結果

問題 No.3033 エルハートの数え上げ
ユーザー 👑 p-adic
提出日時 2025-02-05 17:22:18
言語 C++17(gcc12)
(gcc 12.3.0 + boost 1.87.0)
結果
AC  
実行時間 33 ms / 2,000 ms
コード長 41,829 bytes
コンパイル時間 16,485 ms
コンパイル使用メモリ 319,128 KB
実行使用メモリ 6,820 KB
最終ジャッジ日時 2025-02-21 20:52:06
合計ジャッジ時間 18,057 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge4
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 3
other AC * 20
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ソースコード

diff #
プレゼンテーションモードにする

// 入力制約／フォーマットチェックその１（下限／上限のみチェック）
#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  #define USE_GETLINE
#endif
#ifdef INCLUDE_MAIN
void Solve()
{
  CEXPR( int , bound_NABCD , 1e9 );
  CEXPR( int , bound_M , 20 );
  GETLINE_COUNT( NM_str , 2 , ' ' );
  STOI( NM_str , N , 1 , bound_NABCD );
  STOI( NM_str , M , 4 , bound_M );
  vector<T4<ll>> ABCD( M );
  FOR( m , 0 , M ){
    GETLINE_COUNT( ABCDm_str , 4 , ' ' );
    STOI( ABCDm_str , A , -bound_NABCD , bound_NABCD );
    STOI( ABCDm_str , B , -bound_NABCD , bound_NABCD );
    STOI( ABCDm_str , C , -bound_NABCD , bound_NABCD );
    STOI( ABCDm_str , D , -bound_NABCD , bound_NABCD );
    ABCD[m] = {A,B,C,D};
  }
  auto Belong = [&]( const int& x , const int& y , const int& z , const int& n ){
    bool b = true;
    RUN( ABCD , [A,B,C,D] ){
      b &= A * x + B * y + C * z + D * n >= 0;
    }
    return b;
  };
  auto Naive = [&]( const int& n ){
    int a = 0;
    int bound = 15 * n;
    FOREQ( x , -bound , bound ){
      FOREQ( y , -bound , bound ){
        FOREQ( z , -bound , bound ){
          Belong( x , y , z , n ) ? ++a : a;
        }
      }
    }
    return a;
  };
  vector<MP> arg( 4 );
  vector<MP> val( 4 );
  FOR( i , 0 , 4 ){
    arg[i] = i;
    val[i] = Naive( i );
  }
  auto coef = LagrangeInterpolation( arg , val );
  RETURN( -PolynoimialEvaluation<MP>( coef , -N ) );
}
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。redefinitionを避けるため圧縮元はincludeしない。
/* 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
/* ConstexprModulo (7KB)*/
CEXPR(uint,P,998244353);
#define RP Represent
#define DeRP Derepresent
TE <uint M,TY INT> CE INT Residue(INT n)NE{RE MO(n < 0?((((++n)*= -1)%= M)*= -1)+= M - 1:n < INT(M)?n:n %= M);}TE <TY INT> CE INT& ResidueP(INT& n
    )NE{CE CO uint trunc =(1 << 23)- 1;INT n_u = n >> 23;n &= trunc;INT n_uq =(n_u / 7)/ 17;n_u -= n_uq * 119;n += n_u << 23;RE n < n_uq?n += P - 
    n_uq:n -= n_uq;}
TE <uint M> CL Mod;TE <uint M>CL COantsForMod{PU:COantsForMod()= delete;ST CE CO uint g_memory_bound = 1e6;ST CE CO uint g_memory_LE = M < 
    g_memory_bound?M:g_memory_bound;ST CE uint g_M_minus = M - 1;ST CE int g_order = M - 1;ST CE int g_order_minus = g_order - 1;};
#define SFINAE_FOR_MOD enable_if_t<is_COructible_v<uint,decay_t<T>>>*
#define DC_OF_CM_FOR_MOD(OPR)CE bool OP OPR(CO Mod<M>& n)CO NE
#define DC_OF_AR_FOR_MOD(OPR,EX)CE Mod<M> OP OPR(Mod<M> n)CO EX;
#define DF_OF_CM_FOR_MOD(OPR)TE <uint M> CE bool Mod<M>::OP OPR(CO Mod<M>& n)CO NE{RE m_n OPR n.m_n;}
#define DF_OF_AR_FOR_MOD(OPR,EX,LEFT,OPR2)TE <uint M> CE Mod<M> Mod<M>::OP OPR(Mod<M> n)CO EX{RE MO(LEFT OPR2 ## = *TH);}TE <uint M,TY T
    ,SFINAE_FOR_MOD = nullptr> CE Mod<M> OP OPR(T n0,CO Mod<M>& n1)EX{RE MO(Mod<M>(MO(n0))OPR ## = n1);}
