// 入力制約／フォーマットチェック
#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  #define USE_GETLINE
#endif
#ifdef INCLUDE_MAIN

void Solve()
{
  CEXPR( int , bound_N , 3e6 );
  CEXPR( ll , bound_K , 1e18 );
  GETLINE_COUNT_ASSERT( NK_str , ' ' , 2 );
  STOI( NK_str , N , 1 , bound_N );
  STOI( NK_str , K , 1 , bound_K );
  CEXPR( int , val_limit , 1002 );
  static HeapPrimeEnumeration pe{ val_limit };
  if( K == 1 ){
    RETURN( N );
  }
  int P = 100003;
  vector<vector<int>> div = TotalEnumerateDivisor( P );
  vector<int> C( P );
  FOR( i , 1 , P ){
    C[i] = Sum<ll>( div[i] ) % P;
  }
  LoopDetection ldt{ Sum<ll>( EnumerateDivisor( pe , N ) ) % P , [&]( const int& i ){ return C[i]; } };
  RETURN( ldt.IteratedComposition( K - 2 ) );
}
REPEAT_MAIN(1);

#else // INCLUDE_MAIN
#ifdef INCLUDE_LIBRARY

// https://github.com/p-adic/cpp
// VVV ライブラリは以下に挿入する。

/* 圧縮用 */
#define TE template
#define TY typename
#define US using
#define ST static
#define AS assert
#define IN inline
#define CL class
#define PU public
#define OP operator
#define CE constexpr
#define CO const
#define NE noexcept
#define RE return 
#define WH while
#define VO void
#define VE vector
#define LI list
#define BE begin
#define EN end
#define SZ size
#define LE length
#define PW Power
#define MO move
#define TH this
#define CRI CO int&
#define CRUI CO uint&
#define CRL CO ll&
#define VI virtual 
#define IS basic_istream<char,Traits>
#define OS basic_ostream<char,Traits>
#define ST_AS static_assert
#define reMO_CO remove_const
#define is_COructible_v is_constructible_v
#define rBE rbegin
// CEXPRがCEに依存しているので削除しない。
// redefinitionを避けるため圧縮元はincludeしない。

TE <int val_limit,int le_max = val_limit>CL PrimeEnumeration{PU:bool m_is_composite[val_limit];int m_val[le_max];int m_le;CE PrimeEnumeration();IN CRI OP[](CRI i)CO;CE CRI Get(CRI i)CO;CE CO bool& IsComposite(CRI n)CO;CE CRI length()CO NE;};
TE <int val_limit,int le_max> CE PrimeEnumeration<val_limit,le_max>::PrimeEnumeration():m_is_composite(),m_val(),m_le(0){for(int i = 2;i < val_limit;i++){if(! m_is_composite[i]){for(ll j = ll(i)* i;j < val_limit;j += i){m_is_composite[j]= true;}m_val[m_le++]= i;if(m_le >= le_max){break;}}}}TE <int val_limit,int le_max> IN CRI PrimeEnumeration<val_limit,le_max>::OP[](CRI i)CO{AS(0 <= i && i < m_le);RE m_val[i];}TE <int val_limit,int le_max> CE CRI PrimeEnumeration<val_limit,le_max>::Get(CRI i)CO{RE m_val[i];}TE <int val_limit,int le_max> CE CO bool& PrimeEnumeration<val_limit,le_max>::IsComposite(CRI n)CO{RE m_is_composite[n];}TE <int val_limit,int le_max> CE CRI PrimeEnumeration<val_limit,le_max>::length()CO NE{RE m_le;}
CL HeapPrimeEnumeration{PU:int m_val_limit;VE<bool> m_is_composite;VE<int> m_val;int m_le;IN HeapPrimeEnumeration(CRI val_limit);IN CRI OP[](CRI i)CO;IN CRI Get(CRI i)CO;IN bool IsComposite(CRI n)CO;IN CRI length()CO NE;};
IN HeapPrimeEnumeration::HeapPrimeEnumeration(CRI val_limit):m_val_limit(val_limit),m_is_composite(m_val_limit),m_val(),m_le(0){for(int i = 2;i < m_val_limit;i++){if(! m_is_composite[i]){for(ll j = ll(i)* i;j < val_limit;j += i){m_is_composite[j]= true;}m_val.push_back(i);}}m_le = m_val.SZ();}IN CRI HeapPrimeEnumeration::OP[](CRI i)CO{AS(0 <= i && i < m_le);RE m_val[i];}IN CRI HeapPrimeEnumeration::Get(CRI i)CO{RE OP[](i);}IN bool HeapPrimeEnumeration::IsComposite(CRI n)CO{AS(0 <= n && n < m_val_limit);RE m_is_composite[n];}IN CRI HeapPrimeEnumeration::length()CO NE{RE m_le;}

TE <TY PE> auto CheckPE(CO PE& pe)-> decltype(pe.IsComposite(0),true_type());TE <TY...> false_type CheckPE(...);TE <TY T>CE bool IsPE = decltype(CheckPE(declval<T>()))();

TE <int val_limit>CL LeastDivisor{PU:int m_val[val_limit];CE LeastDivisor()NE;IN CRI OP[](CRI i)CO;CE CRI Get(CRI i)CO;CE int length()CO NE;};
TE <int val_limit> CE LeastDivisor<val_limit>::LeastDivisor()NE:m_val{}{for(int d = 2;d < val_limit;d++){if(m_val[d]== 0){for(int n = d;n < val_limit;n += d){m_val[n]== 0?m_val[n]= d:d;}}}}TE <int val_limit> IN CRI LeastDivisor<val_limit>::OP[](CRI i)CO{AS(0 <= i && i < val_limit);RE m_val[i];}TE <int val_limit> CE CRI LeastDivisor<val_limit>::Get(CRI i)CO{RE m_val[i];}TE <int val_limit> CE int LeastDivisor<val_limit>::length()CO NE{RE val_limit;}
CL HeapLeastDivisor{PU:int m_val_limit;VE<int> m_val;IN HeapLeastDivisor(CRI val_limit)NE;IN CRI OP[](CRI i)CO;IN CRI Get(CRI i)CO;IN CRI length()CO NE;};
IN HeapLeastDivisor::HeapLeastDivisor(CRI val_limit)NE:m_val_limit(val_limit),m_val(m_val_limit){for(int d = 2;d < m_val_limit;d++){if(m_val[d]== 0){for(int n = d;n < m_val_limit;n += d){m_val[n]== 0?m_val[n]= d:d;}}}}IN CRI HeapLeastDivisor::OP[](CRI i)CO{AS(0 <= i && i < m_val_limit);RE m_val[i];}IN CRI HeapLeastDivisor::Get(CRI i)CO{RE m_val[i];}IN CRI HeapLeastDivisor::length()CO NE{RE m_val_limit;}

