mat ) noexcept { m_M = mat.m_M; return *this; } template Matrix& Matrix::operator+=( const Matrix& mat ) { auto itr1y = m_M.begin() , end1y = m_M.end(); auto itr2y = mat.m_M.begin(); while( itr1y != end1y ){ auto itr1xy = itr1y->begin() , end1xy = itr1y->end(); auto itr2xy = itr2y->begin(); while( itr1xy != end1xy ){ *itr1xy += *itr2xy; itr1xy++; itr2xy++; } itr1y++; itr2y++; } return *this; } template Matrix& Matrix::operator-=( const Matrix& mat ) { auto itr1y = m_M.begin() , end1y = m_M.end(); auto itr2y = mat.m_M.begin(); while( itr1y != end1y ){ auto itr1xy = itr1y->begin() , end1xy = itr1y->end(); auto itr2xy = itr2y->begin(); while( itr1xy != end1xy ){ *itr1xy -= *itr2xy; itr1xy++; itr2xy++; } itr1y++; itr2y++; } return *this; } template Matrix& Matrix::operator*=( const T& scalar ) noexcept { for( auto itry = m_M.begin() , endy = m_M.end() ; itry != endy ; itry++ ){ for( auto itrxy = itry->begin() , endxy = itry->end() ; itrxy != endxy ; itrxy++ ){ *itrxy *= scalar; } } return *this; } template inline TableTypeForMatrix& Matrix::RefTable() noexcept { return m_M; } template inline const TableTypeForMatrix& Matrix::GetTable() const noexcept { return m_M; } template inline const Matrix& Matrix::Unit() noexcept { static const Matrix unit = Unit_Body(); return unit; } template Matrix Matrix::Unit_Body() noexcept { TableTypeForMatrix M{}; for( SizeTypeForMatrix y = 0 ; y < Y ; y++ ){ LineTypeForMatrix vec{}; for( SizeTypeForMatrix x = 0 ; x < X ; x++ ){ vec.push_back( x == y ? 1 : 0 ); } M.push_back( vec ); } return Matrix( M ); } template inline void Matrix::ConstructTable( TableTypeForMatrix& M , LineTypeForMatrix& vec ) noexcept { M.push_back( vec ); vec.clear(); } template template void Matrix::ConstructTable( TableTypeForMatrix& M , LineTypeForMatrix& vec , const Arg& arg , const Args&... args ) noexcept { vec.push_back( arg ); if( vec.size() == X ){ ConstructTable( M , vec ); } if( M.size() < Y ){ ConstructTable( M , vec , args... ); } return; } template inline Matrix operator==( const Matrix& mat1 , const Matrix& mat2 ) noexcept { return mat1.GetTable() == mat2.GetTable(); } template inline Matrix operator!=( const Matrix& mat1 , const Matrix& mat2 ) noexcept { return !( mat1 == mat2 ); } template Matrix operator+( const Matrix& mat1 , const Matrix& mat2 ) { Matrix mat1_copy = mat1; mat1_copy += mat2; return mat1_copy; } template Matrix operator-( const Matrix& mat1 , const Matrix& mat2 ) { Matrix mat1_copy = mat1; mat1_copy -= mat2; return mat1_copy; } template inline Matrix operator*( const Matrix& mat1 , const Matrix& mat2 ) { const TableTypeForMatrix& M1 = mat1.GetTable(); const TableTypeForMatrix& M2 = mat2.GetTable(); TableTypeForMatrix M_prod{}; auto begin2x = M2.begin(); for( auto itr1y = M1.begin() , end1y = M1.end() ; itr1y != end1y ; itr1y++ ){ LineTypeForMatrix vec{}; auto begin1yx = itr1y->begin() , end1yx = itr1y->end(); for( SizeTypeForMatrix z = 0 ; z < Z ; z++ ){ auto itr1yx = begin1yx; auto itr2x = begin2x; T inner_product = 0; while( itr1yx != end1yx ){ inner_product += ( *itr1yx ) * ( *itr2x )[z]; itr1yx++; itr2x++; } vec.push_back( inner_product ); } M_prod.push_back( vec ); } return Matrix( M_prod ); } template Matrix operator*( const T& scalar , const Matrix& mat ) { Matrix mat_copy = mat; mat_copy *= scalar; return mat_copy; } template Matrix Transpose( const Matrix& mat ) { const TableTypeForMatrix& M = mat.GetTable(); TableTypeForMatrix M_t{}; auto beginy = M.begin(); for( auto itr1x = beginy->begin() , end1x = beginy->end() ; itr1x != end1x ; itr1x++ ){ M_t.push_back( LineTypeForMatrix() ); } for( auto itry = beginy , endy = M.end() ; itry != endy ; itry++ ){ auto itryx = itry->begin() , endyx = itry->end(); auto itr_ty = M_t.begin(); while( itryx != endyx ){ itr_ty->push_back( *itryx ); itryx++; itr_ty++; } } return Matrix( M_t ); } template T Trace( const Matrix& mat ) { int i = 0; T answer =0; const TableTypeForMatrix& M = mat.GetTable(); for( auto itry = M.begin() , endy = M.end() ; itry != endy ; itry++ ){ answer += ( *itry )[i]; i++; } return answer; } // template inline Matrix<2,2,T> PowerBinaryMethod( const Matrix<2,2,T>& mat , const UINT& num , const Matrix<2,2,T>& init_dummy , const bool& for_right_multiplication_dummy ) { return PowerBinaryMethod( TwoByTwoMatrix( mat ) , num , TwoByTwoMatrix( init_dummy ) , for_right_multiplication_dummy ).GetMatrix22(); } // using INT_TYPE_FOR_MOD = int; using INT_TYPE_FOR_MOD = long long int; void LazyEvaluationOfModularInverse( const INT_TYPE_FOR_MOD& M , const INT_TYPE_FOR_MOD& n , INT_TYPE_FOR_MOD& m ); 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; } // 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; template inline basic_ostream& operator<<( basic_ostream& os , const Mod& n ); 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 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"; } template inline basic_ostream& operator<<( basic_ostream& os , const Mod& n ) { return os << n.Represent(); } int main() { UNTIE; CEXPR( ll , bound_N , 1000000000000000000 ); CIN_ASSERT( N , 2 , bound_N ); // A[2] = 1; // A[n] = 1 + sum( int i = 2 ; i <= n - 2 ; i++ ) A[i]; // A[3] = 1; // A[4] = 2; // A[5] = 3; // A[6] = 5; using MOD = Mod<998244353>; Matrix<2,1,MOD> v { 1 , 1 }; Matrix<2,2,MOD> T { 1 , 1 , 1 , 0 }; Matrix<2,2,MOD> power_T = Matrix<2,2,MOD>::Unit(); Matrix<2,2,MOD> power{ T }; N -= 2; while( N != 0 ){ if( N % 2 == 1 ){ power_T = power_T * power; } power = power * power; N /= 2; } v = power_T * v; const TableTypeForMatrix& v_vec = v.GetTable(); RETURN( ( v_vec[0][0] + v_vec[1][0] - 1 ).Represent() ); }