TE <uint M>CL Mod{PU:uint m_n;CE Mod()NE;CE Mod(CO Mod<M>& n)NE;CE Mod(Mod<M>&& n)NE;TE <TY T,SFINAE_FOR_MOD = nullptr> CE Mod(T n)NE;CE Mod<M>& OP
    =(Mod<M> n)NE;CE Mod<M>& OP+=(CO Mod<M>& n)NE;CE Mod<M>& OP-=(CO Mod<M>& n)NE;CE Mod<M>& OP*=(CO Mod<M>& n)NE;IN Mod<M>& OP/=(Mod<M> n);TE <TY 
    INT> CE Mod<M>& OP<<=(INT n);TE <TY INT> CE Mod<M>& OP>>=(INT n);CE Mod<M>& OP++()NE;CE Mod<M> OP++(int)NE;CE Mod<M>& OP--()NE;CE Mod<M> OP--(int
    )NE;DC_OF_CM_FOR_MOD(==);DC_OF_CM_FOR_MOD(!=);DC_OF_CM_FOR_MOD(<);DC_OF_CM_FOR_MOD(<=);DC_OF_CM_FOR_MOD(>);DC_OF_CM_FOR_MOD(>=);DC_OF_AR_FOR_MOD
    (+,NE);DC_OF_AR_FOR_MOD(-,NE);DC_OF_AR_FOR_MOD(*,NE);DC_OF_AR_FOR_MOD(/,);TE <TY INT> CE Mod<M> OP^(INT EX)CO;TE <TY INT> CE Mod<M> OP<<(INT n)CO
    ;TE <TY INT> CE Mod<M> OP>>(INT n)CO;CE Mod<M> OP-()CO NE;CE Mod<M>& SignInvert()NE;IN Mod<M>& Invert();TE <TY INT> CE Mod<M>& PW(INT EX);CE VO 
    swap(Mod<M>& n)NE;CE CRUI RP()CO NE;ST CE Mod<M> DeRP(uint n)NE;ST IN CO Mod<M>& Inverse(CRUI n);ST IN CO Mod<M>& Factorial(CRUI n);ST IN CO Mod
    <M>& FactorialInverse(CRUI n);ST IN Mod<M> Combination(CRUI n,CRUI i);ST IN CO Mod<M>& zero()NE;ST IN CO Mod<M>& one()NE;ST IN CE uint GetModulo
    ()NE;TE <TY INT> CE Mod<M>& PositivePW(INT EX)NE;TE <TY INT> CE Mod<M>& NonNegativePW(INT EX)NE;US COants = COantsForMod<M>;};
US MP = Mod<P>;
TE <uint M> CE Mod<M>::Mod()NE:m_n(){}TE <uint M> CE Mod<M>::Mod(CO Mod<M>& n)NE:m_n(n.m_n){}TE <uint M> CE Mod<M>::Mod(Mod<M>&& n)NE:m_n(MO(n.m_n
    )){}TE <uint M> TE <TY T,SFINAE_FOR_MOD> CE Mod<M>::Mod(T n)NE:m_n(Residue<M>(MO(n))){}TE <uint M> CE Mod<M>& Mod<M>::OP=(Mod<M> n)NE{m_n = MO(n
    .m_n);RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP+=(CO Mod<M>& n)NE{(m_n += n.m_n)< M?m_n:m_n -= M;RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP-=(CO 
    Mod<M>& n)NE{m_n < n.m_n?(m_n += M)-= n.m_n:m_n -= n.m_n;RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP*=(CO Mod<M>& n)NE{m_n = MO(ull(m_n)* n.m_n)% M
    ;RE *TH;}TE <> CE MP& MP::OP*=(CO MP& n)NE{ull m_n_copy = m_n;m_n = MO((m_n_copy *= n.m_n)< P?m_n_copy:ResidueP(m_n_copy));RE *TH;}TE <uint M> IN 
    Mod<M>& Mod<M>::OP/=(Mod<M> n){RE OP*=(n.Invert());}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::OP<<=(INT n){AS(n >= 0);RE *TH *= Mod<M>(2
    ).NonNegativePW(MO(n));}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::OP>>=(INT n){AS(n >=0);WH(n-- > 0){((m_n & 1)== 0?m_n:m_n += M)>>= 1;}RE *TH
    ;}TE <uint M> CE Mod<M>& Mod<M>::OP++()NE{m_n < COants::g_M_minus?++m_n:m_n = 0;RE *TH;}TE <uint M> CE Mod<M> Mod<M>::OP++(int)NE{Mod<M> n{*TH}
    ;OP++();RE n;}TE <uint M> CE Mod<M>& Mod<M>::OP--()NE{m_n == 0?m_n = COants::g_M_minus:--m_n;RE *TH;}TE <uint M> CE Mod<M> Mod<M>::OP--(int
    )NE{Mod<M> n{*TH};OP--();RE n;}DF_OF_CM_FOR_MOD(==);DF_OF_CM_FOR_MOD(!=);DF_OF_CM_FOR_MOD(>);DF_OF_CM_FOR_MOD(>=);DF_OF_CM_FOR_MOD
    (<);DF_OF_CM_FOR_MOD(<=);DF_OF_AR_FOR_MOD(+,NE,n,+);DF_OF_AR_FOR_MOD(-,NE,n.SignInvert(),+);DF_OF_AR_FOR_MOD(*,NE,n,*);DF_OF_AR_FOR_MOD(/,,n
    .Invert(),*);TE <uint M> TE <TY INT> CE Mod<M> Mod<M>::OP^(INT EX)CO{RE MO(Mod<M>(*TH).PW(MO(EX)));}TE <uint M> TE <TY INT> CE Mod<M> Mod<M>::OP