TE <TY PE,TY INT>auto PrimeFactorisation(CO PE& pe,INT n)-> enable_if_t<IsPE<PE>,pair<VE<INT>,VE<int>>>{AS(n > 0);VE<INT> P{};VE<int> E{};CRI le = pe.length();for(int i = 0;i < le;i++){auto& p = pe[i];if(n % p == 0){int e = 1;WH((n /= p)% p == 0){e++;}P.push_back(p);E.push_back(e);}else if(n / p < p){break;}}if(n != 1){P.push_back(n);E.push_back(1);}RE{MO(P),MO(E)};}TE <TY LD>auto PrimeFactorisation(CO LD& ld,int n)-> enable_if_t<!IsPE<LD>,pair<VE<int>,VE<int>>>{AS(n > 0);VE<int> P{};VE<int> E{};if(n > 1){P.push_back(ld[n]);E.push_back(1);n /= ld[n];}WH(n > 1){if(P.back()!= ld[n]){P.push_back(ld[n]);E.push_back(1);}else{E.back()++;}n /= ld[n];}RE{MO(P),MO(E)};}TE <TY PE,TY INT>auto PrimePowerFactorisation(CO PE& pe,INT n)-> enable_if_t<IsPE<PE>,tuple<VE<INT>,VE<int>,VE<INT>>>{AS(n > 0);VE<INT> P{};VE<int> E{};VE<INT> Q{};CRI le = pe.length();for(int i = 0;i < le;i++){auto& p = pe[i];if(n % p == 0){int e = 1;INT q = p;WH((n /= p)% p == 0){e++;q *= p;}P.push_back(p);E.push_back(e);Q.push_back(q);}else if(n / p < p){break;}}if(n != 1){P.push_back(n);E.push_back(1);Q.push_back(n);}RE{MO(P),MO(E),MO(Q)};}TE <TY LD>auto PrimePowerFactorisation(CO LD& ld,int n)-> enable_if_t<!IsPE<LD>,tuple<VE<int>,VE<int>,VE<int>>>{AS(n > 0);VE<int> P{};VE<int> E{};VE<int> Q{};if(n > 1){P.push_back(ld[n]);E.push_back(1);Q.push_back(ld[n]);n /= ld[n];}WH(n > 1){if(P.back()!= ld[n]){P.push_back(ld[n]);E.push_back(1);Q.push_back(ld[n]);}else{Q.back()*= ld[n];E.back()++;}n /= ld[n];}RE{MO(P),MO(E),MO(Q)};}

TE <TY INT> INT CountDivisorBody(VE<int>& E)NE{CO int LE = E.SZ();INT AN = 1;for(int i = 0;i < LE;i++){AN *= ++E[i];}RE AN;}TE <TY PE,TY INT>INT CountDivisor(CO PE& pe,INT n)NE{auto[P,E]= PrimeFactorisation(pe,MO(n));RE CountDivisorBody<INT>(E);}

TE <TY INT> VE<INT> EnumerateDivisorBody(CO VE<INT>& P,VE<int>& E){CO int le = P.SZ();VE AN(CountDivisorBody<INT>(E),INT(1));int SZ = 1;for(int i = 0;i < le;i++){auto& P_i = P[i];auto& E_i = E[i];INT q = 1;int j_shift = 0;for(int e = 1;e < E_i;e++){q *= P_i;j_shift += SZ;for(int j = 0;j < SZ;j++){AN[j + j_shift]= AN[j]* q;}}SZ *= E_i;}RE AN;}TE <TY PE,TY INT>VE<INT> EnumerateDivisor(CO PE& pe,INT n){auto[P,E]= PrimeFactorisation(pe,MO(n));RE EnumerateDivisorBody(P,E);}TE <TY INT> VE<INT> EnumerateSquareFreeDivisorBody(CO VE<INT>& P){CO int le = P.SZ();CO int SZ = 1 << le;VE AN(SZ,INT(1));for(int i = 0;i < le;i++){CO int s = 1 << i;for(int j = s;j < SZ;(++j)|= s){AN[j]*= P[i];}}RE AN;}TE <TY PE,TY INT>VE<INT> EnumerateSquareFreeDivisor(CO PE& pe,INT n){auto[P,E]= PrimeFactorisation(pe,MO(n));RE EnumerateSquareFreeDivisorBody(P);}TE <TY INT>VE<VE<INT>> TotalEnumerateDivisor(CO INT& n_max)NE{VE<VE<INT>> AN(n_max + 1);for(INT d = 1;d <= n_max;d++){for(INT n = d;n <= n_max;n += d){AN[n].push_back(d);}}RE AN;}TE <TY INT,TY FUNC,TY U>VE<U> TotalEnumerateDivisor(CO INT& n_max,FUNC f,CO U& init)NE{ST_AS(is_invocable_r_v<U,FUNC,U,CO INT&>);VE<U> AN(n_max + 1,init);for(INT d = 1;d <= n_max;d++){for(INT n = 0;n <= n_max;n += d){AN[n]= f(MO(AN[n]),d);}}RE AN;}

template <typename T , typename F>
class LoopDetection_Body
{

private:
  T m_init;
  F m_f;
  int m_search_max;
  
protected:
  // 像の要素数
  int m_length;
  // ループ始点
  int m_loop_start;
  // ループ長取得
  int m_loop_length;

public:
  inline LoopDetection_Body( const T& init , F f , const int& search_max );
  
  template <typename INT> T IteratedComposition( const INT& n );

  // 像の要素数取得
  inline const int& GetLength() noexcept;
  // ループ始点取得
  inline const int& GetLoopStart() noexcept;
  // ループ長取得
  inline const int& GetLoopLength() noexcept;
  
private:
  inline void SetInit();
  inline void SearchLoop();
  virtual T e( const int& i ) = 0;
  virtual int e_inv( const T& t ) = 0;
  virtual void SetValue( const int& i ) = 0;
  virtual const int& GetValue( const int& i ) = 0;

};

template <typename T , typename F>
class ValueCalculatorForLoopDetection :
  public LoopDetection_Body<T,F>
{

private:
  // m_value[n]にf^n(init)を格納する。
  vector<int> m_value;
  // m_value_inv[i]にf^n(init)=iとなる最小のnを格納する。
  unordered_map<int,int> m_value_inv;

public:
  inline ValueCalculatorForLoopDetection( const T& init , F f , const int& search_max );

private:
  // f^0(init),...,f^n(init)までを既に計算した状態でf^{n+1}(init)=iと記録する。
  void SetValue( const int& i );
  // f^i(init)を返す。
  int& RefValue( const int& i );
  // f^i(init)を返す。
  const int& GetValue( const int& i );
  // f^n(init)=iとなる最小のnが見付かっていればそれを、見付かっていなければ-1を返す。
  int& RefValue_inv( const int& i );

};

template <typename F>
class LoopDetection :
  public ValueCalculatorForLoopDetection<int,F>
{
  
public:
  inline LoopDetection( const int& init , F f , const int& search_max = -1 );

private:
  inline int e( const int& i );
  inline int e_inv( const int& t );

};

template <typename T , typename F>
class MemorisationLoopDetection :
  public LoopDetection_Body<T,F>
{

private:
  // m_memory[t]にf^n(init)=tとなる最小のnを格納する。
  Map<T,int> m_memory;
  // m_memory_inv[n]にf^n(init)を格納する。
  vector<T> m_memory_inv;
  // つまりf^n自体をメモ化番号付けに流用する。だからGetValueが恒等関数になる。

public:
  inline MemorisationLoopDetection( const T& init , F f , const int& search_max = -1 );
  inline T e( const int& i );
  inline int e_inv( const T& t );
  // 何もしない。
  void SetValue( const int& i );
  // iを返す。
  const int& GetValue( const int& i );

};

template <typename T , typename Enum_T , typename Enum_T_inv , typename F>
class EnumerationLoopDetection :
  public ValueCalculatorForLoopDetection<T,F>
{

private:
  Enum_T m_enum_T;
  Enum_T_inv m_enum_T_inv;

public:
  inline EnumerationLoopDetection( const T& init , Enum_T enum_T , Enum_T_inv enum_T_inv , F f , const int& search_max = -1 );
  
private:
  inline T e( const int& i );
  inline int e_inv( const T& t );

