#783930 (C++14) No.2122 黄金比で擬似乱数生成

提出ソース

結果

問題	No.2122 黄金比で擬似乱数生成
ユーザー	👑 p-adic
提出日時	2022-08-03 02:15:59
言語	C++14 (gcc 12.3.0 + boost 1.83.0)
結果	AC
実行時間	23 ms / 2,000 ms
コード長	38,074 bytes
コンパイル時間	1,787 ms
コンパイル使用メモリ	103,932 KB
実行使用メモリ	4,384 KB
最終ジャッジ日時	2023-09-26 02:47:08
合計ジャッジ時間	3,047 ms
ジャッジサーバーID （参考情報）	judge12 / judge11

このコードへのチャレンジ
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テストケース

テストケース表示

入力	結果	実行時間実行使用メモリ
testcase_00	AC	2 ms 4,376 KB
testcase_01	AC	2 ms 4,380 KB
testcase_02	AC	2 ms 4,384 KB
testcase_03	AC	2 ms 4,380 KB
testcase_04	AC	2 ms 4,380 KB
testcase_05	AC	2 ms 4,380 KB
testcase_06	AC	2 ms 4,376 KB
testcase_07	AC	2 ms 4,380 KB
testcase_08	AC	2 ms 4,384 KB
testcase_09	AC	2 ms 4,376 KB
testcase_10	AC	2 ms 4,376 KB
testcase_11	AC	2 ms 4,380 KB
testcase_12	AC	13 ms 4,380 KB
testcase_13	AC	12 ms 4,380 KB
testcase_14	AC	2 ms 4,380 KB
testcase_15	AC	2 ms 4,376 KB
testcase_16	AC	2 ms 4,376 KB
testcase_17	AC	2 ms 4,376 KB
testcase_18	AC	1 ms 4,376 KB
testcase_19	AC	1 ms 4,376 KB
testcase_20	AC	2 ms 4,380 KB
testcase_21	AC	2 ms 4,376 KB
testcase_22	AC	23 ms 4,376 KB
testcase_23	AC	23 ms 4,376 KB
testcase_24	AC	20 ms 4,376 KB
testcase_25	AC	22 ms 4,380 KB

権限があれば一括ダウンロードができます

コンパイルメッセージ

main.cpp: 関数 ‘void ShuffleMemorise(MOD&, const ll&, const ll&)’ 内:
main.cpp:1211:20: 警告: ‘num’ may be used uninitialized [-Wmaybe-uninitialized]
 1211 |   const ll d = ( I - num ) % period;
      |                ~~~~^~~~~~~
main.cpp:1191:6: 備考: ‘num’ はここで定義されています
 1191 |   ll num;
      |      ^~~

ソースコード

raw source code

#include <iostream>
#include <list>
#include <vector>
#include <string>
#include <stdio.h>
#include <stdint.h>
using namespace std;

using ll = long long;

// 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 <typename T>
using VLArray = list<T>;

template <typename INT>
INT Residue( const INT& M , const INT& n ) noexcept;

template <typename INT>
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;

}


using INT_TYPE_FOR_ADIC_INT = long long int;

template <INT_TYPE_FOR_ADIC_INT P , INT_TYPE_FOR_ADIC_INT LENGTH = 0>
class AdicInt
{

private:
  VLArray<INT_TYPE_FOR_ADIC_INT> m_expansion;
  INT_TYPE_FOR_ADIC_INT m_n;

public:
  inline AdicInt( const INT_TYPE_FOR_ADIC_INT& n ) noexcept;

  inline const VLArray<INT_TYPE_FOR_ADIC_INT>& GetExpansion() const noexcept;
  inline const INT_TYPE_FOR_ADIC_INT& GetValue() const noexcept;

  static const VLArray<INT_TYPE_FOR_ADIC_INT>& Expand( const INT_TYPE_FOR_ADIC_INT& n ) noexcept;

