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 2025-02-02 16:15:17
言語 C++17(gcc12)
(gcc 12.3.0 + boost 1.87.0)
結果
CE  
(最新)
AC  
(最初)
実行時間 -
コード長 67,796 bytes
コンパイル時間 30,239 ms
コンパイル使用メモリ 7,076 KB
最終ジャッジ日時 2025-02-21 20:50:59
合計ジャッジ時間 30,718 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge1
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ソースコード

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プレゼンテーションモードにする

#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  #define DEBUG_OUTPUT
#endif
#ifdef INCLUDE_MAIN
VO Solve()
{
  // えっ知らない・・・。
  // https://math.mit.edu/~rstan/transparencies/mathclub12.pdfを参考に実装。
  CIN( int , N , M );
  CIN_A( T4<ll> , 0 , M , ABCD );
  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;
  };
  vector<vector<MP>> solution{};
  auto Naive = [&]( const int& n ){
    int a = 0;
    int bound = 15 * n;
    FOREQ( x , -bound , bound ){
      FOREQ( y , -bound , bound ){
        FOREQ( z , -bound , bound ){
          if( Belong( x , y , z , n ) ){
            ++a;
            if( n == 1 ){
              solution += vector<MP>{x,y,z};
            }
          }
        }
      }
    }
    return a;
  };
  vector<MP> arg( 4 );
  vector<MP> val( 4 );
  FOR( i , 0 , 4 ){
    arg[i] = i + 1;
    val[i] = Naive( i + 1 );
  }
  int size = len( solution );
  assert( size > 0 );
  Polynomial<MP> f = LagrangeInterpolation( arg , val );
  CERR( val[0] , f( 1 ) );
  CERR( val[1] , f( 2 ) );
  CERR( val[2] , f( 3 ) );
  CERR( val[3] , f( 4 ) );
  FOR( i , 0 , size - 1 ){
    solution[i] -= solution[size-1];
  }
  solution.pop_back();
  int dimension = Rank( solution );
  CERR( dimension );
  RETURN( f( -N ) * ( ( dimension & 1 ) == 1 ? -1 : 1 ) );  
}
REPEAT_MAIN(1);
#else /* INCLUDE_MAIN */
#ifdef INCLUDE_SUB
/* 圧縮時は中身だけ削除する。*/
IN VO Experiment()
{
}
/* 圧縮時は中身だけ削除する。*/
IN VO SmallTest()
{
}
/* 圧縮時は中身だけ削除する。*/
IN VO RandomTest( const int& test_case_num )
{
}
#define INCLUDE_MAIN
#include __FILE__
#else /* INCLUDE_SUB */
#ifdef INCLUDE_LIBRARY
/* VVV 常設でないライブラリは以下に挿入する。*/
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Polynomial/Truncate/a_Body.hpp"
#else
#define PO Polynomial
#define TRPO TruncatedPolynomial
TE <TY T,int EX_lim>CL PW3PW_CE{PU:T m_val[EX_lim];CE PW3PW_CE(CO T& t);CE CO T& OP[](CRI i)CO;CE CO T(&Get()CO)[EX_lim];};
TE <TY T,int EX_lim> CE PW3PW_CE<T,EX_lim>::PW3PW_CE(CO T& t):m_val(){T PW{t};for(uint EX = EX_lim - 1;EX + 1 > 0;EX--){m_val[EX]= -PW;m_val[EX]*= PW 
    *= PW;}}TE <TY T,int EX_lim> CE CO T& PW3PW_CE<T,EX_lim>::OP[](CRI i)CO{AS(i < EX_lim);RE m_val[i];}TE <TY T,int EX_lim> CE CO T(&PW3PW_CE<T
    ,EX_lim>::Get()CO)[EX_lim]{RE m_val;}
#define PS_FOR_FFT(MOD,LE,BORDER,PR,IPR,MINT)ST_AS((MINT<MOD>::DeRP(PR)*= MINT<MOD>::DeRP(IPR))== MINT<MOD>::DeRP(1));TE <> CE CO uint 
    LimitOfPWForFFT<MINT<MOD> > = LE - 1;TE <> CE CO uint BorderForFFT<MINT<MOD> > = BORDER;TE <> IN CO MINT<MOD>(&PrimitiveRootOfTwoForFFT()NE
    )[LimitOfPWForFFT<MINT<MOD> >]{ST CE PW3PW_CE<MINT<MOD>,LimitOfPWForFFT<MINT<MOD> > > PRT{PR};ST_AS(PRT.m_val[0]== MINT<MOD>::DeRP(1));RE PRT.Get
    ();}TE <> IN CO MINT<MOD>(&InversePrimitiveRootOfTwoForFFT()NE)[LimitOfPWForFFT<MINT<MOD> >]{ST CE PW3PW_CE<MINT<MOD>,LimitOfPWForFFT<MINT<MOD> > 
    > IPRT{IPR};ST_AS(IPRT.m_val[0]== MINT<MOD>::DeRP(1)&&(MINT<MOD>::DeRP(PR)*= MINT<MOD>::DeRP(IPR))== MINT<MOD>::DeRP(1));RE IPRT.Get();}
TE <TY T> CE CO uint LimitOfPWForFFT{};TE <TY T> CE CO uint BorderForFFT{};TE <TY T> IN CO T(&PrimitiveRootOfTwoForFFT()NE)[LimitOfPWForFFT<T>];TE 
    <TY T> IN CO T(&InversePrimitiveRootOfTwoForFFT()NE)[LimitOfPWForFFT<T>];
PS_FOR_FFT(998244353,24,4,31,128805723,Mod);PS_FOR_FFT(167772161,26,4,17,29606852,Mod);PS_FOR_FFT(469762049,27,4,30,15658735,Mod);PS_FOR_FFT
    (754974721,25,4,362,415027540,Mod);
TE <TY T> VO CooleyTukey(VE<T>& f,CRUI N_input_start,CRUI N_input_lim,CRUI N_output_start,CRUI N_output_lim,CRUI two_PW,CRUI EX,CO T(&PRT
    )[LimitOfPWForFFT<T>]){CO uint LE = two_PW + N_input_start + N_output_start;f.reserve(LE);WH(f.SZ()< LE){f.push_back(0);}ST VE<uint> 