};

template <typename T , typename F> inline LoopDetection_Body<T,F>::LoopDetection_Body( const T& init , F f , const int& search_max ) : m_init( init ) , m_f( move( f ) ) , m_search_max( search_max ) , m_length() , m_loop_start( -1 ) , m_loop_length( -1 ) { static_assert( is_invocable_r_v<T,F,T> ); }
template <typename T , typename F> inline ValueCalculatorForLoopDetection<T,F>::ValueCalculatorForLoopDetection( const T& init , F f , const int& search_max ) : LoopDetection_Body<T,F>( init , move( f ) , search_max ) , m_value() , m_value_inv() {}
template <typename F> inline LoopDetection<F>::LoopDetection( const int& init , F f , const int& search_max ) : ValueCalculatorForLoopDetection<int,F>( init , move( f ) , search_max ) {}
template <typename T , typename F> inline MemorisationLoopDetection<T,F>::MemorisationLoopDetection( const T& init , F f , const int& search_max ) : LoopDetection_Body<T,F>( init , move( f ) , search_max ) , m_memory() , m_memory_inv() {}
template <typename T , typename Enum_T , typename Enum_T_inv , typename F> inline EnumerationLoopDetection<T,Enum_T,Enum_T_inv,F>::EnumerationLoopDetection( const T& init , Enum_T enum_T , Enum_T_inv enum_T_inv , F f , const int& search_max ) : ValueCalculatorForLoopDetection<T,F>( init , move( f ) , search_max ) , m_enum_T( enum_T ) , m_enum_T_inv( enum_T_inv ) { static_assert( is_invocable_r_v<T,Enum_T,int> && is_invocable_r_v<int,Enum_T_inv,T> ); }

template <typename T , typename F> template <typename INT>
T LoopDetection_Body<T,F>::IteratedComposition( const INT& n )
{

  if( m_length == 0 ){

    SetInit();

  }

  if( n < m_length ){

    return e( GetValue( n ) );
    
  }

  if( m_loop_start != -1 ){

    return e( GetValue( m_loop_start + ( n - m_loop_start ) % m_loop_length ) );

  }
  
  SetValue( e_inv( m_f( e( GetValue( m_length - 1 ) ) ) ) );
  return IteratedComposition( n );

}

template <typename T , typename F> inline const int& LoopDetection_Body<T,F>::GetLength() noexcept { if( m_loop_length == -1 ){ SearchLoop(); } return m_length; }
template <typename T , typename F> inline const int& LoopDetection_Body<T,F>::GetLoopStart() noexcept { if( m_loop_start == -1 ){ SearchLoop(); } return m_loop_start; }
template <typename T , typename F> inline const int& LoopDetection_Body<T,F>::GetLoopLength() noexcept { if( m_loop_length == -1 ){ SearchLoop(); } return m_loop_length; }

template <typename T , typename F> inline void LoopDetection_Body<T,F>::SetInit() { assert( m_length == 0 ); SetValue( e_inv( m_init ) ); }

template <typename T , typename F> inline void LoopDetection_Body<T,F>::SearchLoop() { assert( m_loop_length == -1 ); int n = 0; while( m_loop_length == -1 && ( m_search_max == -1 || n <= m_search_max ) ){ IteratedComposition( n++ ); } }

template <typename F> inline int LoopDetection<F>::e( const int& i ) { return i; }
template <typename T , typename F> inline T MemorisationLoopDetection<T,F>::e( const int& i ) { using base = LoopDetection_Body<T,F>; assert( i < base::m_length ); return m_memory_inv[i]; }
template <typename T , typename Enum_T , typename Enum_T_inv , typename F> inline T EnumerationLoopDetection<T,Enum_T,Enum_T_inv,F>::e( const int& i ) { return m_enum_T( i ); }

template <typename F> inline int LoopDetection<F>::e_inv( const int& t ) { return t; }
template <typename T , typename F> inline int MemorisationLoopDetection<T,F>::e_inv( const T& t ) { using base = LoopDetection_Body<T,F>; if( m_memory.count( t ) == 0 ){ m_memory_inv.push_back( t ); return m_memory[t] = base::m_length++; } base::m_loop_length = base::m_length - ( base::m_loop_start = m_memory[t] ); return base::m_loop_start; }
template <typename T , typename Enum_T , typename Enum_T_inv , typename F> inline int EnumerationLoopDetection<T,Enum_T,Enum_T_inv,F>::e_inv( const T& t ) { return m_enum_T_inv( t ); }

template <typename T , typename F> inline void ValueCalculatorForLoopDetection<T,F>::SetValue( const int& i )
{

  using base = LoopDetection_Body<T,F>;
  int& m_value_inv_i = RefValue_inv( i );

  if( m_value_inv_i != -1 ){

    base::m_loop_length = base::m_length - ( base::m_loop_start = m_value_inv_i );

  } else {
  
    RefValue( base::m_length ) = i;
    m_value_inv_i = base::m_length++;

  }

  return;

}

template <typename T , typename F> inline void MemorisationLoopDetection<T,F>::SetValue( const int& i ) {}

template <typename T , typename F> inline int& ValueCalculatorForLoopDetection<T,F>::RefValue( const int& i ) { const int size = m_value.size(); if( size <= i ){ assert( size == i ); m_value.push_back( -1 ); } return m_value[i]; }
template <typename T , typename F> inline const int& ValueCalculatorForLoopDetection<T,F>::GetValue( const int& i ) { return RefValue( i ); }
template <typename T , typename F> inline const int& MemorisationLoopDetection<T,F>::GetValue( const int& i ) { return i; }

template <typename T , typename F> inline int& ValueCalculatorForLoopDetection<T,F>::RefValue_inv( const int& i ) { if( m_value_inv.count( i ) == 0 ){ return m_value_inv[i] = -1; } return m_value_inv[i]; }

// AAA ライブラリは以上に挿入する。

#define INCLUDE_MAIN
#include __FILE__
#else // INCLUDE_LIBRARY
#ifdef DEBUG
  #define _GLIBCXX_DEBUG
  #define SIGNAL signal( SIGABRT , &AlertAbort );
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE2 )
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); assert( ( MIN ) <= A && A <= ( MAX ) )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define CERRNS( ... ) VariadicCoutNonSep( cerr , __VA_ARGS__ )
  #define CERR_A( I , N , A ) CoutArray( cerr , I , N , A ) << endl
  int exec_mode = 0;
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define SIGNAL 
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define CERR( ... ) 
  #define CERRNS( ... ) 
  #define CERR_A( I , N , A ) 
#endif
#ifdef REACTIVE
  #ifdef DEBUG
    #define RSET( A , ... ) A = __VA_ARGS__
  #else
    #define RSET( A , ... ) cin >> A
  #endif
  #define RCIN( LL , A , ... ) LL A; RSET( A , __VA_ARGS__ )
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#ifdef USE_GETLINE
  #define SET_SEPARATE( SEPARATOR , ... ) VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define SET( ... ) SET_SEPARATE( '\n' , __VA_ARGS__ )
  #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; SET_SEPARATE( SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
  #define FINISH_MAIN GETLINE( test_case_num_str ); test_case_num = stoi( test_case_num_str ); ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } CHECK_REDUNDANT_INPUT; }
#else
  #define SET( ... ) VariadicCin( cin , __VA_ARGS__ )
  #define CIN( LL , ... ) LL __VA_ARGS__; SET( __VA_ARGS__ )
  #define SET_A( I , N , ... ) VariadicResize( N + I , __VA_ARGS__ ); FOR( VARIABLE_FOR_SET_A , 0 , N ){ VariadicSet( cin , VARIABLE_FOR_SET_A + I , __VA_ARGS__ ); }
  #define CIN_A( LL , I , N , ... ) VE<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
  #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) VE<VE<LL>> VAR( N0 + I0 ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , VAR[VARIABLE_FOR_CIN_AA + I0] ); }
  #define FINISH_MAIN SET_ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } CHECK_REDUNDANT_INPUT; }
#endif
#include <bits/stdc++.h>
using namespace std;
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr ); SIGNAL;
#define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , test_case_num_bound , BOUND ); int test_case_num = 1; if constexpr( test_case_num_bound > 1 ){ CERR( "テストケースの個数を入力してください。" ); FINISH_MAIN;
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now(); double loop_average_time = 0.0 , loop_start_time = loop_average_time , current_time = loop_start_time; int loop_count = current_time; assert( loop_count == 0 )
#define CURRENT_TIME ( current_time = static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 ) )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME , loop_count == 0 ? loop_start_time = current_time : loop_average_time = ( current_time - loop_start_time ) / loop_count , ++loop_count , current_time < TL_MS - loop_average_time * 2 - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_ASSERT( A , MIN , MAX ) SET( A ); ASSERT( A , MIN , MAX )
#define SET_A_ASSERT( I , N , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A + I] , MIN , MAX ); }
#define SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_AA0 , 0 , N0 ){ FOR( VARIABLE_FOR_SET_AA1 , 0 , N1 ){ SET_ASSERT( A[VARIABLE_FOR_SET_AA0 + I0][VARIABLE_FOR_SET_AA1 + I1] , MIN , MAX ); } }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); SET_A_ASSERT( I , N , A , MIN , MAX )
#define CIN_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) vector A( N0 + I0 , vector<decldecay_t( MAX )>( N1 + I1 ) ); SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX )
#define PR1( A1 , ... ) A1
#define PR2( A1 , A2 , ... ) A2
#define PR3( A1 , A2 , A3 , ... ) A3
#define FOR_( VAR , INITIAL , FINAL , UPPER , COMP , INCR ) for( decldecay_t( UPPER ) VAR = INITIAL ; VAR COMP ( FINAL ) ; VAR INCR )
#define FOR( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , < , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQ( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , <= , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQINV( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , INITIAL , + 1 > , PR3( __VA_ARGS__ , -= PR2( __VA_ARGS__ , ? ) , -- ) )
#define ITR( ARRAY ) auto begin_ ## ARRAY = ARRAY .BE() , itr_ ## ARRAY = begin_ ## ARRAY , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define RUN( ARRAY , ... ) for( auto&& __VA_ARGS__ : ARRAY )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS ); cerr << fixed << setprecision( DECIMAL_DIGITS )
#define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
#define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ )
#define COUT_A( I , N , A ) CoutArray( cout , I , N , A ) << ENDL
#define DERR( ... ) 
#define DERRNS( ... ) 
#define DERR_A( I , N , A ) 
#define WHAT( ... ) 
#define RETURN( ... ) COUT( __VA_ARGS__ ); return