    <<(INT n)CO{RE MO(Mod<M>(*TH)<<= MO(n));}TE <uint M> TE <TY INT> CE Mod<M> Mod<M>::OP>>(INT n)CO{RE MO(Mod<M>(*TH)>>= MO(n));}TE <uint M> CE Mod
    <M> Mod<M>::OP-()CO NE{RE MO(Mod<M>(*TH).SignInvert());}TE <uint M> CE Mod<M>& Mod<M>::SignInvert()NE{m_n > 0?m_n = M - m_n:m_n;RE *TH;}TE <uint 
    M> IN Mod<M>& Mod<M>::Invert(){AS(m_n != 0);uint m_n_neg;RE m_n < COants::g_memory_LE?(m_n = Inverse(m_n).m_n,*TH):((m_n_neg = M - m_n)< COants
    ::g_memory_LE)?(m_n = M - Inverse(m_n_neg).m_n,*TH):NonNegativePW(COants::g_order_minus);}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::PositivePW
    (INT EX)NE{Mod<M> PW{*TH};EX--;WH(EX != 0){(EX & 1)== 1?*TH *= PW:*TH;EX >>= 1;PW *= PW;}RE *TH;}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M
    >::NonNegativePW(INT EX)NE{RE EX == 0?(m_n = 1,*TH):PositivePW(MO(EX));}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::PW(INT EX){bool neg = EX < 0
    ;AS(!(neg && m_n == 0));RE NonNegativePW(MO(neg?(EX %= COants::g_order)== 0?EX:EX += COants::g_order:EX));}TE <uint M> CE VO Mod<M>::swap(Mod<M>& 
    n)NE{std::swap(m_n,n.m_n);}TE <uint M> IN CO Mod<M>& Mod<M>::Inverse(CRUI n){AS(n < M);ST VE<Mod<M>> memory ={zero(),one()};ST uint LE_curr = 2
    ;WH(LE_curr <= n){memory.push_back(DeRP(M - memory[M % LE_curr].m_n * ull(M / LE_curr)% M));LE_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& 
    Mod<M>::Factorial(CRUI n){if(M <= n){RE zero();}ST VE<Mod<M>> memory ={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory.push_back
    (memory[LE_curr - 1]* LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::FactorialInverse(CRUI n){ST VE<Mod<M>> memory ={one
    (),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory.push_back(memory[LE_curr - 1]* Inverse(LE_curr));LE_curr++;}RE memory[n];}TE <uint M> IN 
    Mod<M> Mod<M>::Combination(CRUI n,CRUI i){RE i <= n?Factorial(n)* FactorialInverse(i)* FactorialInverse(n - i):zero();}TE <uint M> CE CRUI Mod<M
    >::RP()CO NE{RE m_n;}TE <uint M> CE Mod<M> Mod<M>::DeRP(uint n)NE{Mod<M> n_copy{};n_copy.m_n = MO(n);RE n_copy;}TE <uint M> IN CO Mod<M>& Mod<M
    >::zero()NE{ST CE CO Mod<M> z{};RE z;}TE <uint M> IN CO Mod<M>& Mod<M>::one()NE{ST CE CO Mod<M> o{1};RE o;}TE <uint M> IN CE uint Mod<M
    >::GetModulo()NE{RE M;}TE <uint M> IN Mod<M> Inverse(CO Mod<M>& n){RE MO(Mod<M>(n).Invert());}TE <uint M,TY INT> CE Mod<M> PW(Mod<M> n,INT EX){RE 
    MO(n.PW(MO(EX)));}TE <uint M> CE VO swap(Mod<M>& n0,Mod<M>& n1)NE{n0.swap(n1);}TE <uint M> IN string to_string(CO Mod<M>& n)NE{RE to_string(n.RP
    ())+ " + " + to_string(M)+ "Z";}TE <uint M,CL Traits> IN IS& OP>>(IS& is,Mod<M>& n){ll m;is >> m;n = m;RE is;}TE <uint M,CL Traits> IN OS& OP
    <<(OS& os,CO Mod<M>& n){RE os << n.RP();}
#define DF_OF_EXTENED_REDUCED_ROW_ECHELON_FORM_FOR_MOD(DECL_J)CO MODINT& zero = MODINT::zero();CO int M_N = L == 0?0:A[0].SZ(),M = M_N - N;AS(M >= 0
    );int rank = RowEchelonForm(A);VE<bool> solvable(N,true);int i = rank;WH(--i >= 0){auto& A_i = A[i];DECL_J;WH(++j < M){if(A_i[j]!= zero){break
    ;}}if(j == M){WH(j < M_N){solvable[j]= solvable[j]&& A_i[j]== zero;j++;}rank--;}else{int i_curr = i;WH(--i_curr >= 0){auto& A_i_curr = A[i_curr]