    bit_reverse[32]={VE<uint>(1)};ST uint e_next = 1;ST uint two_PW_next = 1;ST uint two_PW_next2 = 2;ST VE<uint>* p_bit_reverse_prev = bit_reverse
    ;ST VE<uint>* p_bit_reverse_curr = p_bit_reverse_prev + 1;WH(e_next <= EX){*p_bit_reverse_curr = VE<uint>(two_PW_next2);uint* 
    p_bit_reverse_curr_i = &((*p_bit_reverse_curr)[0]);uint* p_bit_reverse_curr_i_plus = p_bit_reverse_curr_i + two_PW_next;uint* 
    p_bit_reverse_prev_i = &((*p_bit_reverse_prev)[0]);for(uint i = 0;i < two_PW_next;i++){(*(p_bit_reverse_curr_i_plus++)= *(p_bit_reverse_curr_i
    ++)= *(p_bit_reverse_prev_i++)* 2)+= 1;}e_next++;swap(two_PW_next,two_PW_next2);two_PW_next2 *= 4;p_bit_reverse_prev++;p_bit_reverse_curr++;}CO 
    VE<uint>& bit_reverse_EX = bit_reverse[EX];uint bit_num = 0;CO uint* p_bit_num_reverse = &(bit_reverse_EX[bit_num]);WH(bit_num < two_PW){if
    (*p_bit_num_reverse < bit_num){swap(f[*p_bit_num_reverse + N_input_start],f[bit_num + N_input_start]);}bit_num++;p_bit_num_reverse++;}CO T& one = 
    PRT[0];T zeta,diff;uint i,j,j_lim,two_PW_curr = 1,two_PW_curr_2 = 2;WH(two_PW_curr < two_PW){CO uint N_input_start_plus = N_input_start + 
    two_PW_curr;bit_num = i = 0;zeta = one;WH(i < two_PW){j = i;j_lim = i + two_PW_curr;WH(j < j_lim){diff = f[j + N_input_start] - f[j + 
    N_input_start_plus];f[j + N_input_start] += f[j + N_input_start_plus];f[j + N_input_start_plus] = zeta * diff;j++;}bit_num++;i += two_PW_curr_2;j 
    = 0;WH(true){if(((bit_num >> j)& 1)== 1){zeta *= PRT[j+1];break;}j++;}}two_PW_curr <<= 1;two_PW_curr_2 <<= 1;}CO uint LE_fixed = N_output_lim + 
    N_input_start;WH(f.SZ()> LE_fixed){f.pop_back();}for(uint i = 0;i < N_output_start;i++){f[N_input_start + i]= 0;}RE;}
TE <TY T> IN VO FFT(VE<T>& f,CRUI N_input_start,CRUI N_input_lim,CRUI two_PW,CRUI EX){CooleyTukey<T>(f,N_input_start,N_input_lim,0,two_PW,two_PW,EX
    ,PrimitiveRootOfTwoForFFT<T>());}TE <TY T> IN VO FFT(VE<T>& f,CRUI N_input_start,CRUI N_input_lim,CRUI N_output_start,CRUI N_output_lim,CRUI 
    two_PW,CRUI EX){CooleyTukey<T>(f,N_input_start,N_input_lim,N_output_start,N_output_lim,two_PW,EX,PrimitiveRootOfTwoForFFT<T>());}TE <TY T> IN VO 
    IFFT(VE<T>& f,CRUI N_input_start,CRUI N_input_lim,CRUI two_PW,CO T& two_PW_inv,CRUI EX){CooleyTukey<T>(f,N_input_start,N_input_lim,0,two_PW
    ,two_PW,EX,InversePrimitiveRootOfTwoForFFT<T>());CO uint SZ = two_PW + N_input_start;for(uint i = N_input_start;i < SZ;i++){f[i]*= two_PW_inv
    ;}}TE <TY T> IN VO IFFT(VE<T>& f,CRUI N_input_start,CRUI N_input_lim,CRUI N_output_start,CRUI N_output_lim,CRUI two_PW,CO T& two_PW_inv,CRUI EX
    ){CooleyTukey<T>(f,N_input_start,N_input_lim,N_output_start,N_output_lim,two_PW,EX,InversePrimitiveRootOfTwoForFFT<T>());CO uint SZ = 
    N_output_lim + N_input_start;for(uint i = N_output_start + N_input_start;i < SZ;i++){f[i]*= two_PW_inv;}}
#define DC_OF_AR_FOR_PO(FUNC)IN PO<T> OP FUNC(PO<T> f)CO;IN PO<T> OP FUNC(T t)CO
#define DF_OF_AR_FOR_PO(FUNC,DEF)TE <TY T> IN PO<T> PO<T>::OP FUNC(PO<T> f)CO{RE MO(DEF);};TE <TY T> IN PO<T> PO<T>::OP FUNC(T t)CO{RE *TH FUNC PO<T
    >(MO(t));}
TE <TY T>CL TRPO;TE <TY T>CL PO{PU:VE<T> m_f;uint m_SZ;IN PO();IN PO(CO PO<T>& f);IN PO(PO<T>&& f);IN PO(VE<T> f);IN PO(T t);IN PO(CRUI i,T t);IN PO
    <T>& OP=(T n);IN PO<T>& OP=(PO<T> f);IN PO<T>& OP=(VE<T> f);IN CO T& OP[](CRUI i)CO;IN T& OP[](CRUI i);T OP()(CO T& t)CO;PO<T>& OP+=(CO PO<T>& f
    );PO<T>& OP-=(CO PO<T>& f);PO<T>& OP*=(CO PO<T>& f);PO<T>& OP*=(PO<T>&& f);IN PO<T>& OP/=(CO PO<T>& f);PO<T>& OP/=(CO T& t);PO<T>& OP%=(CO PO<T>& 
    f);PO<T>& OP%=(CO T& t);bool OP==(CO PO<T>& f)CO;bool OP==(CO T& t)CO;TE <TY P> IN bool OP!=(CO P& f)CO;DC_OF_AR_FOR_PO(+);IN PO<T> OP-()CO
    ;DC_OF_AR_FOR_PO(-);DC_OF_AR_FOR_PO(*);IN PO<T> OP/(CO PO<T>& f)CO;IN PO<T> OP/(CO T& t)CO;IN PO<T> OP%(CO PO<T>& f)CO;IN PO<T> OP%(CO T& t)CO;IN 
    CO VE<T>& GetCoefficient()CO NE;IN CRUI SZ()CO NE;IN VO resize(CRUI deg_plus)NE;IN VO swap(PO<T>& f);IN VO swap(VE<T>& f);VO ReMORedundantZero