// 型のエイリアス
#define decldecay_t( VAR ) decay_t<decltype( VAR )>
template <typename F , typename...Args> using ret_t = decltype( declval<F>()( declval<Args>()... ) );
template <typename T> using inner_t = typename T::type;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
using ld = long double;
using lld = __float128;
using path = pair<int,ll>;

/* VVV 常設ライブラリの非圧縮版は以下に挿入する。*/
// BinarySearch
constexpr bool reactive =
#ifdef REACTIVE
true;
#else
false;
#endif
// EXPRESSIONがANSWERの広義単調関数の時、EXPRESSION >= CONST_TARGETの整数解を格納。
#define BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , DESIRED_INEQUALITY , CONST_TARGET , INEQUALITY_FOR_CHECK , UPDATE_U , UPDATE_L , UPDATE_ANSWER , EXTERNAL ) \
  static_assert( ! is_same<decldecay_t( CONST_TARGET ),uint>::value && ! is_same<decldecay_t( CONST_TARGET ),ull>::value ); \
  ll ANSWER = MINIMUM;							\
  {									\
    ll ANSWER ## _L = MINIMUM;                                          \
    ll ANSWER ## _R = MAXIMUM;                                          \
    ANSWER = UPDATE_ANSWER;						\
    ll EXPRESSION_BS;							\
    const ll CONST_TARGET_BS = ( CONST_TARGET );			\
    ll DIFFERENCE_BS;							\
    while( ANSWER ## _L < ANSWER ## _R ){                               \
      DIFFERENCE_BS = ( EXPRESSION_BS = ( EXPRESSION ) ) - CONST_TARGET_BS; \
      if( DIFFERENCE_BS INEQUALITY_FOR_CHECK 0 ){			\
	ANSWER ## _R = UPDATE_U;                                        \
      } else {								\
	ANSWER ## _L = UPDATE_L;                                        \
      }									\
      ANSWER = UPDATE_ANSWER;						\
    }									\
    if( ANSWER ## _L > ANSWER ## _R || !( reactive || ( EXPRESSION ) DESIRED_INEQUALITY CONST_TARGET_BS ) ){ \
      ANSWER = EXTERNAL;						\
    }									\
  }									\

// 単調増加の時にEXPRESSION >= CONST_TARGETの最小解を格納。
#define MIN_GEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CONST_TARGET , >= , ANSWER , ANSWER + 1 , Mid( ANSWER ## _L , ANSWER ## _R ) , ( MAXIMUM ) + 1 )
// 単調増加の時にEXPRESSION <= CONST_TARGETの最大解を格納。
#define MAX_LEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CONST_TARGET , > , ANSWER - 1 , ANSWER , Mid( ANSWER ## _L + 1 , ANSWER ## _R ) , ( MINIMUM ) - 1 )
// 単調減少の時にEXPRESSION >= CONST_TARGETの最大解を格納。
#define MAX_GEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CONST_TARGET , < , ANSWER - 1 , ANSWER , Mid( ANSWER ## _L + 1 , ANSWER ## _R ) , ( MINIMUM ) - 1 )
// 単調減少の時にEXPRESSION <= CONST_TARGETの最小解を格納。
#define MIN_LEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CONST_TARGET , <= , ANSWER , ANSWER + 1 , Mid( ANSWER ## _L , ANSWER ## _R ) , ( MAXIMUM ) + 1 )
template <typename INT> inline constexpr INT Mid( const INT& l , const INT& r ) { return l + ( ( r - l ) >> 1 ); }

// Random
ll GetRand( const ll& Rand_min , const ll& Rand_max ) { assert( Rand_min <= Rand_max ); ll answer = time( NULL ); return answer * rand() % ( Rand_max + 1 - Rand_min ) + Rand_min; }

// Set
#define DC_OF_HASH( ... ) DECLARATION_OF_HASH( __VA_ARGS__ )

#define DECLARATION_OF_HASH( ... )				\
  struct hash<__VA_ARGS__>					\
  {								\
								\
    inline size_t operator()( const __VA_ARGS__& n ) const;	\
								\
  };								\

#define DEFINITION_OF_POP_FOR_SET( SET )                                \
  template <typename T> inline T pop_max( SET& S ) { assert( !S.empty() ); auto itr = --S.end(); T answer = *itr; S.erase( itr ); return answer; } \
  template <typename T> inline T pop_min( SET& S ) { assert( !S.empty() ); auto itr = S.begin(); T answer = *itr; S.erase( itr ); return answer; } \
  template <typename T> inline SET& operator<<=( SET& S , T t ) { S.insert( move( t ) ); return S; } \
  template <typename T , typename U> inline SET& operator<<=( SET& S , U&& u ) { S.insert( T{ forward<U>( u ) } ); return S; } \
  template <typename T> inline SET& operator>>=( SET& S , const T& t ) { auto itr = S.lower_bound( t ); assert( itr != S.end() && *itr == t ); S.erase( itr ); return S; } \
  template <typename T , typename U> inline SET& operator>>=( SET& S , const U& u ) { return S >>= T{ u }; } \
  template <typename T> inline const T& Get( const SET& S , int i ) { auto begin = S.begin() , end = S.end(); auto& itr = i < 0 ? ( ++i , --end ) : begin; while( i > 0 && itr != end ){ --i; ++itr; } while( i < 0 && itr != begin ){ ++i; --itr; } assert( i == 0 ); return *itr; } \

#define DEFINITION_OF_UNION_FOR_SET( SET )                              \
  template <typename T> inline SET& operator|=( SET& S0 , SET S1 ) { S0.merge( move( S1 ) ); return S0; } \
  template <typename T> inline SET operator|( SET S0 , SET S1 ) { return move( S0.size() < S1.size() ? S1 |= move( S0 ) : S0 |= move( S1 ) ); } \

class is_ordered
{

private:
  is_ordered() = delete;
  template <typename T> static constexpr auto Check( const T& t ) -> decltype( t < t , true_type() );
  static constexpr false_type Check( ... );

public:
  template <typename T> static constexpr const bool value = is_same_v< decltype( Check( declval<T>() ) ) , true_type >;