};

template <INT_TYPE_FOR_ADIC_INT P , INT_TYPE_FOR_ADIC_INT LENGTH> inline AdicInt<P,LENGTH>::AdicInt( const INT_TYPE_FOR_ADIC_INT& n ) noexcept : m_expansion( Expand( n ) ) , m_n( n ) {}

template <INT_TYPE_FOR_ADIC_INT P , INT_TYPE_FOR_ADIC_INT LENGTH> inline const VLArray<INT_TYPE_FOR_ADIC_INT>& AdicInt<P,LENGTH>::GetExpansion() const noexcept { return m_expansion; }
template <INT_TYPE_FOR_ADIC_INT P , INT_TYPE_FOR_ADIC_INT LENGTH> inline const INT_TYPE_FOR_ADIC_INT& AdicInt<P,LENGTH>::GetValue() const noexcept { return m_n; }

template <INT_TYPE_FOR_ADIC_INT P , INT_TYPE_FOR_ADIC_INT LENGTH>
const VLArray<INT_TYPE_FOR_ADIC_INT>& AdicInt<P,LENGTH>::Expand( const INT_TYPE_FOR_ADIC_INT& n ) noexcept
{

  static VLArray<INT_TYPE_FOR_ADIC_INT> memory_n{};
  static VLArray<VLArray<INT_TYPE_FOR_ADIC_INT> > 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<INT_TYPE_FOR_ADIC_INT> answer{};

  if( LENGTH == 0 ){
  
    for( INT_TYPE_FOR_ADIC_INT i = 0 ; n_copy != 0 ; i++ ){

      const INT_TYPE_FOR_ADIC_INT d = Residue<INT_TYPE_FOR_ADIC_INT>( 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<INT_TYPE_FOR_ADIC_INT>( 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 <typename T , typename UINT>
T Power( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true , const string& method = "normal" );

template <typename T , typename UINT> inline T PowerNormalMethod( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true );
template <typename T , typename UINT>
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 <typename T> 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<T,UINT>とはしない。
template <typename T , typename UINT>
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 <typename T , typename UINT> 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 <typename T , typename UINT>
T PowerBinaryMethod( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication )
{

  const VLArray<UINT>& 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<T>としない。
    power = Square( power );

  }

  return answer;

}

template <typename T> inline T Square( const T& t ) { return t * t; }

using INT_TYPE_FOR_MOD = long long int;

// ここをtempate <typename INT , INT M>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 <INT_TYPE_FOR_MOD M>
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<M>& n ) noexcept;
  inline Mod<M>& operator=( const INT_TYPE_FOR_MOD& n ) noexcept;
  Mod<M>& operator=( const Mod<M>& n ) noexcept;
  Mod<M>& operator+=( const INT_TYPE_FOR_MOD& n ) noexcept;
  inline Mod<M>& operator+=( const Mod<M>& n ) noexcept;
  inline Mod<M>& operator-=( const INT_TYPE_FOR_MOD& n ) noexcept;
  inline Mod<M>& operator-=( const Mod<M>& n ) noexcept;
  Mod<M>& operator*=( const INT_TYPE_FOR_MOD& n ) noexcept;
  Mod<M>& operator*=( const Mod<M>& n ) noexcept;

  // INT_TYPE_FOR_MODでの割り算ではないことに注意
  virtual Mod<M>& operator/=( const INT_TYPE_FOR_MOD& n );
  virtual Mod<M>& operator/=( const Mod<M>& n );
  
  Mod<M>& operator%=( const INT_TYPE_FOR_MOD& n );
  inline Mod<M>& operator%=( const Mod<M>& 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<M>& operator++() noexcept;
  inline Mod<M>& operator++( int ) noexcept;
  inline Mod<M>& operator--() noexcept;
  inline Mod<M>& 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 <INT_TYPE_FOR_MOD M> inline bool operator==( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator==( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator==( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator==( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;

template <INT_TYPE_FOR_MOD M> inline bool operator!=( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator!=( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator!=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator!=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;

template <INT_TYPE_FOR_MOD M> inline bool operator<( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator<( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator<( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;

template <INT_TYPE_FOR_MOD M> inline bool operator<=( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator<=( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator<=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator<=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;

template <INT_TYPE_FOR_MOD M> inline bool operator>( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator>( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator>( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator>( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;

template <INT_TYPE_FOR_MOD M> inline bool operator>=( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator>=( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator>=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline bool operator>=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;