    ();IN string Display()CO NE;ST PO<T> Quotient(CO PO<T>& f0,CO PO<T>& f1);ST PO<T> TP(CO PO<T>& f,CRUI f_TP_SZ);ST IN CO PO<T>& zero();ST IN CO PO
    <T>& one();ST IN CO PO<T>& x();ST IN CO T& c_zero();ST IN CO T& c_one();ST IN CO T& c_minus_one();IN PO<T>& SignInvert();};
#define DF_BODY_OF_PS_OF_MU_OF_PO_PROTH_MOD(TYPE,ARG,RHS)TE <> IN PO<TYPE>& PO<TYPE>::OP*=(ARG f){if(m_SZ != 0){VE<TYPE> v{};v.swap(m_f);TRPO<TYPE> 
    TH_copy{m_SZ + f.m_SZ - 1,MO(v)};TH_copy *= RHS;m_f = MO(TH_copy.PO<TYPE>::m_f);m_SZ = m_f.SZ();}RE *TH;}
#define RE_ZERO_FOR_MU_FOR_TR_PO_IF(CONDITION)if(CONDITION){RE OP=(zero);}
#define RE_ZERO_FOR_TR_MU_CO_FOR_TR_PO_IF(CONDITION)if(CONDITION){RE TRPO<T>(m_N);}
#define RE_ZERO_FOR__FOR_TR_PO_IF(MU,CONDITION)RE_ZERO_FOR_ ## MU ## _FOR_TR_PO_IF(CONDITION)
#define SET_VE_FOR_AN_OF_MU_FOR_TR_PO(N_OUTPUT_LIM)if(PO<T>::m_SZ < N_OUTPUT_LIM){for(uint i = PO<T>::m_SZ;i < N_OUTPUT_LIM;i++){PO<T>::m_f.push_back
    (0);}PO<T>::m_SZ = N_OUTPUT_LIM;}
#define SET_VE_FOR_AN_OF_TR_MU_CO_FOR_TR_PO(N_OUTPUT_LIM)VE<T> AN(N_OUTPUT_LIM)
#define SET_VE_FOR_AN_OF__FOR_TR_PO(MU,N_OUTPUT_LIM)SET_VE_FOR_AN_OF_ ## MU ## _FOR_TR_PO(N_OUTPUT_LIM)
#define SET_SUM_OF_MU_FOR_TR_PO PO<T>::m_f[i]= sum
#define SET_SUM_OF_TR_MU_CO_FOR_TR_PO AN[i]= sum
#define SET_SUM_OF__FOR_TR_PO(MU)SET_SUM_OF_ ## MU ## _FOR_TR_PO
#define SET_N_INPUT_START_FOR_MU_FOR_TR_PO(F,SZ,N_INPUT_START_NUM)uint N_INPUT_START_NUM{};for(uint i = 0;i < SZ && searching;i++){if(F[i]!= zero
    ){N_INPUT_START_NUM = i;searching = false;}}
#define SET_N_INPUT_MAX_FOR_MU_FOR_TR_PO(F,SZ,N_INPUT_MAX_NUM)uint N_INPUT_MAX_NUM{};searching = true;for(uint i =(SZ)- 1;searching;i--){if(F[i]!= 
    zero){N_INPUT_MAX_NUM = i;searching = false;}}
#define CN_FOR_MU_FOR_TR_PO(J_MIN)CO uint j_max = i < N_input_max_0_start_1?i - N_input_start_1:N_input_max_0;T sum{zero};for(uint j = J_MIN;j <= 
    j_max;j++){sum += PO<T>::m_f[j]* f.PO<T>::m_f[i - j];}PO<T>::m_f[i]= sum;
#define CN_FOR_TR_MU_CO_FOR_TR_PO(J_MIN)CO uint j_max = i < N_input_max_0_start_1?i - N_input_start_1:N_input_max_0;T& m_fi = AN[i];for(uint j = 
    J_MIN;j <= j_max;j++){m_fi += PO<T>::m_f[j]* f.PO<T>::m_f[i - j];}
#define CN_FOR__FOR_TR_PO(MU,J_MIN)CN_FOR_ ## MU ## _FOR_TR_PO(J_MIN)
#define ZEROIFICATION_FOR_MU_FOR_TR_PO for(uint i = 0;i < N_input_start_0_start_1;i++){PO<T>::m_f[i]= 0;}
#define ZEROIFICATION_FOR_TR_MU_CO_FOR_TR_PO CRUI N_output_start_fixed = N_output_start < N_input_start_0_start_1?N_output_start
    :N_input_start_0_start_1;for(uint i = 0;i < N_output_start_fixed;i++){AN[i]= 0;}
#define ZEROIFICATION_FOR__FOR_TR_PO(MU)ZEROIFICATION_FOR_ ## MU ## _FOR_TR_PO
#define DF_0_OF__FOR_TR_PO(MU,ACCESS_ENTRY,N_OUTPUT_START)RE_ZERO_FOR__FOR_TR_PO_IF(MU,PO<T>::m_SZ == 0);uint N_output_max = PO<T>::m_SZ + f.PO<T
    >::m_SZ - 2;if(N_output_max >= m_N){N_output_max = m_N - 1;}CO uint N_output_lim = N_output_max + 1;SET_VE_FOR_AN_OF__FOR_TR_PO(MU,N_output_lim
    );for(uint i = N_output_max;searching;i--){T sum{zero};for(uint j = 0;j <= i;j++){sum += ACCESS_ENTRY * f.PO<T>::OP[](i - j
    );}SET_SUM_OF__FOR_TR_PO(MU);searching = i > N_OUTPUT_START;}
#define DF_1_OF__FOR_TR_PO(MU)SET_N_INPUT_START_FOR_MU_FOR_TR_PO(PO<T>::m_f,PO<T>::m_SZ,N_input_start_0);RE_ZERO_FOR__FOR_TR_PO_IF(MU,searching
    );searching = true;SET_N_INPUT_START_FOR_MU_FOR_TR_PO(f,f.PO<T>::m_SZ,N_input_start_1);
#define SET_N_INPUT_RANGE SET_N_INPUT_MAX_FOR_MU_FOR_TR_PO(PO<T>::m_f,PO<T>::m_SZ,N_input_max_0);SET_N_INPUT_MAX_FOR_MU_FOR_TR_PO(f,f.PO<T>::m_SZ < 
    m_N?f.PO<T>::m_SZ:m_N,N_input_max_1);CO uint N_input_max_0_max_1 = N_input_max_0 + N_input_max_1;CO uint N_input_start_0_start_1 = 
    N_input_start_0 + N_input_start_1;uint N_output_lim_fixed = N_input_max_0_max_1 < m_N?N_input_max_0_max_1 + 1:m_N;
#define DF_3_OF__FOR_TR_PO(MU)CO uint N_input_start_0_max_1 = N_input_start_0 + N_input_max_1;CO uint N_input_max_0_start_1 = N_input_max_0 + 
    N_input_start_1;CO uint N_output_max_fixed = N_output_lim_fixed - 1;SET_VE_FOR_AN_OF__FOR_TR_PO(MU,N_output_lim_fixed);for(uint i = 