    ;CO MODINT A_i_curr_j = A_i_curr[j];for(int j_curr = j;j_curr < M_N;j_curr++){A_i_curr[j_curr]-= A_i_curr_j * A_i[j_curr];}}}}
TE <TY MODINT>int RowEchelonForm(VE<VE<MODINT>>& A){CO MODINT& zero = MODINT::zero();CO int L = A.SZ(),M = L == 0?0:A[0].SZ();int i_min = 0,i_curr
    ,j_curr = 0;WH(i_min < L && j_curr < M){i_curr = i_min;WH(i_curr < L && A[i_curr][j_curr]== zero){i_curr++;}if(i_curr < L){swap(A[i_min]
    ,A[i_curr]);auto& A_i_min = A[i_min];MODINT inv = 1 / A_i_min[j_curr];for(int j = j_curr;j < M;j++){A_i_min[j]*= inv;}for(int i = i_min + 1;i < L
    ;i++){auto& A_i = A[i];CO MODINT& A_i_j_curr = A_i[j_curr];if(A_i_j_curr != zero){for(int j = M - 1;j >= j_curr;j--){A_i[j]-= A_i_j_curr * 
    A_i_min[j];}}}i_min++;}j_curr++;}RE i_min;}TE <TY MODINT>pair<int,VE<MODINT>> ExtendedReducedRowEchelonForm(VE<VE<MODINT>>& A){CO int L = A.SZ
    ();CE int N = 1;VE<int> left(L,-1);DF_OF_EXTENED_REDUCED_ROW_ECHELON_FORM_FOR_MOD(int& j = left[i]);VE<MODINT> solution{};if(solvable[0]
    ){solution.resize(M);i = rank;WH(--i >= 0){auto& A_i = A[i];CRI j = left[i];solution[j]= A_i[M];}}RE{rank,MO(solution)};}TE <TY MODINT>tuple<int
    ,VE<bool>,VE<VE<MODINT>>> MultiExtendedReducedRowEchelonForm(VE<VE<MODINT>>& A,CRI N){CO int L = A.SZ();VE<int> left(L,-1
    );DF_OF_EXTENED_REDUCED_ROW_ECHELON_FORM_FOR_MOD(int& j = left[i]);VE<VE<MODINT>> solutions(M,VE<MODINT>(N));i = rank;WH(--i >= 0){auto& A_i = 
    A[i];CRI j = left[i];auto& solutions_j = solutions[j];for(int k = 0;k < N;k++){solutions_j[k]= A_i[M + k];}}RE{rank,MO(solvable),MO(solutions)}
    ;}TE <TY MODINT> IN int ReducedRowEchelonForm(VE<VE<MODINT>>& A){RE get<0>(MultiExtendedReducedRowEchelonForm(A,0));}TE <TY MODINT> IN int Rank
    (VE<VE<MODINT>> A){RE ReducedRowEchelonForm(A);}TE <TY MODINT>VE<VE<MODINT>> Inverse(CO VE<VE<MODINT>>& A){CO int L = A.SZ();VE A_copy(L,VE
    <MODINT>(L + L));for(int i = 0;i < L;i++){auto& A_i = A[i];auto& A_copy_i = A_copy[i];for(int j = 0;j < L;j++){A_copy_i[j]= A_i[j];}for(int j = 0
    ;j < L;j++){A_copy_i[L + j]= i == j?1:0;}}auto[rank,solvable,AN]= MultiExtendedReducedRowEchelonForm(A_copy,AN,L);if(rank != L){AN.clear();}RE AN
    ;}TE <TY MODINT>pair<int,VE<MODINT>> LinearRelation(VE<VE<MODINT>>& A){CO int L = A.SZ();CO int M = A.empty()?0:A[0].SZ();CO int rank = 
    ReducedRowEchelonForm(A);VE<MODINT> coeff{};if(rank < M){coeff.resize(M);CO MODINT& zero = MODINT::zero();int j = 0;WH(j < L && A[j][j]!= zero){j
    ++;}for(int i = 0;i < j;i++){coeff[i]= -A[i][j];}coeff[j]= 1;}RE{rank,MO(coeff)};}
// Tに合わせたExtendedReducedRowEchelonForm関数とRank関数を先にincludeする。
TE <TY T>VE<T> LagrangeInterpolation(CO VE<T>& arg,CO VE<T>& val){CO int M = arg.SZ();AS(M > 0 && M == int(val.SZ()));VE A(M,VE<T>(M + 1,1));for(int 
    i = 0;i < M;i++){for(int j = 1;j < M;j++){A[i][j]= A[i][j-1]* arg[i];}A[i][M]= val[i];}auto[rank,coef]= ExtendedReducedRowEchelonForm(A);AS(rank 
    == M);RE coef;}TE <TY T>T PolynoimialEvaluation(CO VE<T>& coef,CO T& arg){T AN{},pw{1};for(auto& c:coef){AN += c * pw;pw *= arg;}RE AN;}
// 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 DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE2 )
  #define SIGNAL signal( SIGABRT , &AlertAbort );