};

template <typename T>
using Set = conditional_t<is_constructible_v<unordered_set<T>>,unordered_set<T>,conditional_t<is_ordered::value<T>,set<T>,void>>;

template <typename SET , typename T> inline typename SET::const_iterator MaximumLeq( const SET& S , const T& t ) { auto itr = S.upper_bound( t ); return itr == S.begin() ? S.end() : --itr; }
template <typename SET , typename T> inline typename SET::const_iterator MaximumLt( const SET& S , const T& t ) { auto itr = S.lower_bound( t ); return itr == S.begin() ? S.end() : --itr; }
template <typename SET , typename T> inline typename SET::const_iterator MinimumGeq( const SET& S , const T& t ) { return S.lower_bound( t ); }
template <typename SET , typename T> inline typename SET::const_iterator MinimumGt( const SET& S , const T& t ) { return S.upper_bound( t ); }

template <typename SET , typename ITERATOR> inline void EraseBack( SET& S , ITERATOR& itr ) { itr = S.erase( itr ); }
template <typename SET , typename ITERATOR> inline void EraseFront( SET& S , ITERATOR& itr ) { itr = S.erase( itr ); itr == S.begin() ? itr = S.end() : --itr; }

template <template <typename...> typename SET , typename T , typename...Args> inline bool In( const T& t , const SET<T,Args...>& S ) { return S.count( t ) > 0; }

DEFINITION_OF_POP_FOR_SET( set<T> );
DEFINITION_OF_POP_FOR_SET( unordered_set<T> );
DEFINITION_OF_POP_FOR_SET( multiset<T> );
DEFINITION_OF_POP_FOR_SET( unordered_multiset<T> );

DEFINITION_OF_UNION_FOR_SET( set<T> );
DEFINITION_OF_UNION_FOR_SET( unordered_set<T> );
DEFINITION_OF_UNION_FOR_SET( multiset<T> );
DEFINITION_OF_UNION_FOR_SET( unordered_multiset<T> );
DEFINITION_OF_UNION_FOR_SET( vector<T> );
DEFINITION_OF_UNION_FOR_SET( list<T> );

// Tuple
#define DEFINITION_OF_ARITHMETIC_FOR_TUPLE( OPR )			\
  template <typename T , typename U , template <typename...> typename PAIR> inline auto operator OPR ## =( PAIR<T,U>& t0 , const PAIR<T,U>& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = get<0>( t1 ); get<1>( t0 ) OPR ## = get<1>( t1 ); return t0; } \
  template <typename T , typename U , typename V , template <typename...> typename TUPLE> inline auto operator OPR ## =( TUPLE<T,U,V>& t0 , const TUPLE<T,U,V>& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = get<0>( t1 ); get<1>( t0 ) OPR ## = get<1>( t1 ); get<2>( t0 ) OPR ## = get<2>( t1 ); return t0; } \
    template <typename T , typename U , typename V , typename W , template <typename...> typename TUPLE> inline auto operator OPR ## =( TUPLE<T,U,V,W>& t0 , const TUPLE<T,U,V,W>& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = get<0>( t1 ); get<1>( t0 ) OPR ## = get<1>( t1 ); get<2>( t0 ) OPR ## = get<2>( t1 ); get<3>( t0 ) OPR ## = get<3>( t1 ); return t0; } \
  template <typename ARG , typename T , typename U , template <typename...> typename PAIR> inline auto operator OPR ## =( PAIR<T,U>& t0 , const ARG& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = t1; get<1>( t0 ) OPR ## = t1; return t0; } \
  template <typename ARG , typename T , typename U , typename V , template <typename...> typename TUPLE> inline auto operator OPR ## =( TUPLE<T,U,V>& t0 , const ARG& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = t1; get<1>( t0 ) OPR ## = t1; get<2>( t0 ) OPR ## = t1; return t0; } \
    template <typename ARG , typename T , typename U , typename V , typename W , template <typename...> typename TUPLE> inline auto operator OPR ## =( TUPLE<T,U,V,W>& t0 , const ARG& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = t1; get<1>( t0 ) OPR ## = t1; get<2>( t0 ) OPR ## = t1; get<3>( t0 ) OPR ## = t1; return t0; } \
  template <template <typename...> typename TUPLE , typename...ARGS , typename ARG> inline auto operator OPR( const TUPLE<ARGS...>& t0 , const ARG& t1 ) -> decldecay_t( ( get<0>( t0 ) , t0 ) ) { auto t = t0; return move( t OPR ## = t1 ); } \

#define DEFINITION_OF_INCREMENT_FOR_TUPLE( INCR )			\
  template <typename T , typename U , template <typename...> typename PAIR> inline auto operator INCR( PAIR<T,U>& t ) -> decltype( ( get<0>( t ) , t ) )& { INCR get<0>( t ); INCR get<1>( t ); return t; } \
  template <typename T , typename U , typename V , template <typename...> typename TUPLE> inline auto operator INCR ( TUPLE<T,U,V>& t ) -> decltype( ( get<0>( t ) , t ) )& { INCR get<0>( t ); INCR get<1>( t ); INCR get<2>( t ); return t; } \
  template <typename T , typename U , typename V , typename W , template <typename...> typename TUPLE> inline auto operator INCR ( TUPLE<T,U,V,W>& t ) -> decltype( ( get<0>( t ) , t ) )& { INCR get<0>( t ); INCR get<1>( t ); INCR get<2>( t ); INCR get<3>( t ); return t; } \

DEFINITION_OF_ARITHMETIC_FOR_TUPLE( + );
template <typename T , typename U , template <typename...> typename V> inline auto operator-( const V<T,U>& t ) -> decltype( get<0>( t ) , t ) { return { -get<0>( t ) , -get<1>( t ) }; }
template <typename T , typename U , typename V> inline tuple<T,U,V> operator-( const tuple<T,U,V>& t ) { return { -get<0>( t ) , -get<1>( t ) , -get<2>( t ) }; }
template <typename T , typename U , typename V , typename W> inline tuple<T,U,V,W> operator-( const tuple<T,U,V,W>& t ) { return { -get<0>( t ) , -get<1>( t ) , -get<2>( t ) , -get<3>( t ) }; }
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( - );
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( * );
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( / );
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( % );

DEFINITION_OF_INCREMENT_FOR_TUPLE( ++ );
DEFINITION_OF_INCREMENT_FOR_TUPLE( -- );

template <class Traits , typename T> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , tuple<T>& arg ){ return is >> get<0>( arg ); }
template <class Traits , typename T , typename U , template <typename...> typename V> inline auto operator>>( basic_istream<char,Traits>& is , V<T,U>& arg ) -> decltype((get<0>(arg),is))& { return is >> get<0>( arg ) >> get<1>( arg ); }
template <class Traits , typename T , typename U , typename V> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , tuple<T,U,V>& arg ) { return is >> get<0>( arg ) >> get<1>( arg ) >> get<2>( arg ); }
template <class Traits , typename T , typename U , typename V , typename W> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , tuple<T,U,V,W>& arg ) { return is >> get<0>( arg ) >> get<1>( arg ) >> get<2>( arg ) >> get<3>( arg ); }

template <class Traits , typename T> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const tuple<T>& arg ) { return os << get<0>( arg ); }
template <class Traits , typename T , typename U , template <typename...> typename V> inline auto operator<<( basic_ostream<char,Traits>& os , const V<T,U>& arg ) -> decltype((get<0>(arg),os))& { return os << get<0>( arg ) << " " << get<1>( arg ); }
template <class Traits , typename T , typename U , typename V> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const tuple<T,U,V>& arg ) { return os << get<0>( arg ) << " " << get<1>( arg ) << " " << get<2>( arg ); }
template <class Traits , typename T , typename U , typename V , typename W> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const tuple<T,U,V,W>& arg ) { return os << get<0>( arg ) << " " << get<1>( arg ) << " " << get<2>( arg ) << " " << get<3>( arg ); }

template <int n>
class TupleAccessIndex
{};

template <typename...Types>
class Tuple :
public tuple<Types...>
{

public:
  inline Tuple( Types&&... args );
  template <typename...Args> inline Tuple( Args&&... args );
  
  template <int n> inline auto& operator[]( const TupleAccessIndex<n>& i ) noexcept;
  template <int n> inline const auto& operator[]( const TupleAccessIndex<n>& i ) const noexcept;