    N_output_max_fixed;i > N_input_start_0_max_1;i--){CN_FOR__FOR_TR_PO(MU,i - N_input_max_1);}searching = true;for(uint i = N_input_start_0_max_1 < 
    N_output_max_fixed?N_input_start_0_max_1:N_output_max_fixed;searching;i--){CN_FOR__FOR_TR_PO(MU,N_input_start_0);searching = i > 
    N_input_start_0_start_1;}ZEROIFICATION_FOR__FOR_TR_PO(MU);
#define SET_SHIFTED_VE_FOR_MU(V,F,I_START,I_MAX,I_SHIFT)VE<T> V(product_LE);for(uint i = I_START;i <= I_MAX;i++){V[I_SHIFT + i]= F[i];}
#define DF_OF_MU_FOR_TR_PO(RE_LINE_0,RE_LINE_1,RE_LINE_2,RE_LINE_3,RE_LINE_4,MU,ACCESS_ENTRY,N_OUTPUT_START,FIX_N_OUTPUT_LIM)CE CRUI border_0 = 
    FFT_MU_border_0<T>;CO T& zero = PO<T>::c_zero();bool searching = true;if(PO<T>::m_SZ < border_0 && f.PO<T>::m_SZ < border_0){RE_LINE_0
    ;DF_0_OF__FOR_TR_PO(MU,ACCESS_ENTRY,N_OUTPUT_START);RE_LINE_1;}DF_1_OF__FOR_TR_PO(MU);RE_LINE_2;SET_N_INPUT_RANGE;FIX_N_OUTPUT_LIM;RE_LINE_3
    ;DF_3_OF__FOR_TR_PO(MU);RE_LINE_4;
#define DF_OF_FFT_MU_FOR_TR_PO(RE_LINE_0,RE_LINE_1,RE_LINE_2,RE_LINE_3,RE_LINE_4,RE_LINE_5,MU,ACCESS_ENTRY,N_OUTPUT_START,N_OUTPUT_START_SHIFTED
    ,FIX_N_OUTPUT_LIM,DC_OF_F0,N_INPUT_START_0,N_INPUT_LIM_0,DC_OF_F1,N_INPUT_START_1,N_INPUT_LIM_1,VE_FOR_IFFT,RESZ_VE_FOR_IFFT,I_START,MU_FORMULA
    ,SET_AN)CE CRUI border_0 = FFT_MU_border_0<T>;CO T& zero = PO<T>::c_zero();bool searching = true;if(PO<T>::m_SZ < border_0 && f.PO<T>::m_SZ < 
    border_0){RE_LINE_0;DF_0_OF__FOR_TR_PO(MU,ACCESS_ENTRY,N_OUTPUT_START);RE_LINE_1;}DF_1_OF__FOR_TR_PO(MU);RE_LINE_2;SET_N_INPUT_RANGE
    ;FIX_N_OUTPUT_LIM;RE_LINE_3;CO uint N_input_TR_deg_0_deg_1 = N_input_max_0 - N_input_start_0 + N_input_max_1 - N_input_start_1;CE CRUI border_1 = 
    FFT_MU_border_1<T>;if(N_input_TR_deg_0_deg_1 < border_1){DF_3_OF__FOR_TR_PO(MU);RE_LINE_4;}uint two_PW = FFT_MU_border_1_2<T>;uint EX = 
    FFT_MU_border_1_2_EX<T>;T two_PW_inv{FFT_MU_border_1_2_inv<T>};WH(N_input_TR_deg_0_deg_1 >= two_PW){two_PW *= 2;two_PW_inv /= 2;EX++;}CO uint 
    product_LE = N_input_start_0_start_1 + two_PW;DC_OF_F0;DC_OF_F1;FFT<T>(f0,N_INPUT_START_0,N_INPUT_LIM_0,two_PW,EX);FFT<T>(f1,N_INPUT_START_1
    ,N_INPUT_LIM_1,two_PW,EX);RESZ_VE_FOR_IFFT;for(uint i = I_START + two_PW - 1;i + 1 > I_START;i--){MU_FORMULA;}CO uint N_output_lim_shifted = 
    N_output_lim_fixed - N_input_start_0_start_1;CO uint N_output_start_shifted = min(N_output_lim_shifted,uint(N_OUTPUT_START_SHIFTED));IFFT<T
    >(VE_FOR_IFFT,N_input_start_0_start_1,product_LE,N_output_start_shifted,N_output_lim_shifted,two_PW,two_PW_inv,EX);SET_AN;RE_LINE_5;
#define DF_OF_INVERSE_FOR_TR_PO(TYPE,RECURSION)CRUI N = f.GetTruncation();uint PW;uint PW_2 = 1;TRPO< TYPE > f_inv{PW_2,PO< TYPE >::c_one()/ f[0]};WH
    (PW_2 < N){PW = PW_2;PW_2 *= 2;f_inv.SetTruncation(PW_2);RECURSION;}f_inv.SetTruncation(N);RE f_inv
#define DF_OF_EXP_FOR_TR_PO(TYPE,RECURSION)AS(f[0]== PO< TYPE >::c_zero());CRUI N = f.GetTruncation();uint PW;uint PW_2 = 1;TRPO< TYPE > f_exp{PW_2
    ,PO< TYPE >::c_one()};WH(PW_2 < N){PW = PW_2;PW_2 *= 2;f_exp.SetTruncation(PW_2);RECURSION;}f_exp.SetTruncation(N);RE f_exp
#define DF_OF_PS_OF_MU_OF_TR_PO(TYPE,BORDER_0,BORDER_1,BORDER_1_2,BORDER_1_2_EX,BORDER_1_2_INV)TE <> CE CO uint FFT_MU_border_0< TYPE > = BORDER_0;TE 
    <> CE CO uint FFT_MU_border_1< TYPE > = BORDER_1;ST_AS(FFT_MU_border_0< TYPE > <= FFT_MU_border_1< TYPE >);TE <> CE CO uint FFT_MU_border_1_2< 
    TYPE > = BORDER_1_2;ST_AS(FFT_MU_border_1< TYPE > < FFT_MU_border_1_2< TYPE > && FFT_MU_border_1_2< TYPE > <= FFT_MU_border_1< TYPE > * 2 );TE <> 
    CE CO uint FFT_MU_border_1_2_EX< TYPE > = BORDER_1_2_EX;ST_AS(FFT_MU_border_1_2< TYPE > == 1 << FFT_MU_border_1_2_EX< TYPE > );TE <> CE CO uint 
    FFT_MU_border_1_2_inv< TYPE > = BORDER_1_2_INV;ST_AS((TYPE::DeRP(FFT_MU_border_1_2< TYPE >)*= TYPE::DeRP(FFT_MU_border_1_2_inv< TYPE >))== TYPE
    ::DeRP(1));TE <> IN TRPO< TYPE >& TRPO< TYPE >::OP*=(CO PO< TYPE >& f){RE TRPO< TYPE >::FFT_MU(f);}TE <> IN TRPO< TYPE >& TRPO< TYPE >::OP*=(PO< 