template <INT_TYPE_FOR_MOD M> Mod<M> operator+( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator+( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator+( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> inline Mod<M> operator-( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator-( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator-( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator*( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator*( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator*( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept;
template <INT_TYPE_FOR_MOD M> Mod<M> operator/( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 );
template <INT_TYPE_FOR_MOD M> Mod<M> operator/( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 );
template <INT_TYPE_FOR_MOD M> Mod<M> operator/( const Mod<M>& n0 , const Mod<M>& n1 );
template <INT_TYPE_FOR_MOD M> Mod<M> operator%( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 );
template <INT_TYPE_FOR_MOD M> inline Mod<M> operator%( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 );
template <INT_TYPE_FOR_MOD M> inline Mod<M> operator%( const Mod<M>& n0 , const Mod<M>& n1 );
template <INT_TYPE_FOR_MOD M> Mod<M> Inverse( const Mod<M>& n );

template <INT_TYPE_FOR_MOD M> Mod<M> Power( const Mod<M>& 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 <INT_TYPE_FOR_MOD M> inline Mod<M> Power( const Mod<M>& n , const Mod<M>& 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 <typename T> inline T Square( const T& t );
template <> inline Mod<2> Square<Mod<2> >( const Mod<2>& t );

void LazyEvaluationOfModularInverse( const INT_TYPE_FOR_MOD& M , const INT_TYPE_FOR_MOD& n , INT_TYPE_FOR_MOD& m );


template <INT_TYPE_FOR_MOD M> inline Mod<M>::Mod() noexcept : m_n( 0 ) , m_inv( M ){}

template <INT_TYPE_FOR_MOD M> inline Mod<M>::Mod( const INT_TYPE_FOR_MOD& n ) noexcept : m_n( Residue<INT_TYPE_FOR_MOD>( M , n ) ) , m_inv( 0 ){}

template <INT_TYPE_FOR_MOD M> inline Mod<M>::Mod( const Mod<M>& n ) noexcept : m_n( n.m_n ) , m_inv( 0 ){}

template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator=( const INT_TYPE_FOR_MOD& n ) noexcept { return operator=( Mod<M>( n ) ); }

template <INT_TYPE_FOR_MOD M>
Mod<M>& Mod<M>::operator=( const Mod<M>& n ) noexcept
{

  m_n = n.m_n;
  m_inv = n.m_inv;
  return *this;

}

template <INT_TYPE_FOR_MOD M>
Mod<M>& Mod<M>::operator+=( const INT_TYPE_FOR_MOD& n ) noexcept
{

  m_n = Residue<INT_TYPE_FOR_MOD>( M , m_n + n );
  m_inv = 0;
  return *this;

}

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

template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator-=( const INT_TYPE_FOR_MOD& n ) noexcept { return operator+=( -n ); }

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

template <INT_TYPE_FOR_MOD M>
Mod<M>& Mod<M>::operator*=( const INT_TYPE_FOR_MOD& n ) noexcept
{

  m_n = Residue<INT_TYPE_FOR_MOD>( M , m_n * n );
  m_inv = 0;
  return *this;

}

template <INT_TYPE_FOR_MOD M>
Mod<M>& Mod<M>::operator*=( const Mod<M>& n ) noexcept
{

  m_n = Residue<INT_TYPE_FOR_MOD>( 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<INT_TYPE_FOR_MOD>( M , m_inv * n.m_inv );

  }
  
  return *this;

}

// 仮想関数なのでinline指定しない。
template <INT_TYPE_FOR_MOD M>
Mod<M>& Mod<M>::operator/=( const INT_TYPE_FOR_MOD& n )
{

  return operator/=( Mod<M>( n ) );

}

template <INT_TYPE_FOR_MOD M>
Mod<M>& Mod<M>::operator/=( const Mod<M>& n )
{
  
  return operator*=( Inverse( n ) );
  
}

template <INT_TYPE_FOR_MOD M>
Mod<M>& Mod<M>::operator%=( const INT_TYPE_FOR_MOD& n )
{

  m_n %= Residue<INT_TYPE_FOR_MOD>( M , n );
  m_inv = 0;
  return *this;