};

// structural binding用
template <typename...Types>
class tuple_size<Tuple<Types...>> :
  public tuple_size<tuple<Types...>>
{};

template <size_t n , typename...Types>
class tuple_element<n,Tuple<Types...>> :
  public tuple_element<n,tuple<Types...>>
{};

template <typename T , typename U> using Pair = Tuple<T,U>;
template <typename INT> using T2 = Tuple<INT,INT>;
template <typename INT> using T3 = Tuple<INT,INT,INT>;
template <typename INT> using T4 = Tuple<INT,INT,INT,INT>;

constexpr TupleAccessIndex<0> O{};
constexpr TupleAccessIndex<1> I{};
constexpr TupleAccessIndex<2> II{};
constexpr TupleAccessIndex<3> III{};

template <typename...Types> inline Tuple<Types...>::Tuple( Types&&... args ) : tuple<Types...>( move( args )... ) {}
template <typename...Types> template <typename...Args> inline Tuple<Types...>::Tuple( Args&&... args ) : tuple<Types...>( forward<Args>( args )... ) {}

template <typename...Types> template <int n> inline auto& Tuple<Types...>::operator[]( const TupleAccessIndex<n>& i ) noexcept { return get<n>( *this ); }
template <typename...Types> template <int n> inline const auto& Tuple<Types...>::operator[]( const TupleAccessIndex<n>& i ) const noexcept { return get<n>( *this ); }

template <typename RET , template <typename...> typename PAIR , typename INT> T2<RET> cast( const PAIR<INT,INT>& t ) { return {get<0>(t),get<1>(t)}; }
template <typename RET , typename INT> T3<RET> cast( const tuple<INT,INT,INT>& t ) { return {get<0>(t),get<1>(t),get<2>(t)}; }
template <typename RET , typename INT> T4<RET> cast( const tuple<INT,INT,INT,INT>& t ) { return {get<0>(t),get<1>(t),get<2>(t),get<3>(t)}; }

#define DEFINITION_OF_HASH_FOR_TUPLE( PAIR )				\
  template <typename T , typename U> inline size_t hash<PAIR<T,U>>::operator()( const PAIR<T,U>& n ) const { static const size_t seed = ( GetRand( 1e3 , 1e8 ) << 1 ) | 1; static const hash<T> h0; static const hash<U> h1; return ( h0( get<0>( n ) ) * seed ) ^ h1( get<1>( n ) ); } \

template <typename T> DECLARATION_OF_HASH( tuple<T> );
template <typename T , typename U> DECLARATION_OF_HASH( pair<T,U> );
template <typename T , typename U> DECLARATION_OF_HASH( tuple<T,U> );
template <typename T , typename U , typename V> DECLARATION_OF_HASH( tuple<T,U,V> );
template <typename T , typename U , typename V , typename W> DECLARATION_OF_HASH( tuple<T,U,V,W> );

template <typename T> inline size_t hash<tuple<T>>::operator()( const tuple<T>& n ) const { static const hash<T> h; return h(get<0>( n ) ); }
DEFINITION_OF_HASH_FOR_TUPLE( pair );
DEFINITION_OF_HASH_FOR_TUPLE( tuple );
template <typename T , typename U , typename V> inline size_t hash<tuple<T,U,V>>::operator()( const tuple<T,U,V>& n ) const { static const size_t seed = ( GetRand( 1e3 , 1e8 ) << 1 ) | 1; static const hash<pair<T,U>> h01; static const hash<V> h2; return ( h01( { get<0>( n ) , get<1>( n ) } ) * seed ) ^ h2( get<2>( n ) ); }
template <typename T , typename U , typename V , typename W> inline size_t hash<tuple<T,U,V,W>>::operator()( const tuple<T,U,V,W>& n ) const { static const size_t seed = ( GetRand( 1e3 , 1e8 ) << 1 ) | 1; static const hash<pair<T,U>> h01; static const hash<pair<V,W>> h23; return ( h01( { get<0>( n ) , get<1>( n ) } ) * seed ) ^ h23( { get<2>( n ) , get<3>( n ) } ); }

// Vector
#define DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , OPR )			\
  template <typename T> inline V<T>& operator OPR ## = ( V<T>& a0 , const V<T>& a1 ) { assert( a0.size() <= a1.size() ); auto itr0 = a0.begin() , end0 = a0.end(); auto itr1 = a1.begin(); while( itr0 != end0 ){ *( itr0++ ) OPR ## = *( itr1++ ); } return a0; } \
  template <typename T> inline V<T>& operator OPR ## = ( V<T>& a , const T& t ) { for( auto& x : a ){ x OPR## = t; } return a; } \
  template <typename T , typename U> inline V<T> operator OPR( V<T> a , const U& u ) { return move( a OPR ## = u ); } \

#define DEFINITION_OF_INCREMENT_FOR_VECTOR( V , INCR )			\
  template <typename T> inline V<T>& operator INCR( V<T>& a ) { for( auto& i : a ){ INCR i; } return a; } \

#define DEFINITION_OF_SHIFT_FOR_VECTOR( V )			\
  template <typename T> inline V<T>& operator<<=( V<T>& a , T t ) { a.push_back( move( t ) ); return a; } \
  template <typename T , typename U> inline V<T>& operator<<=( V<T>& a , U&& u ) { return a <<= T{ forward<U>( u ) }; } \
  template <typename T> inline T pop( V<T>& a ) { assert( !a.empty() ); T answer = move( a.back() ); a.pop_back(); return answer; } \

#define DEFINITION_OF_ARITHMETICS_FOR_VECTOR( V )			\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , + );                         \
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , - );				\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , * );				\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , / );				\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , % );				\
  DEFINITION_OF_INCREMENT_FOR_VECTOR( V , ++ );				\
  DEFINITION_OF_INCREMENT_FOR_VECTOR( V , -- );				\
  template <typename T> inline V<T> operator-( V<T> a ) { return move( a *= T( -1 ) );} \
  template <typename T> inline V<T> operator*( const T& t , V<T> v ) { return move( v *= t ); } \
  DEFINITION_OF_SHIFT_FOR_VECTOR( V );                                  \

DEFINITION_OF_ARITHMETICS_FOR_VECTOR( vector );
DEFINITION_OF_ARITHMETICS_FOR_VECTOR( list );
DEFINITION_OF_SHIFT_FOR_VECTOR( basic_string );

template <typename V> inline auto Get( V& a ) { return [&]( const int& i = 0 ) -> const decldecay_t( a[0] )& { return a[i]; }; }
template <typename T> inline vector<T> id( const int& size ) { vector<T> answer( size ); for( int i = 0 ; i < size ; i++ ){ answer[i] = i; } return answer; }

template <typename V> inline void Sort( V& a , const bool& reversed = false ) { using T = decldecay_t(a[0]); if( reversed ){ static auto comp = []( const T& t0 , const T& t1 ) { return t1 < t0; }; sort( a.begin() , a.end() , comp ); } else { sort( a.begin() , a.end() ); } }
template <typename V0 , typename V1> inline void Sort( V0& a , V1& b , const bool& reversed = false ) { const int size = a.size(); assert( size == int( b.size() ) ); vector<pair<decldecay_t(a[0]),decldecay_t(b[0])>> v( size ); for( int i = 0 ; i < size ; i++ ){ v[i] = { move( a[i] ) , move( b[i] ) }; } Sort( v , reversed ); for( int i = 0 ; i < size ; i++ ){ a[i] = move( v[i].first ); b[i] = move( v[i].second ); } }
template <typename V> inline pair<vector<int>,vector<int>> IndexSort( const V& a , const bool& reversed = false ) { const int size = a.size(); auto index = id<int>( size ) , ord = index; sort( index.begin() , index.end() , [&]( const int& i , const int& j ) { const pair ti{ a[i] , i } , tj{ a[j] , j }; return reversed ? tj < ti : ti < tj; } ); for( int i = 0 ; i < size ; i++ ){ ord[index[i]] = i; } return { move( index ) , move( ord ) }; }

template <typename V> inline int len( const V& a ) { return a.size(); }

template <typename V> inline void Reverse( V& a ) { const int size = len( a ) , half = size / 2; for( int i = 0 ; i < half ; i++ ){ swap( a[i] , a[size-1-i] ); } }
;
template <typename V> inline V Reversed( V a ) { Reverse( a ); return move( a ); }

template <typename RET , template <typename...> typename V , typename T> inline V<RET> cast( const V<T>& a ) { V<RET> answer{}; for( auto& x : a ){ answer <<= a; } }