    TYPE >&& f){RE TRPO< TYPE >::FFT_MU(MO(f));}TE <> TRPO< TYPE > IN Inverse(CO TRPO< TYPE >& f){DF_OF_INVERSE_FOR_TR_PO(TYPE,f_inv.TRMinus(f_inv
    .FFT_TRMU_CO(f,PW,PW_2).FFT_TRMU(f_inv,PW,PW_2),PW,PW_2));}TE <> TRPO< TYPE > IN Exp(CO TRPO< TYPE >& f){DF_OF_EXP_FOR_TR_PO(TYPE,f_exp.TRMinus
    ((TRIntegral(Differential(f_exp).FFT_TRMU_CO(Inverse(f_exp),PW - 1,PW_2),PW).TRMinus(f,PW,PW_2)).FFT_TRMU(f_exp,PW,PW_2),PW,PW_2));}
#define DF_OF_PS_OF_MU_OF_PO_PROTH_MOD(MOD,BORDER_1_2_INV,MINT)DF_OF_PS_OF_MU_OF_TR_PO(MINT<MOD>,17,512,1024,10,BORDER_1_2_INV
    );DF_BODY_OF_PS_OF_MU_OF_PO_PROTH_MOD(MINT<MOD>,CO PO<MINT<MOD> >&,TH == &f?TH_copy:f);DF_BODY_OF_PS_OF_MU_OF_PO_PROTH_MOD(MINT<MOD>,PO<MINT<MOD> 
    >&&,MO(f));
TE <TY T>CL TRPO:PU PO<T>{PU:uint m_N;PU:IN TRPO(CRUI N = 0);IN TRPO(CO TRPO<T>& f);IN TRPO(TRPO<T>&& f);IN TRPO(CRUI N,T t);IN TRPO(CRUI N,CO PO<T>& 
    f);IN TRPO(CRUI N,PO<T>&& f);IN TRPO(CRUI N,VE<T>&& f);IN TRPO(CRUI N,CRUI i,T t);IN TRPO<T>& OP=(TRPO<T> f);IN TRPO<T>& OP=(T n);IN TRPO<T>& OP
    =(PO<T> f);IN TRPO<T>& OP+=(CO T& t);IN TRPO<T>& OP+=(CO PO<T>& f);IN TRPO<T>& OP+=(CO TRPO<T>& f);TRPO<T>& TRPlus(CO PO<T>& f,CRUI N_input_start
    ,CRUI N_input_limit);IN TRPO<T>& OP-=(CO T& t);IN TRPO<T>& OP-=(CO PO<T>& f);IN TRPO<T>& OP-=(CO TRPO<T>& f);TRPO<T>& TRMinus(CO PO<T>& f,CRUI 
    N_input_start,CRUI N_input_limit);IN TRPO<T>& OP*=(CO T& t);TRPO<T>& OP*=(CO PO<T>& f);IN TRPO<T>& OP*=(PO<T>&& f);TRPO<T>& FFT_MU(CO PO<T>& f
    );TRPO<T>& TRMU(CO PO<T>& f,CRUI N_output_start,CRUI N_output_lim);TRPO<T>& FFT_TRMU(CO PO<T>& f,CRUI N_output_start,CRUI N_output_lim);TRPO<T>& 
    FFT_TRMU(PO<T>&& f,CRUI N_output_start,CRUI N_output_lim);TRPO<T> TRMU_CO(CO PO<T>& f,CRUI N_output_start,CRUI N_output_lim)CO;TRPO<T> 
    FFT_TRMU_CO(CO PO<T>& f,CRUI N_output_start,CRUI N_output_lim)CO;TRPO<T> FFT_TRMU_CO(PO<T>&& f,CRUI N_output_start,CRUI N_output_lim)CO;IN TRPO<T
    >& OP/=(CO T& t);IN TRPO<T>& OP/=(CO TRPO<T>& t);IN TRPO<T>& OP%=(CO T& t);TE <TY P> IN TRPO<T> OP+(CO P& f)CO;IN TRPO<T> OP-()CO;TE <TY P> IN 
    TRPO<T> OP-(CO P& f)CO;TE <TY P> IN TRPO<T> OP*(CO P& f)CO;TE <TY P> IN TRPO<T> OP/(CO P& f)CO;IN TRPO<T> OP%(CO T& t)CO;IN VO SetTruncation(CRUI 
    N)NE;IN CRUI GetTruncation()CO NE;IN TRPO<T>& TruncateInitial(CRUI N)NE;IN TRPO<T>& TruncateFinal(CRUI N)NE;};TE <TY T> CE CO uint 
    FFT_MU_border_0 = 17;TE <TY T> CE CO uint FFT_MU_border_1{};TE <TY T> CE CO uint FFT_MU_border_1_2{};TE <TY T> CE CO uint FFT_MU_border_1_2_EX{}
    ;TE <TY T> CE CO uint FFT_MU_border_1_2_inv{};
TE <TY T> IN TRPO<T>::TRPO(CRUI N):PO<T>(),m_N(N){AS(m_N>0);}TE <TY T> IN TRPO<T>::TRPO(CO TRPO<T>& f):PO<T>(f),m_N(f.m_N){}TE <TY T> IN TRPO<T
    >::TRPO(TRPO<T>&& f):PO<T>(MO(f.m_f)),m_N(f.m_N){}TE <TY T> IN TRPO<T>::TRPO(CRUI N,T t):PO<T>(MO(t)),m_N(N){AS(m_N>0);}TE <TY T> IN TRPO<T
    >::TRPO(CRUI N,CO PO<T>& f):PO<T>(),m_N(N){AS(m_N>0);PO<T>::m_SZ = f.PO<T>::m_SZ < m_N?f.PO<T>::m_SZ:m_N;PO<T>::m_f = VE<T>(PO<T>::m_SZ);for(uint 
    i = 0;i < PO<T>::m_SZ;i++){PO<T>::m_f[i]= f.PO<T>::m_f[i];}}TE <TY T> IN TRPO<T>::TRPO(CRUI N,PO<T>&& f):PO<T>(),m_N(N){AS(m_N>0);if(f.PO<T
    >::m_SZ < m_N * 2){PO<T>::OP=(MO(f));if(f.PO<T>::m_SZ > m_N){TruncateFinal(m_N);}}else{PO<T>::m_f = VE<T>(m_N);for(uint i = 0;i < m_N;i++){PO<T
    >::m_f[i]= MO(f.PO<T>::m_f[i]);}PO<T>::m_SZ = m_N;}}TE <TY T> IN TRPO<T>::TRPO(CRUI N,VE<T>&& f):PO<T>(),m_N(N){AS(m_N>0);CO uint f_SZ = f.SZ
    ();if(f_SZ < m_N * 2){PO<T>::OP=(MO(f));if(f_SZ > m_N){TruncateFinal(m_N);}}else{PO<T>::m_f = VE<T>(m_N);for(uint i = 0;i < m_N;i++){PO<T
    >::m_f[i]= MO(f[i]);}}}TE <TY T> IN TRPO<T>::TRPO(CRUI N,CRUI i,T t):PO<T>(),m_N(N){AS(m_N>0);if(i < m_N?t != PO<T>::c_zero():false){PO<T>::OP[]
    (i)= MO(t);}}TE <TY T> IN TRPO<T>& TRPO<T>::OP=(TRPO<T> f){PO<T>::OP=(MO(f.m_f));m_N = f.m_N;RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T>::OP=(T n){PO<T
    >::OP=(MO(n));RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T>::OP=(PO<T> f){RE OP=(TRPO<T>(m_N,MO(f)));}TE <TY T> IN TRPO<T>& TRPO<T>::OP+=(CO T& t){PO<T