}

template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator%=( const Mod<M>& n ) { return operator%=( n.m_n ); }

template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator++() noexcept { return operator+=( 1 ); }
template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator++( int ) noexcept { return operator++(); }
template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator--() noexcept { return operator-=( 1 ); }
template <INT_TYPE_FOR_MOD M> inline Mod<M>& Mod<M>::operator--( int ) noexcept { return operator-=(); }

template <INT_TYPE_FOR_MOD M> inline const INT_TYPE_FOR_MOD& Mod<M>::Represent() const noexcept { return m_n; }

template <INT_TYPE_FOR_MOD M>
void Mod<M>::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 <INT_TYPE_FOR_MOD M>
bool Mod<M>::CheckInvertible() noexcept
{

  if( m_inv == 0 ){

    LazyEvaluationOfModularInverse( M , m_n , m_inv );

  }

  return m_inv != M;
  
}

template <INT_TYPE_FOR_MOD M> inline bool Mod<M>::IsSmallerThan( const INT_TYPE_FOR_MOD& n ) const noexcept { return m_n < Residue<INT_TYPE_FOR_MOD>( M , n ); }
template <INT_TYPE_FOR_MOD M> inline bool Mod<M>::IsBiggerThan( const INT_TYPE_FOR_MOD& n ) const noexcept { return m_n > Residue<INT_TYPE_FOR_MOD>( M , n ); }

template <INT_TYPE_FOR_MOD M> inline bool operator==( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0 == Mod<M>( n1 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator==( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return Mod<M>( n0 ) == n0; }
template <INT_TYPE_FOR_MOD M> inline bool operator==( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return n0.Represent() == n1.Represent(); }

template <INT_TYPE_FOR_MOD M> inline bool operator!=( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n0 == n1 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator!=( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return !( n0 == n1 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator!=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return !( n0 == n1 ); }

template <INT_TYPE_FOR_MOD M> inline bool operator<( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0.IsSmallerThan( n1 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator<( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return n1.IsBiggerThan( n0 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator<( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return n0.Represent() < n1.Represent(); }

template <INT_TYPE_FOR_MOD M> inline bool operator<=( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n1 < n0 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator<=( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return !( n1 < n0 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator<=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return !( n1 < n0 ); }

template <INT_TYPE_FOR_MOD M> inline bool operator>( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n1 < n0; }
template <INT_TYPE_FOR_MOD M> inline bool operator>( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return n1 < n0; }
template <INT_TYPE_FOR_MOD M> inline bool operator>( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return n1 < n0; }

template <INT_TYPE_FOR_MOD M> inline bool operator>=( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n0 < n1 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator>=( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return !( n0 < n1 ); }
template <INT_TYPE_FOR_MOD M> inline bool operator>=( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return !( n0 < n1 ); }

template <INT_TYPE_FOR_MOD M>
Mod<M> operator+( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept
{

  auto n = n0;
  n += n1;
  return n;

}

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator+( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return n1 + n0; }

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator+( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return n0 + n1.Represent(); }

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator-( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0 + ( -n1 ); }

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator-( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return Mod<M>( n0 - n1.Represent() ); }

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator-( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept { return n0 - n1.Represent(); }

template <INT_TYPE_FOR_MOD M>
Mod<M> operator*( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept
{

  auto n = n0;
  n *= n1;
  return n;

}

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator*( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) noexcept { return n1 * n0; }

template <INT_TYPE_FOR_MOD M>
Mod<M> operator*( const Mod<M>& n0 , const Mod<M>& n1 ) noexcept
{

  auto n = n0;
  n *= n1;
  return n;

}

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator/( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 ) { return n0 / Mod<M>( n1 ); }

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator/( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) { return Mod<M>( n0 ) / n1; }

template <INT_TYPE_FOR_MOD M>
Mod<M> operator/( const Mod<M>& n0 , const Mod<M>& n1 )
{

  auto n = n0;
  n /= n1;
  return n;

}

template <INT_TYPE_FOR_MOD M>
Mod<M> operator%( const Mod<M>& n0 , const INT_TYPE_FOR_MOD& n1 )
{

  auto n = n0;
  n %= n1;
  return n;