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) 
      ); assert( ( MIN ) <= A && A <= ( MAX ) )
  #define COUT( ... ) VariadicCout( cout << "出力：" , __VA_ARGS__ ) << endl
  #define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ ) << flush
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define CERRNS( ... ) VariadicCout( cerr , __VA_ARGS__ ) << flush
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout << "出力："  , A , N ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << 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 ) assert( ( MIN ) <= A && A <= ( MAX ) )
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ )
  #define CERR( ... ) 
  #define CERRNS( ... ) 
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout , A , N ) << ENDL
  #define CERR_A( A , N ) 
#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_LL( A ) { GETLINE( A ## _str ); A = stoll( A ## _str ); }
  #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
#else
  #define SET_LL( A ) cin >> A
  #define CIN( LL , ... ) LL __VA_ARGS__; VariadicCin( cin , __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 , ... ) vector<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
  #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) vector<vector<LL>> VAR( N0 + I0 ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , 
      VAR[VARIABLE_FOR_CIN_AA + I0] ); }
#endif
#include <bits/stdc++.h>
using namespace std;
#define REPEAT_MAIN( BOUND ) int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr ); SIGNAL; CEXPR( int , bound_test_case_num , BOUND ); 
    int test_case_num = 1; if constexpr( bound_test_case_num > 1 ){ CERR( "テストケースの個数を入力してください。" ); SET_ASSERT( test_case_num , 1 , 
    bound_test_case_num ); } FOR( test_case , 0 , test_case_num ){ if constexpr( bound_test_case_num > 1 ){ CERR( "testcase" , test_case , ":" ); } 
    Solve(); CERR( "" ); } CHECK_REDUNDANT_INPUT; }
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now()
#define 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 < TL_MS - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) constexpr LL BOUND = VALUE
#define SET_ASSERT( A , MIN , MAX ) SET_LL( 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 FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( decldecay_t( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( decldecay_t( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( decldecay_t( INITIAL ) VAR = INITIAL ; VAR + 1 > FINAL ; VAR -- )
#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 )
#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。*/
// Random
ll GetRand( const int& Rand_min , const int& Rand_max ) { assert( Rand_min <= Rand_max ); ll answer = time( NULL ); return answer * rand() % ( 
    Rand_max + 1 - Rand_min ) + Rand_min; }
// Set
#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(); const T answer = move( *itr ); S.erase( itr ); 
      return answer; } \
  template <typename T> inline T pop_min( SET& S ) { assert( !S.empty() ); auto itr = S.begin(); const T answer = move( *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> inline SET& operator-=( SET& S , const T& t ) { S.erase( t ); return S; } \