    >::OP+=(t);RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T>::OP+=(CO PO<T>& f){RE TRPlus(f,0,f.m_SZ);}TE <TY T> IN TRPO<T>& TRPO<T>::OP+=(CO TRPO<T>& f){RE 
    TRPlus(f,0,f.PO<T>::m_SZ);}TE <TY T>TRPO<T>& TRPO<T>::TRPlus(CO PO<T>& f,CRUI N_input_start,CRUI N_input_lim){CRUI SZ = N_input_lim < m_N
    ?N_input_lim < f.PO<T>::m_SZ?N_input_lim:f.PO<T>::m_SZ:m_N < f.PO<T>::m_SZ?m_N:f.PO<T>::m_SZ;if(PO<T>::m_SZ < SZ){PO<T>::m_f.reserve(SZ);for(uint 
    i = N_input_start;i < PO<T>::m_SZ;i++){PO<T>::m_f[i]+= f.PO<T>::m_f[i];}for(uint i = PO<T>::m_SZ;i < SZ;i++){PO<T>::m_f.push_back(f.PO<T>::m_f[i]
    );}PO<T>::m_SZ = SZ;}else{for(uint i = N_input_start;i < SZ;i++){PO<T>::m_f[i]+= f.PO<T>::m_f[i];}}RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T>::OP-=(CO 
    T& t){PO<T>::OP-=(t);RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T>::OP-=(CO PO<T>& f){RE TRMinus(f,0,f.m_SZ);}TE <TY T> IN TRPO<T>& TRPO<T>::OP-=(CO TRPO
    <T>& f){RE TRMinus(f,0,f.PO<T>::m_SZ);}TE <TY T>TRPO<T>& TRPO<T>::TRMinus(CO PO<T>& f,CRUI N_input_start,CRUI N_input_lim){CRUI SZ = N_input_lim 
    < m_N?N_input_lim < f.PO<T>::m_SZ?N_input_lim:f.PO<T>::m_SZ:m_N < f.PO<T>::m_SZ?m_N:f.PO<T>::m_SZ;if(PO<T>::m_SZ < SZ){PO<T>::m_f.reserve(SZ);for
    (uint i = N_input_start;i < PO<T>::m_SZ;i++){PO<T>::m_f[i]-= f.PO<T>::m_f[i];}for(uint i = PO<T>::m_SZ;i < SZ;i++){PO<T>::m_f.push_back(- f.PO<T
    >::m_f[i]);}PO<T>::m_SZ = SZ;}else{for(uint i = N_input_start;i < SZ;i++){PO<T>::m_f[i]-= f.PO<T>::m_f[i];}}RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T
    >::OP*=(CO T& t){PO<T>::OP*=(t);RE *TH;}TE <TY T>TRPO<T>& TRPO<T>::OP*=(CO PO<T>& f){DF_OF_MU_FOR_TR_PO(RE_ZERO_FOR_MU_FOR_TR_PO_IF(f.PO<T>::m_SZ 
    == 0),RE *TH,RE_ZERO_FOR_MU_FOR_TR_PO_IF(searching),RE_ZERO_FOR_MU_FOR_TR_PO_IF(N_input_start_0_start_1 >= m_N),RE *TH,MU,PO<T>::m_f[j],0,);}TE 
    <TY T> IN TRPO<T>& TRPO<T>::OP*=(PO<T>&& f){RE OP*=(f);}TE <TY T>TRPO<T>& TRPO<T>::FFT_MU(CO PO<T>& f){DF_OF_FFT_MU_FOR_TR_PO
    (RE_ZERO_FOR_MU_FOR_TR_PO_IF(f.PO<T>::m_SZ == 0),RE *TH,RE_ZERO_FOR_MU_FOR_TR_PO_IF(searching),RE_ZERO_FOR_MU_FOR_TR_PO_IF
    (N_input_start_0_start_1 >= N_output_lim_fixed),RE *TH,RE *TH,MU,PO<T>::m_f[j],0,0,,VE<T>& f0 = PO<T>::m_f,N_input_start_0,N_input_max_0 + 1
    ,SET_SHIFTED_VE_FOR_MU(f1,f.PO<T>::m_f,N_input_start_1,N_input_max_1,N_input_start_0),N_input_start_0_start_1,N_input_start_0 + N_input_max_1 + 1
    ,f1,,N_input_start_0,f1[N_input_start_1 + i]*= f0[i],OP=(TRPO<T>(m_N,MO(f1))));}TE <TY T>TRPO<T>& TRPO<T>::TRMU(CO PO<T>& f,CRUI N_output_start
    ,CRUI N_output_lim){DF_OF_MU_FOR_TR_PO(,RE *TH,,RE_ZERO_FOR_MU_FOR_TR_PO_IF(N_input_start_0_start_1 >= N_output_lim_fixed),RE *TH,MU,PO<T
    >::m_f[j],N_output_start,if(N_output_lim_fixed > N_output_lim){N_output_lim_fixed = N_output_lim;});}TE <TY T>TRPO<T>& TRPO<T>::FFT_TRMU(CO PO<T
    >& f,CRUI N_output_start,CRUI N_output_lim){DF_OF_FFT_MU_FOR_TR_PO(,RE *TH,,RE_ZERO_FOR_MU_FOR_TR_PO_IF(N_input_start_0_start_1 >= 
    N_output_lim_fixed),RE *TH,RE *TH,MU,PO<T>::m_f[j],N_output_start,N_output_start < N_input_start_0_start_1?0:N_output_start - 
    N_input_start_0_start_1,if(N_output_lim_fixed > N_output_lim){N_output_lim_fixed = N_output_lim;},VE<T>& f0 = PO<T>::m_f,N_input_start_0
    ,N_input_max_0 + 1,SET_SHIFTED_VE_FOR_MU(f1,f.PO<T>::m_f,N_input_start_1,N_input_max_1,N_input_start_0),N_input_start_0_start_1,N_input_start_0 + 
    N_input_max_1 + 1,f1,,N_input_start_0,f1[N_input_start_1 + i]*= f0[i],OP=(TRPO<T>(m_N,MO(f1))));}TE <TY T>TRPO<T>& TRPO<T>::FFT_TRMU(PO<T>&& f
    ,CRUI N_output_start,CRUI N_output_lim){DF_OF_FFT_MU_FOR_TR_PO(,RE *TH,,RE_ZERO_FOR_MU_FOR_TR_PO_IF(N_input_start_0_start_1 >= N_output_lim_fixed
    ),RE *TH,RE *TH,MU,PO<T>::m_f[j],N_output_start,N_output_start < N_input_start_0_start_1?0:N_output_start - N_input_start_0_start_1,if
    (N_output_lim_fixed > N_output_lim){N_output_lim_fixed = N_output_lim;},VE<T>& f0 = PO<T>::m_f,N_input_start_0,N_input_max_0 + 1,VE<T>&& f1 = MO