#define DEFINITION_OF_COUT_FOR_VECTOR( V ) template <class Traits , typename Arg> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const V<Arg>& arg ) { auto begin = arg.begin() , end = arg.end(); auto itr = begin; while( itr != end ){ ( itr == begin ? os : os << " " ) << *itr; itr++; } return os; }

DEFINITION_OF_COUT_FOR_VECTOR( vector );
DEFINITION_OF_COUT_FOR_VECTOR( list );
DEFINITION_OF_COUT_FOR_VECTOR( set );
DEFINITION_OF_COUT_FOR_VECTOR( unordered_set );
DEFINITION_OF_COUT_FOR_VECTOR( multiset );

inline void VariadicResize( const int& size ) {}
template <typename Arg , typename... ARGS> inline void VariadicResize( const int& size , Arg& arg , ARGS&... args ) { arg.resize( size ); VariadicResize( size , args... ); }

// Map
#define DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , OPR )			\
  template <typename T , typename U> inline MAP<T,U>& operator OPR ## = ( MAP<T,U>& a , const pair<T,U>& v ) { a[v.first] OPR ## = v.second; return a; } \
  template <typename T , typename U> inline MAP<T,U>& operator OPR ## = ( MAP<T,U>& a0 , const MAP<T,U>& a1 ) { for( auto& [t,u] : a1 ){ a0[t] OPR ## = u; } return a0; } \
  template <typename T , typename U , typename ARG> inline MAP<T,U> operator OPR( MAP<T,U> a , const ARG& arg ) { return move( a OPR ## = arg ); } \

#define DEFINITION_OF_ARITHMETICS_FOR_MAP( MAP ) \
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , + );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , - );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , * );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , / );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , % );	\

template <typename T , typename U>
using Map = conditional_t<is_constructible_v<unordered_map<T,int>>,unordered_map<T,U>,conditional_t<is_ordered::value<T>,map<T,U>,void>>;

DEFINITION_OF_ARITHMETICS_FOR_MAP( map );
DEFINITION_OF_ARITHMETICS_FOR_MAP( unordered_map );

// StdStream
template <class Traits> inline basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is ) { return is; }
template <class Traits , typename Arg , typename... ARGS> inline basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is , Arg& arg , ARGS&... args ) { return VariadicCin( is >> arg , args... ); }
template <class Traits> inline basic_istream<char,Traits>& VariadicSet( basic_istream<char,Traits>& is , const int& i ) { return is; }
template <class Traits , typename Arg , typename... ARGS> inline basic_istream<char,Traits>& VariadicSet( basic_istream<char,Traits>& is , const int& i , Arg& arg , ARGS&... args ) { return VariadicSet( is >> arg[i] , i , args... ); }

template <class Traits> inline basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , const char& separator ) { return is; }
template <class Traits , typename Arg , typename... ARGS> inline basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , const char& separator , Arg& arg , ARGS&... args ) { return VariadicGetline( getline( is , arg , separator ) , separator , args... ); }

template <class Traits , typename Arg> inline basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , Arg&& arg ) { return os << forward<Arg>( arg ); }
template <class Traits , typename Arg1 , typename Arg2 , typename... ARGS> inline basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , Arg1&& arg1 , Arg2&& arg2 , ARGS&&... args ) { return VariadicCout( os << forward<Arg1>( arg1 ) << " " , forward<Arg2>( arg2 ) , forward<ARGS>( args )... ); }

template <class Traits , typename Arg> inline basic_ostream<char,Traits>& VariadicCoutNonSep( basic_ostream<char,Traits>& os , Arg&& arg ) { return os << forward<Arg>( arg ); }
template <class Traits , typename Arg1 , typename Arg2 , typename... ARGS> inline basic_ostream<char,Traits>& VariadicCoutNonSep( basic_ostream<char,Traits>& os , Arg1&& arg1 , Arg2&& arg2 , ARGS&&... args ) { return VariadicCoutNonSep( os << forward<Arg1>( arg1 ) , forward<Arg2>( arg2 ) , forward<ARGS>( args )... ); }

template <class Traits , typename ARRAY> inline basic_ostream<char,Traits>& CoutArray( basic_ostream<char,Traits>& os , const int& i_start , const int& i_ulim , ARRAY&& a ) { for( int i = i_start ; i < i_ulim ; i++ ){ ( i == i_start ? os : ( os << " " ) ) << a[i]; } return os; }

// Arithmetic/Iteration
#define SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( TYPE ) \
  template <> inline TYPE ArithmeticProgressionSum( const TYPE& l , const TYPE& r , const TYPE& d ) { return SpecialisedArithmeticProgressionSum( l , r , d ); } \

template <typename T , typename U , template <typename...> typename V , typename OPR> T LeftConnectiveProd( T t , const V<U>& f , OPR opr ) { for( auto& u : f ){ t = opr( move( t ) , u ); } return move( t ); }
template <typename T , typename U , template <typename...> typename V> inline T Sum( const V<U>& f ) { return LeftConnectiveProd( T{ 0 } , f , []( T t0 , const U& u1 ){ return move( t0 += u1 ); } ); }
template <typename T , typename U , template <typename...> typename V> inline T Prod( const V<U>& f ) { return LeftConnectiveProd( T{ 1 } , f , []( T t0 , const U& u1 ){ return move( t0 *= u1 ); } ); }

template <typename T> inline T& SetMax( T& t ) { return t; }
template <typename T , typename U , typename... Args> inline T& SetMax( T& t0 , const U& u1 , const Args&... args ) { return SetMax( t0 < u1 ? t0 = u1 : t0 , args... ); }
template <typename T> inline T& SetMin( T& t ) { return t; }
template <typename T , typename U , typename... Args> inline T& SetMin( T& t0 , const U& u1 , const Args&... args ) { return SetMin( u1 < t0 ? t0 = u1 : t0 , args... ); }

template <typename T> inline const T& Max( const vector<T>& f ) { return *max_element( f.begin() , f.end() ); }
template <typename T , template <typename...> typename SET> inline const T& Max( const SET<T>& f ) { return *--f.end(); }
template <typename T , typename U , typename...Args> inline T Max( T t0 , const U& t1 , const Args&... args ) { return move( SetMax( t0 , t1 , args... ) ); }
template <typename T> inline const T& Min( const vector<T>& f ) { return *min_element( f.begin() , f.end() ); }
template <typename T , template <typename...> typename SET> inline const T& Min( const SET<T>& f ) { return *f.begin(); }
template <typename T , typename U , typename...Args> inline T Min( T t0 , const U& t1 , const Args&... args ) { return move( SetMin( t0 , t1 , args... ) ); }

template <typename T , typename UINT>
T Power( const T& t , const UINT& exponent , T init = 1 )
{

  return exponent > 1 ? Power( t * t , exponent >> 1 , move( exponent & 1 ? init *= t : init ) ) : move( exponent > 0 ? init *= t : ( assert( exponent == 0 ) , init ) );