}

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator%( const INT_TYPE_FOR_MOD& n0 , const Mod<M>& n1 ) { return Mod<M>( n0 ) % n1.Represent(); }

template <INT_TYPE_FOR_MOD M> inline Mod<M> operator%( const Mod<M>& n0 , const Mod<M>& n1 ) { return n0 % n1.Represent(); }

template <INT_TYPE_FOR_MOD M>
Mod<M> Inverse( const Mod<M>& n )
{

  auto n_copy = n;
  n_copy.Invert();
  return n_copy;

}

template <INT_TYPE_FOR_MOD M>
Mod<M> Power( const Mod<M>& n , const INT_TYPE_FOR_MOD& p , const string& method )
{

  if( p >= 0 ){

    return Power<Mod<M>,INT_TYPE_FOR_MOD>( n , p , 1 , true , true , method );

  }

  return Inverse( Power<M>( 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 <INT_TYPE_FOR_MOD M> inline Mod<M> Power( const Mod<M>& n , const Mod<M>& p , const string& method ) { return Power<Mod<M>,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<Mod<2> >( const Mod<2>& t ) { return t; }


void LazyEvaluationOfModularInverse( const INT_TYPE_FOR_MOD& M , const INT_TYPE_FOR_MOD& n , INT_TYPE_FOR_MOD& m )
{

  static VLArray<INT_TYPE_FOR_MOD> 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<vector<INT_TYPE_FOR_MOD> > memory_inverse{};

  auto itr_M = memory_M.begin() , end_M = memory_M.end();
  auto itr_inverse = memory_inverse.begin();

  vector<INT_TYPE_FOR_MOD>* 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<INT_TYPE_FOR_MOD>() );
    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 <typename T>
using LineTypeForMatrix = vector<T>;

template <typename T>
using TableTypeForMatrix = LineTypeForMatrix<LineTypeForMatrix<T> >;

using SizeTypeForMatrix = ll;

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
class Matrix
{

private:
  TableTypeForMatrix<T>  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 <typename... Args> Matrix( const Args&... args ) noexcept;

  inline Matrix( const Matrix<Y,X,T>& 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 <typename... Args> inline Matrix( const TableTypeForMatrix<T>& M ) noexcept;

  Matrix<Y,X,T>& operator=( const Matrix<Y,X,T>& mat ) noexcept;
  Matrix<Y,X,T>& operator+=( const Matrix<Y,X,T>& mat );
  Matrix<Y,X,T>& operator-=( const Matrix<Y,X,T>& mat );
  Matrix<Y,X,T>& 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<T>& RefTable() noexcept;
  inline const TableTypeForMatrix<T>& GetTable() const noexcept;

  static inline const Matrix<Y,X,T>& Unit() noexcept;

private:
  static inline void ConstructTable( TableTypeForMatrix<T>& M , LineTypeForMatrix<T>& vec ) noexcept;
  template <typename Arg , typename... Args> static void ConstructTable( TableTypeForMatrix<T>& M , LineTypeForMatrix<T>& vec , const Arg& arg , const Args&... args ) noexcept;
  
  static Matrix<Y,X,T> Unit_Body() noexcept;

};

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline Matrix<Y,X,T> operator==( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 ) noexcept;

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline Matrix<Y,X,T> operator!=( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 ) noexcept;

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T> operator+( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 );

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T> operator-( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 );

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , SizeTypeForMatrix Z , typename T>
Matrix<Y,Z,T> operator*( const Matrix<Y,X,T>& mat1 , const Matrix<X,Z,T>& mat2 );

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T> operator*( const T& scalar , const Matrix<Y,X,T>& mat );

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<X,Y,T> Transpose( const Matrix<Y,X,T>& mat );

template <SizeTypeForMatrix X , typename T>
T Trace( const Matrix<X,X,T>& mat );


// ../Arithmetic/Power/a_Body.hppにて定義
// template <typename T , typename UINT>
// T PowerBinaryMethod( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication );
template <typename T , typename UINT>
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 <typename T>
class TwoByTwoMatrix
{

private:
  T m_M00;
  T m_M01;
  T m_M10;
  T m_M11;

public:
  inline TwoByTwoMatrix( const T& M00 , const T& M01 , const T& M10 , const T& M11 ) noexcept;
  TwoByTwoMatrix( const Matrix<2,2,T>& mat );
  TwoByTwoMatrix<T>& operator=( const TwoByTwoMatrix<T>& mat ) noexcept;

  inline Matrix<2,2,T> GetMatrix22() const noexcept;

  static inline TwoByTwoMatrix<T> Multiply( const TwoByTwoMatrix<T>& mat1 , const TwoByTwoMatrix<T>& mat2 );
  static inline TwoByTwoMatrix<T> Square( const TwoByTwoMatrix<T>& mat );
  