    (f.PO<T>::m_f),N_input_start_1,N_input_max_1 + 1,f0,f0.reserve(product_LE),0,f1[N_input_start_0_start_1 + i]= f0[N_input_start_0 + i]* 
    f1[N_input_start_1 + i],for(uint i = N_input_start_0;i < N_input_start_0_start_1;i++){f0[i]= 0;}PO<T>::m_SZ = f0.SZ();SetTruncation(m_N););}TE 
    <TY T>TRPO<T> TRPO<T>::TRMU_CO(CO PO<T>& f,CRUI N_output_start,CRUI N_output_lim)CO{DF_OF_MU_FOR_TR_PO(,RE TRPO<T>(m_N,MO(AN
    )),,RE_ZERO_FOR_TR_MU_CO_FOR_TR_PO_IF(N_input_start_0_start_1 >= N_output_lim_fixed),RE TRPO<T>(m_N,MO(AN)),TR_MU_CO,PO<T>::OP[](j
    ),N_output_start,if(N_output_lim_fixed > N_output_lim){N_output_lim_fixed = N_output_lim;});}TE <TY T>TRPO<T> TRPO<T>::FFT_TRMU_CO(CO PO<T>& f
    ,CRUI N_output_start,CRUI N_output_lim)CO{DF_OF_FFT_MU_FOR_TR_PO(,RE TRPO<T>(m_N,MO(AN)),,RE_ZERO_FOR_TR_MU_CO_FOR_TR_PO_IF
    (N_input_start_0_start_1 >= N_output_lim_fixed),RE TRPO<T>(m_N,MO(AN)),RE TRPO<T>(m_N,MO(f0)),TR_MU_CO,PO<T>::OP[](j),N_output_start
    ,N_output_start < N_input_start_0_start_1?0:N_output_start - N_input_start_0_start_1,if(N_output_lim_fixed > N_output_lim){N_output_lim_fixed = 
    N_output_lim;},SET_SHIFTED_VE_FOR_MU(f0,PO<T>::m_f,N_input_start_0,N_input_max_0,N_input_start_1),N_input_start_0_start_1,N_input_start_1 + 
    N_input_max_0 + 1,VE<T> f1 = f.PO<T>::m_f,N_input_start_1,N_input_max_1 + 1,f0,,N_input_start_1,f0[N_input_start_0 + i]*= f1[i],);}TE <TY T>TRPO
    <T> TRPO<T>::FFT_TRMU_CO(PO<T>&& f,CRUI N_output_start,CRUI N_output_lim)CO{DF_OF_FFT_MU_FOR_TR_PO(,RE TRPO<T>(m_N,MO(AN
    )),,RE_ZERO_FOR_TR_MU_CO_FOR_TR_PO_IF(N_input_start_0_start_1 >= N_output_lim_fixed),RE TRPO<T>(m_N,MO(AN)),RE TRPO<T>(m_N,MO(f0)),TR_MU_CO,PO<T
    >::OP[](j),N_output_start,N_output_start < N_input_start_0_start_1?0:N_output_start - N_input_start_0_start_1,if(N_output_lim_fixed > 
    N_output_lim){N_output_lim_fixed = N_output_lim;},SET_SHIFTED_VE_FOR_MU(f0,PO<T>::m_f,N_input_start_0,N_input_max_0,N_input_start_1
    ),N_input_start_0_start_1,N_input_start_1 + N_input_max_0 + 1,VE<T>&& f1 = MO(f.PO<T>::m_f),N_input_start_1,N_input_max_1 + 1,f0,,N_input_start_1
    ,f0[N_input_start_0 + i]*= f1[i],);}TE <TY T> IN TRPO<T>& TRPO<T>::OP/=(CO T& t){PO<T>::OP/=(t);RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T>::OP/=(CO 
    TRPO<T>& f){AS(m_N <= f.m_N);RE OP*=(m_N == f.m_N?Inverse(f):Inverse(TRPO<T>(m_N,f)));}TE <TY T> IN TRPO<T>& TRPO<T>::OP%=(CO T& t){PO<T>::OP%=(t
    );RE *TH;}TE <TY T> TE <TY P> IN TRPO<T> TRPO<T>::OP+(CO P& f)CO{RE MO(TRPO<T>(*TH)+= f);}TE <TY T> IN TRPO<T> TRPO<T>::OP-()CO{RE MO(TRPO<T>(m_N
    )-= *TH);}TE <TY T> TE <TY P> IN TRPO<T> TRPO<T>::OP-(CO P& f)CO{RE MO(TRPO<T>(*TH)-= f);}TE <TY T> TE <TY P> IN TRPO<T> TRPO<T>::OP*(CO P& f
    )CO{RE MO(TRPO<T>(*TH)*= f);}TE <TY T> TE <TY P> IN TRPO<T> TRPO<T>::OP/(CO P& f)CO{RE MO(TRPO<T>(*TH)/= f);}TE <TY T> IN TRPO<T> TRPO<T>::OP%(CO 
    T& t)CO{RE MO(TRPO<T>(*TH)%= t);}TE <TY T> IN VO TRPO<T>::SetTruncation(CRUI N)NE{if(N < m_N){TruncateFinal(N);}m_N = N;}TE <TY T> IN CRUI TRPO<T
    >::GetTruncation()CO NE{RE m_N;}TE <TY T> IN TRPO<T>& TRPO<T>::TruncateInitial(CRUI N)NE{CRUI SZ = N < PO<T>::m_SZ?N:PO<T>::m_SZ;for(uint i = 0;i 
    < SZ;i++){PO<T>::m_f[i]= 0;}RE *TH;}TE <TY T> IN TRPO<T>& TRPO<T>::TruncateFinal(CRUI N)NE{WH(PO<T>::m_SZ > N){PO<T>::m_f.pop_back();PO<T>::m_SZ
    --;}RE *TH;}TE <TY T>TRPO<T> Differential(CRUI n,CO TRPO<T>& f){if(f.PO<T>::m_SZ < n){RE TRPO<T>(n < f.m_N?f.m_N - n:1,PO<T>::zero());}VE<T> df(f
    .PO<T>::m_SZ - n);T coef = T::Factorial(n);uint i = n;WH(i < f.PO<T>::m_SZ){df[i - n]= f[i]* coef;i++;(coef *= i)/=(i - n);}RE TRPO<T>(n < f.m_N
    ?f.m_N - n:1,MO(df));}TE <TY T>TRPO<T> TRDifferential(CO TRPO<T>& f,CRUI N_output_start_plus_one){TRPO<T> f_dif{1 < f.m_N?f.m_N - 1:1};if
    (N_output_start_plus_one < f.PO<T>::m_SZ){CO uint SZ = f.PO<T>::m_SZ - 1;f_dif.PO<T>::m_f = VE<T>(SZ);for(uint i = N_output_start_plus_one;i < f
    .PO<T>::m_SZ;i++){f_dif.PO<T>::m_f[i-1]= f.PO<T>::m_f[i]* i;}f_dif.PO<T>::m_SZ = SZ;}RE f_dif;}TE <TY T> IN TRPO<T> Differential(CO TRPO<T>& f