}

template <typename T> inline T PowerMemorisation( const T& t , const int& exponent ) { assert( exponent >= 0 ); static Map<T,vector<T>> memory{}; auto& answer = memory[t]; if( answer.empty() ){ answer.push_back( 1 ); } while( int( answer.size() ) <= exponent ){ answer.push_back( answer.back() * t ); } return answer[exponent]; }

template <typename INT> inline INT ArithmeticProgressionSum( const INT& l , const INT& r , const INT& d = 1 ) { return ( l + r ) * ( ( r - l ) / d + 1 ) / 2; }
template <typename INT> inline INT SpecialisedArithmeticProgressionSum( const INT& l , const INT& r , const INT& d = 1 ) { assert( l - 1 <= r ); const INT c = ( r - l ) / d; return l - 1 == r ? 0 : ( c & 1 ) == 0 ? ( c + 1 ) * ( l + d * ( c >> 1 ) ) : ( ( c + 1 ) >> 1 ) * ( ( l << 1 ) + d * c ); }
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( int );
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( uint );
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( ll );
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( ull );
template <typename INT> inline INT ArithmeticProgressionSum( const INT& r ) { return ArithmeticProgressionSum( INT{} , r ); }

template <typename INT> inline INT SquareSum( const INT& r ) { return r * ( r + 1 ) * ( 2 * r + 1 ) / 6; }

template <typename T , typename UINT> inline T GeometricProgressionSum( T rate , UINT exponent_max , const T& init ) { T rate_minus = rate - 1; return rate_minus == 0 ? init * ++exponent_max : ( Power( move( rate ) , move( ++exponent_max ) ) - 1 ) / move( rate_minus ) * init; }

template <typename T , typename UINT>
T GeometricProgressionLinearCombinationSum( vector<T> rate , vector<UINT> exponent_max , const vector<T>& init )
{

  const int size = init.size();
  assert( int( rate.size() ) == size && int( exponent_max.size() ) == size );
  T answer{};

  for( int i = 0 ; i < size ; i++ ){

    answer += GeometricProgressionSum( move( rate[i] ) , move( exponent_max[i] ) , init[i] );

  }

  return answer;

}

// Arithmetic/Sqrt
template <typename INT>
INT RoundDownSqrt( const INT& n )
{

  static_assert( is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull> );
  assert( n >= 0 );
  
  if( n <= 1 ){

    return n;
    
  }

  constexpr INT r_max = is_same_v<INT,int> ? 46341 : is_same_v<INT,uint> ? 65536 : is_same_v<INT,ll> ? 3037000500 : 4294967296;
  INT l = 1 , r = min( r_max , n );

  while( l < r - 1 ){

    const INT m = ( l + r ) >> 1;
    // m * m <= nか否かを判定。
    ( m <= n / m ? l : r ) = m;

  }

  return l;

}

template <typename INT>
INT RoundUpSqrt( const INT& n )
{
  
  static_assert( is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull> );
  assert( n >= 0 );

  if( n <= 2 ){

    return n;
    
  }

  constexpr INT r_max = is_same_v<INT,int> ? 46341 : is_same_v<INT,uint> ? 65536 : is_same_v<INT,ll> ? 3037000500 : 4294967296;
  const INT n_minus = n - 1;
  INT l = 1 , r = min( r_max , n );

  while( l + 1 < r ){

    const INT m = ( l + r ) >> 1;
    // m * m < nか否かを判定。
    ( m <= n_minus / m ? l : r ) = m;

  }

  return r;

}

template <typename INT> bool IsSquare( const INT& n ) { const INT r = RoundDownSqrt( n ); return n == r * r; }

/* AAA 常設ライブラリの非圧縮版は以上に挿入する。*/

// デバッグ用
#ifdef DEBUG
  inline void AlertAbort( int n ) { CERR( "abort関数が呼ばれました。assertマクロのメッセージが出力されていない場合はオーバーフローの有無を確認をしてください。" ); }
#endif

// 入力フォーマットチェック用
// 1行中の変数の個数をSEPARATOR区切りで確認
#define GETLINE_COUNT( S , SEPARATOR ) GETLINE( S ); int S ## _count = 0; int VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S = 0; int VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S = S.size(); for( int i = 0 ; i < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ; i++ ){ if( S[i] == SEPARATOR ){ S ## _count++; } } if( VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S > 0 ){ S ## _count++; }
#define GETLINE_COUNT_ASSERT( S , SEPARATOR , COUNT ) GETLINE_COUNT( S , SEPARATOR ); assert( S ## _count == COUNT )
// 余計な入力の有無を確認
#if defined( DEBUG ) || defined( REACTIVE )
  #define CHECK_REDUNDANT_INPUT 
#else
  #ifdef USE_GETLINE
    #define CHECK_REDUNDANT_INPUT string VARIABLE_FOR_CHECK_REDUNDANT_INPUT = ""; getline( cin , VARIABLE_FOR_CHECK_REDUNDANT_INPUT ); assert( VARIABLE_FOR_CHECK_REDUNDANT_INPUT == "" ); assert( ! cin )
  #else
    #define CHECK_REDUNDANT_INPUT string VARIABLE_FOR_CHECK_REDUNDANT_INPUT = ""; cin >> VARIABLE_FOR_CHECK_REDUNDANT_INPUT; assert( VARIABLE_FOR_CHECK_REDUNDANT_INPUT == "" ); assert( ! cin )
  #endif
#endif
// MIN <= N <= MAXを満たすNをSから構築
#define STOI( S , N , MIN , MAX ) decldecay_t( MAX ) N = 0; decldecay_t( MAX ) BOUND ## N = max( decldecay_t( MAX )( abs( MIN ) ) , abs( MAX ) ); { bool VARIABLE_FOR_POSITIVITY_FOR_GETLINE = true; assert( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ); if( S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) == "-" ){ VARIABLE_FOR_POSITIVITY_FOR_GETLINE = false; VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; assert( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ); } assert( S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) != " " ); string VARIABLE_FOR_LETTER_FOR_GETLINE{}; int VARIABLE_FOR_DIGIT_FOR_GETLINE{}; while( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ? ( VARIABLE_FOR_LETTER_FOR_GETLINE = S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) ) != " " : false ){ VARIABLE_FOR_DIGIT_FOR_GETLINE = stoi( VARIABLE_FOR_LETTER_FOR_GETLINE ); assert( N < BOUND ## N / 10 ? true : N == BOUND ## N / 10 && VARIABLE_FOR_DIGIT_FOR_GETLINE <= BOUND ## N % 10 ); N = N * 10 + VARIABLE_FOR_DIGIT_FOR_GETLINE; VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } if( ! VARIABLE_FOR_POSITIVITY_FOR_GETLINE ){ N *= -1; } if( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ){ VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } ASSERT( N , MIN , MAX ); }
#define STOI_A( S , I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); FOR( VARIABLE_FOR_STOI_A , 0 , N ){ STOI( S , A ##_VARIABLE_FOR_STOI_A , MIN , MAX ); A[VARIABLE_FOR_STOI_A + I] = A ##_VARIABLE_FOR_STOI_A; }
// Sをstring SEPARATORで区切りTを構築
#define SEPARATE( S , T , SEPARATOR ) string T{}; { assert( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ); int VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S_prev = VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S; assert( S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) != SEPARATOR ); string VARIABLE_FOR_LETTER_FOR_GETLINE{}; while( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ? ( VARIABLE_FOR_LETTER_FOR_GETLINE = S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) ) != SEPARATOR : false ){ VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } T = S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S_prev , VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S - VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S_prev ); if( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ){ VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } }

#define INCLUDE_LIBRARY
#include __FILE__
#endif // INCLUDE_LIBRARY
#endif // INCLUDE_MAIN