};

template <typename T> inline TwoByTwoMatrix<T> operator*( const TwoByTwoMatrix<T>& mat1 , const TwoByTwoMatrix<T>& mat2 );

// ../../Arithmetic/Power/a_Body.hppにて定義
// template <typename T> inline T Square( const T& t );
template <typename T> inline TwoByTwoMatrix<T> Square( const TwoByTwoMatrix<T>& mat );


template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> template <typename... Args>
Matrix<Y,X,T>::Matrix( const Args&... args ) noexcept
  : m_M()
{

  TableTypeForMatrix<T> M{};
  LineTypeForMatrix<T> vec{};
  ConstructTable( M , vec , args... );
  m_M = M;

}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline Matrix<Y,X,T>::Matrix( const Matrix<Y,X,T>& mat ) noexcept : m_M( mat.m_M ) {}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> template <typename... Args> inline Matrix<Y,X,T>::Matrix( const TableTypeForMatrix<T>& M ) noexcept : m_M( M ) {}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T>& Matrix<Y,X,T>::operator=( const Matrix<Y,X,T>& mat ) noexcept
{

  m_M = mat.m_M;
  return *this;

}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T>& Matrix<Y,X,T>::operator+=( const Matrix<Y,X,T>& 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 <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T>& Matrix<Y,X,T>::operator-=( const Matrix<Y,X,T>& 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 <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> Matrix<Y,X,T>& Matrix<Y,X,T>::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 <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline TableTypeForMatrix<T>& Matrix<Y,X,T>::RefTable() noexcept { return m_M; }
template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline const TableTypeForMatrix<T>& Matrix<Y,X,T>::GetTable() const noexcept { return m_M; }

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline const Matrix<Y,X,T>& Matrix<Y,X,T>::Unit() noexcept { static const Matrix<Y,X,T> unit = Unit_Body(); return unit; }

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T> Matrix<Y,X,T>::Unit_Body() noexcept
{

  TableTypeForMatrix<T> M{};
  
  for( SizeTypeForMatrix y = 0 ; y < Y ; y++ ){

    LineTypeForMatrix<T> vec{};

    for( SizeTypeForMatrix x = 0 ; x < X ; x++ ){

      vec.push_back( x == y ? 1 : 0 );

    }

    M.push_back( vec );

  }

  return Matrix<Y,X,T>( M );

}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline void Matrix<Y,X,T>::ConstructTable( TableTypeForMatrix<T>& M , LineTypeForMatrix<T>& vec ) noexcept { M.push_back( vec ); vec.clear(); }

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> template <typename Arg , typename... Args> void Matrix<Y,X,T>::ConstructTable( TableTypeForMatrix<T>& M , LineTypeForMatrix<T>& 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 <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline Matrix<Y,X,T> operator==( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 ) noexcept { return mat1.GetTable() == mat2.GetTable(); }

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T> inline Matrix<Y,X,T> operator!=( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 ) noexcept { return !( mat1 == mat2 ); }

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T> operator+( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 )
{

  Matrix<Y,X,T> mat1_copy = mat1;
  mat1_copy += mat2;
  return mat1_copy;

}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T> operator-( const Matrix<Y,X,T>& mat1 , const Matrix<Y,X,T>& mat2 )
{

  Matrix<Y,X,T> mat1_copy = mat1;
  mat1_copy -= mat2;
  return mat1_copy;

}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , SizeTypeForMatrix Z , typename T> inline Matrix<Y,Z,T> operator*( const Matrix<Y,X,T>& mat1 , const Matrix<X,Z,T>& mat2 )
{

  const TableTypeForMatrix<T>& M1 = mat1.GetTable();
  const TableTypeForMatrix<T>& M2 = mat2.GetTable();
  TableTypeForMatrix<T> M_prod{};
  auto begin2x = M2.begin();
  
  for( auto itr1y = M1.begin() , end1y = M1.end() ; itr1y != end1y ; itr1y++ ){

    LineTypeForMatrix<T> 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<Y,Z,T>( M_prod );

