#include #include #include #include #include #include #include using namespace std; using ll = long long; using INT_TYPE_FOR_MOD = ll; using INT_TYPE_FOR_ADIC_INT = ll; template using VLArray = list; #define PRIME_NUMBER 1000000007 #define MOD Mod // 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 class AdicInt { private: VLArray m_expansion; INT_TYPE_FOR_ADIC_INT m_n; public: inline AdicInt( const INT_TYPE_FOR_ADIC_INT& n ) noexcept; inline const VLArray& GetExpansion() const noexcept; inline const INT_TYPE_FOR_ADIC_INT& GetValue() const noexcept; static const VLArray& Expand( const INT_TYPE_FOR_ADIC_INT& n ) noexcept; }; template inline AdicInt::AdicInt( const INT_TYPE_FOR_ADIC_INT& n ) noexcept : m_expansion( Expand( n ) ) , m_n( n ) {} template inline const VLArray& AdicInt::GetExpansion() const noexcept { return m_expansion; } template inline const INT_TYPE_FOR_ADIC_INT& AdicInt::GetValue() const noexcept { return m_n; } template const VLArray& AdicInt::Expand( const INT_TYPE_FOR_ADIC_INT& n ) noexcept { static VLArray memory_n{}; static VLArray > memory_answer{}; if( P == 0 ){ // ダミー return memory_n; } auto itr_n = memory_n.begin() , end_n = memory_n.end(); auto itr_answer = memory_answer.begin(); while( itr_n != end_n ){ if( *itr_n == n ){ return *itr_answer; } itr_n++; itr_answer++; } INT_TYPE_FOR_ADIC_INT n_copy = n; VLArray answer{}; if( LENGTH == 0 ){ for( INT_TYPE_FOR_ADIC_INT i = 0 ; n_copy != 0 ; i++ ){ const INT_TYPE_FOR_ADIC_INT d = Residue( P , n_copy ); answer.push_back( d ); n_copy -= d; n_copy /= P; } } else { for( INT_TYPE_FOR_ADIC_INT i = 0 ; i < LENGTH && n_copy != 0 ; i++ ){ const INT_TYPE_FOR_ADIC_INT d = Residue( P , n_copy ); answer.push_back( d ); n_copy -= d; n_copy /= P; } } memory_n.push_back( n ); memory_answer.push_back( answer ); return memory_answer.back(); } // init * ( t ^ num ) template T Power( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true , const string& method = "normal" ); template inline T PowerNormalMethod( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true ); template T PowerBinaryMethod( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true ); // 単なる２乗だが、T次第ではオーバーロードしてより高速なものに置き換える template inline T Square( const T& t ); // PowerBinaryMetodの呼び出しは部分特殊化ではなくオーバーロードで処理できるようにするためにPowerBinaryMethodとはしない。 template inline T Power( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication , const string& method ) { return method == "binary" ? PowerBinaryMethod( t , num , init , for_right_multiplication ) : PowerNormalMethod( t , num , init , for_right_multiplication ); } template inline T PowerNormalMethod( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication ) { return num == 0 ? init : ( for_right_multiplication ? PowerNormalMethod( t , num - 1 , init ) * t : t * PowerNormalMethod( t , num - 1 , init ) ); } template T PowerBinaryMethod( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication ) { const VLArray& num_binary = AdicInt<2>::Expand( num ); T answer = init; T power = t; for( auto itr = num_binary.begin() , end = num_binary.end() ; itr != end ; itr++ ){ if( *itr == 1 ){ answer = for_right_multiplication ? answer * power : power * answer; } // 部分特殊化ではなくオーバーロードで処理できるようにするためにSquareとしない。 power = Square( power ); } return answer; } template inline T Square( const T& t ) { return t * t; } // 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 ); // 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" ); // 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" ); 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++() 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 Power( const Mod& n , const Mod& p , const string& method ) { return Power,INT_TYPE_FOR_MOD>( n , p.Represent() , method ); } 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; } // 階乗（INT = Modの時にMでの値が1であることに注意） template inline INT Factorial( const INT& n , const INT& n_min = 1 , const string& mode = "normal" ); // modular階乗（INT1 = Modの時にMでの値が0であることに注意） template inline INT1 ModularFactorial( const INT2& n , const INT2& n_min = 1 , const string& mode = "normal" ); // 再帰式（呼び出し順によっては再帰深度が大きい） template const INT1& ModularFactorialNormalMethod( const INT2& n ); template const INT1& ModularFactorialNormalMethod( const INT2& n , const INT2& n_min ); template const INT1& ModularFactorialNormalMethod_Body( const INT2& n , const INT2& n_min ); // ループ template INT1 ModularFactorialLoopMethod( const INT2& n , const INT2& n_min = 1 ); // modular階乗の逆数（INT1 = Modの時にMでの値がサポート外であることに注意） template inline INT1 ModularFactorialInverse( const INT2& n , const INT2& n_min = 1 , const string& mode = "normal" ); // 再帰式（呼び出し順によっては再帰深度が大きい） template const INT1& ModularFactorialInverseNormalMethod( const INT2& n ); template const INT1& ModularFactorialInverseNormalMethod( const INT2& n , const INT2& n_min ); template const INT1& ModularFactorialInverseNormalMethod_Body( const INT2& n , const INT2& n_min ); // ループ template INT1 ModularFactorialInverseLoopMethod( const INT2& n , const INT2& n_min = 1 ); // 場合の数 template INT Combination( const INT& n , const INT& m , const string& mode = "normal" ); // 再帰式（呼び出し順によっては再帰深度が大きい） template const INT& CombinationNormalMethod( const INT& n , const INT& m ); // ループ（割り算回数が大きい） template INT CombinationLoopMethod( const INT& n , const INT& m ); // 階乗の比（modular演算でない時はオーバーフローしやすい） template inline INT CombinationFactorialNormalMethod( const INT& n , const INT& m ); template inline INT CombinationFactorialLoopMethod( const INT& n , const INT& m ); template inline INT CombinationModularFactorialInverseNormalMethod( const INT& n , const INT& m ); template