  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 , const SET& S1 ) { for( auto& t : S1 ){ S0 += t; } return S0; } \
  template <typename T> inline SET operator|( SET S0 , const SET& S1 ) { return move( S0 |= 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 SET<T,Args...>& S , const T& t ) { return S.count
    ( t ) == 1; }
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( + );
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 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 ); }
#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_SCALAR_ACTION_FOR_VECTOR( V , OPR )           \
  template <typename T> inline V<T>& operator OPR ## = ( V<T>& a , const T& t ) { for( auto& s : a ){ a OPR ## = t; } return a; } \
#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 , 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_ARITHMETICS_FOR_VECTOR( V )           \
  template <typename T> inline V<T>& operator+=( V<T>& a , const T& t ) { a.push_back( t ); return a; } \
  DEFINITION_OF_SCALAR_ACTION_FOR_VECTOR( V , * );                      \
  DEFINITION_OF_SCALAR_ACTION_FOR_VECTOR( V , / );                      \
  DEFINITION_OF_SCALAR_ACTION_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*( const T& scalar , V<T> v ) { for( auto& t : v ){ t *= scalar; } return move( v ); } \
  template <typename T> inline T pop( V<T>& a ) { assert( !a.empty() ); T answer = move( a.back() ); a.pop_back(); return answer; } \
DEFINITION_OF_ARITHMETICS_FOR_VECTOR( vector );
DEFINITION_OF_ARITHMETICS_FOR_VECTOR( list );
template <typename V> inline auto Get( V& a ) { return [&]( const int& i = 0 ) -> const decldecay_t( a[0] )& { return a[i]; }; }
template <typename T = int> 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 T> inline void Sort( vector<T>& a , const bool& reversed = false ) { 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 T0 , typename T1> inline void Sort( vector<T0>& a , vector<T1>& b , const bool& reversed = false ) { const int size = a.size(); 
    assert( size == int( b.size() ) ); vector<pair<T0,T1>> 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 T> inline vector<int> IndexSort( const vector<T>& a , const bool& reversed = false ) { auto index = id<int>( a.size() ); if( 
    reversed ){ sort( index.begin() , index.end() , [&]( const int& i , const int& j ) { return a[j] < a[i]; } ); } else { sort( index.begin() , 
    index.end() , [&]( const int& i , const int& j ) { return a[i] < a[j]; } ); } return index; }
template <typename V> inline int len( const V& a ) { return a.size(); }
#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 );
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; }
/* 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 , VARIABLE_NUMBER , SEPARATOR ) GETLINE( S ); int VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S = 0; int 
    VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S  = S.size(); { int size = S.size(); int count = 0; for( int i = 0 ; i < size ; i++ ){ if( S[i] == 
    SEPARATOR ){ count++; } } assert( VARIABLE_NUMBER == 0 ? size == 0 : count + 1 == VARIABLE_NUMBER ); }
// 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
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