}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<Y,X,T> operator*( const T& scalar , const Matrix<Y,X,T>& mat )
{

  Matrix<Y,X,T> mat_copy = mat;
  mat_copy *= scalar;
  return mat_copy;

}

template <SizeTypeForMatrix Y , SizeTypeForMatrix X , typename T>
Matrix<X,Y,T> Transpose( const Matrix<Y,X,T>& mat )
{

  const TableTypeForMatrix<T>& M = mat.GetTable();
  TableTypeForMatrix<T> M_t{};

  auto beginy = M.begin();

  for( auto itr1x = beginy->begin() , end1x = beginy->end() ; itr1x != end1x ; itr1x++ ){

    M_t.push_back( LineTypeForMatrix<T>() );

  }

  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<X,Y,T>( M_t );

}

template <SizeTypeForMatrix X , typename T>
T Trace( const Matrix<X,X,T>& mat )
{

  int i = 0;
  T answer =0;
  const TableTypeForMatrix<T>& M = mat.GetTable();

  for( auto itry = M.begin() , endy = M.end() ; itry != endy ; itry++ ){

    answer += ( *itry )[i];
    i++;

  }

  return answer;

}

template <typename T , typename UINT> 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<T>( mat ) , num , TwoByTwoMatrix<T>( init_dummy ) , for_right_multiplication_dummy ).GetMatrix22(); }

template <typename T> inline TwoByTwoMatrix<T>::TwoByTwoMatrix( const T& M00 , const T& M01 , const T& M10 , const T& M11 ) noexcept : m_M00( M00 ) , m_M01( M01 ) , m_M10( M10 ) , m_M11( M11 ) {}

template <typename T>
TwoByTwoMatrix<T>::TwoByTwoMatrix( const Matrix<2,2,T>& mat )
  : m_M00() , m_M01() , m_M10() , m_M11()
{

  const TableTypeForMatrix<T>& M = mat.GetTable();
  const LineTypeForMatrix<T>& M0 = M[0];
  const LineTypeForMatrix<T>& M1 = M[1];
  m_M00 = M0[0];
  m_M01 = M0[1];
  m_M10 = M1[0];
  m_M11 = M1[1];

}

template <typename T>
TwoByTwoMatrix<T>& TwoByTwoMatrix<T>::operator=( const TwoByTwoMatrix<T>& mat ) noexcept
{

  if( &mat != this ){

    m_M00 = mat.m_M00;
    m_M01 = mat.m_M01;
    m_M10 = mat.m_M10;
    m_M11 = mat.m_M11;

  }

  return *this;

}

template <typename T> inline Matrix<2,2,T> TwoByTwoMatrix<T>::GetMatrix22() const noexcept { return Matrix<2,2,T>( m_M00 , m_M01 , m_M10 , m_M11 ); }

template <typename T> inline TwoByTwoMatrix<T> TwoByTwoMatrix<T>::Multiply( const TwoByTwoMatrix<T>& mat1 , const TwoByTwoMatrix<T>& mat2 ) { return TwoByTwoMatrix<T>( mat1.m_M00 * mat2.m_M00 + mat1.m_M01 * mat2.m_M10 , mat1.m_M00 * mat2.m_M01 + mat1.m_M01 * mat2.m_M11 , mat1.m_M10 * mat2.m_M00 + mat1.m_M11 * mat2.m_M10 , mat1.m_M10 * mat2.m_M01 + mat1.m_M11 * mat2.m_M11 ); }

template <typename T> inline TwoByTwoMatrix<T> TwoByTwoMatrix<T>::Square( const TwoByTwoMatrix<T>& mat ) { return TwoByTwoMatrix<T>( mat.m_M00 * mat.m_M00 + mat.m_M01 * mat.m_M10 , ( mat.m_M00 + mat.m_M11 ) * mat.m_M01 , mat.m_M10 * ( mat.m_M00 + mat.m_M11 ) , mat.m_M10 * mat.m_M01 + mat.m_M11 * mat.m_M11 ); }

template <typename T> inline TwoByTwoMatrix<T> operator*( const TwoByTwoMatrix<T>& mat1 , const TwoByTwoMatrix<T>& mat2 ) { return TwoByTwoMatrix<T>::Multiply( mat1 , mat2 ); }

template <typename T> inline TwoByTwoMatrix<T> Square( const TwoByTwoMatrix<T>& mat ) { return TwoByTwoMatrix<T>::Square( mat ); }



#define MODULO 10000 
using MOD = Mod< MODULO >;


void Solve();

int main()
{

  Solve();
  return 0;

}

void Shuffle( MOD& n , const ll& power );
void ShuffleMemorise( MOD& n , const ll& power , const ll& I );

void Solve()
{

  string S;
  ll P;
  ll I;
  cin >> S;
  cin >> P;
  cin >> I;

  MOD answer{ stoll( S ) };

  if( P == 0 ){

    if( I != 0 ){

      answer = 0;

    }

  } else {

    // 素数pと正整数dに対するGL( 2 , p ^ d )の位数
    // = ( p ^ d - p ^ ( d - 1 ) ) ^ 2 * ( p ^ d ) ^ 2 + p ^ ( d - 1 ) * ( p ^ d - p ^ ( d - 1 ) ) ^ 2 * p ^ d
    // = p ^ ( 4 * d - 2 ) * ( p - 1 ) ^ 2 + p ^ ( 4 * d - 3 ) * ( p - 1 ) ^ 2
    // = p ^ ( 4 * d - 3 ) * ( p - 1 ) ^ 2 * ( p + 1 )

    // GL( 2 , 2 ^ 4 )の位数 = 2 ^ ( 4 * 4 - 3 ) * ( 2 - 1 ) ^ 2 * ( 2 + 1 ) = 2 ^ 13 * 3
    // GL( 2 , 5 ^ 4 )の位数 = 5 ^ ( 4 * 4 - 3 ) * ( 5 - 1 ) ^ 2 * ( 5 + 1 ) = 5 ^ 13 * 4 ^ 2 * 6 = 2 ^ 5 * 3 * 5 ^ 13
    // lcm( GL( 2 , 2 ^ 4 )の位数 , GL( 2 , 5 ^ 4 )の位数 ) = 2 ^ 13 * 3 * 5 ^ 13 = 3 * 10 ^ 13;
    const ll power = ( P - 1 ) % 30000000000000;
    
    if( I < MODULO ){

      for( ll i = 0 ; i < I ; i++ ){

	Shuffle( answer , power );

      }

    } else {

      ShuffleMemorise( answer , power , I );

    }

  }

  const string answer_str = to_string( MODULO + answer.Represent() ).substr( 1 );
  cout << answer_str << endl;
  return;

}

void Shuffle( MOD& n , const ll& power )
{

  if( n == 0 ){

    return;

  }
  
  static const Matrix<1,2,MOD> A{ 1 , 0 };
  const Matrix<2,2,MOD> B
  {
    n , 1 ,
      1 , 0
      };
  static const Matrix<2,1,MOD> C{ 1 , 0 };
  static const Matrix<2,2,MOD> E
  {
    1 , 0 ,
      0 , 1
      };
  static const string method = "binary";

  MOD answer = Trace<1,MOD>( A * Power<Matrix<2,2,MOD>,ll>( B , power , E , true , method ) * C );

  if( power % 2 == 0 ){

    answer -= 1;
    
  }

  n = answer;
  return;

}

void ShuffleMemorise( MOD& n , const ll& power , const ll& I )
{

  VLArray<MOD> answer_memory{};

  for( ll i = 0 ; i < MODULO ; i++ ){

    answer_memory.push_back( n );
    Shuffle( n , power );

  }
  
  const MOD& last_input = answer_memory.back();
  answer_memory.push_back( n );

  bool not_found = true;
  ll num;

  auto itr = answer_memory.begin();

  for( ll i = 0 ; i <= MODULO && not_found ; i++ ){

    if( *itr == last_input ){

	not_found = false;
	num = i;

    } else {

      itr++;

    }

  }
    
  const ll period = MODULO - num - 1;
  const ll d = ( I - num ) % period;

  for( ll i = 0 ; i < d ; i++ ){

    itr++;

  }
    
  n = *itr;
  return;
  
}

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