    ){RE TRDifferential<T>(f,1);}TE <TY T>TRPO<T> TRIntegral(CO TRPO<T>& f,CRUI N_output_start){TRPO<T> f_int{f.m_N + 1};if(N_output_start <= f.PO<T
    >::m_SZ){CO uint SZ = f.PO<T>::m_SZ + 1;f_int.PO<T>::m_f = VE<T>(SZ);for(uint i = N_output_start;i <= f.PO<T>::m_SZ;i++){f_int.PO<T>::m_f[i]= f
    .PO<T>::m_f[i - 1]/ T(i);}f_int.PO<T>::m_SZ = SZ;}RE f_int;}TE <TY T> IN TRPO<T> Integral(CO TRPO<T>& f){RE TRIntegral<T>(f,1);}TE <TY T>TRPO<T> 
    Inverse(CO TRPO<T>& f){DF_OF_INVERSE_FOR_TR_PO(T,f_inv.TRMinus(f_inv.TRMU_CO(f,PW,PW_2).TRMU(f_inv,PW,PW_2),PW,PW_2));}TE <TY T>TRPO<T> Exp(CO 
    TRPO<T>& f){DF_OF_EXP_FOR_TR_PO(T,f_exp.TRMinus((TRIntegral(Differential(f_exp).TRMU_CO(Inverse(f_exp),PW - 1,PW_2),PW).TRMinus(f,PW,PW_2)).TRMU
    (f_exp,PW),PW,PW_2));}TE <TY T> IN TRPO<T> Log(CO TRPO<T>& f){AS(f[0]== PO<T>::c_one());RE Integral<T>(Differential<T>(f)/= f);}
DF_OF_PS_OF_MU_OF_PO_PROTH_MOD(P,997269505,Mod);
TE <TY T> IN PO<T>::PO():m_f(),m_SZ(0){}TE <TY T> IN PO<T>::PO(CO PO<T>& f):m_f(f.m_f),m_SZ(f.m_SZ){}TE <TY T> IN PO<T>::PO(PO<T>&& f):m_f(MO(f.m_f
    )),m_SZ(f.m_SZ){}TE <TY T> IN PO<T>::PO(VE<T> f):m_f(MO(f)),m_SZ(m_f.SZ()){}TE <TY T> IN PO<T>::PO(T t):PO(){if(t != c_zero()){OP[](0)= MO(t
    );}}TE <TY T> IN PO<T>::PO(CRUI i,T t):PO(){if(t != c_zero()){OP[](i)= MO(t);}}TE <TY T> IN PO<T>& PO<T>::OP=(T n){m_f.clear();m_SZ = 0;OP[](0)= 
    MO(n);RE *TH;}TE <TY T> IN PO<T>& PO<T>::OP=(PO<T> f){m_f = MO(f.m_f);m_SZ = f.m_SZ;RE *TH;}TE <TY T> IN PO<T>& PO<T>::OP=(VE<T> f){m_f = MO(f
    );m_SZ = m_f.SZ();RE *TH;}TE <TY T> IN CO T& PO<T>::OP[](CRUI i)CO{RE m_SZ <= i?c_zero():m_f[i];}TE <TY T> IN T& PO<T>::OP[](CRUI i){if(m_SZ <= i
    ){CO T& z = c_zero();WH(m_SZ <= i){m_f.push_back(z);m_SZ++;}}RE m_f[i];}TE <TY T> T PO<T>::OP()(CO T& t)CO{T AN =(*TH)[0];T t_pw = c_one();for
    (uint d = 1;d < m_SZ;d++){AN += m_f[d]*(t_pw *= t);}RE AN;}TE <TY T>PO<T>& PO<T>::OP+=(CO PO<T>& f){if(m_SZ < f.m_SZ){m_f.reserve(f.m_SZ);for
    (uint i = 0;i < m_SZ;i++){m_f[i]+= f.m_f[i];}for(uint i = m_SZ;i < f.m_SZ;i++){m_f.push_back(f.m_f[i]);}m_SZ = f.m_SZ;}else{for(uint i = 0;i < f
    .m_SZ;i++){m_f[i]+= f.m_f[i];}}RE *TH;}TE <TY T>PO<T>& PO<T>::OP-=(CO PO<T>& f){if(m_SZ < f.m_SZ){m_f.reserve(f.m_SZ);for(uint i = 0;i < m_SZ;i
    ++){m_f[i]-= f.m_f[i];}for(uint i = m_SZ;i < f.m_SZ;i++){m_f.push_back(- f.m_f[i]);}m_SZ = f.m_SZ;}else{for(uint i = 0;i < f.m_SZ;i++){m_f[i]-= f
    .m_f[i];}}RE *TH;}TE <TY T>PO<T>& PO<T>::OP*=(CO PO<T>& f){if(m_SZ == 0){RE *TH;}if(f.m_SZ == 0){m_f.clear();m_SZ = 0;RE *TH;}CO uint SZ = m_SZ + 
    f.m_SZ - 1;PO<T> product{};for(uint i = 0;i < SZ;i++){T& product_i = product[i];CO uint j_min = m_SZ > i?0:i - m_SZ + 1;CO uint j_lim = i < f
    .m_SZ?i + 1:f.m_SZ;for(uint j = j_min;j < j_lim;j++){product_i += m_f[i - j]* f.m_f[j];}}RE *TH = MO(product);}TE <TY T> IN PO<T>& PO<T>::OP*=(PO
    <T>&& f){RE *TH *= f;};TE <TY T>PO<T>& PO<T>::OP/=(CO T& t){if(t == c_one()){RE *TH;}CO T t_inv{c_one()/ t};for(uint i = 0;i < m_SZ;i++){OP[](i
    )*= t_inv;}RE *TH;}TE <TY T>PO<T> PO<T>::TP(CO PO<T>& f,CRUI f_TP_SZ){VE<T> f_TP(f_TP_SZ);for(uint d = 0;d < f_TP_SZ;d++){f_TP[d]= f.m_f[f.m_SZ - 
    1 - d];}RE PO<T>(MO(f_TP));}TE <TY T>PO<T>& PO<T>::OP%=(CO T& t){if(t == c_one()){RE *TH = zero();}for(uint i = 0;i < m_SZ;i++){m_f[i]%= t;}RE 
    *TH;}TE <TY T>bool PO<T>::OP==(CO PO<T>& f)CO{CRUI SZ0 = SZ();CRUI SZ1 = f.SZ();CRUI SZ_max = SZ0 < SZ1?SZ1:SZ0;for(uint i = 0;i < SZ_max;i++){if
    (OP[](i)!= f[i]){RE false;}}RE true;}TE <TY T>bool PO<T>::OP==(CO T& t)CO{CRUI SZ_max = SZ();CO T& zero = PO<T>::c_zero();for(uint i = 1;i < 
    SZ_max;i++){if(m_f[i]!= zero){RE false;}}RE OP[](0)== t;}TE <TY T> TE<TY P> IN bool PO<T>::OP!=(CO P& f)CO{RE !(*TH == f);}DF_OF_AR_FOR_PO(+,f += 
    *TH);TE <TY T> IN PO<T>& PO<T>::SignInvert(){ReMORedundantZero();for(auto& fi:m_f){fi = -fi;}RE *TH;}TE <TY T> IN PO<T> PO<T>::OP-()CO{RE MO(PO<T
    >(*TH).SignInvert());}DF_OF_AR_FOR_PO(-,f.SignInvert()+= *TH);DF_OF_AR_FOR_PO(*,f *= *TH);TE <TY T> IN PO<T> PO<T>::OP/(CO T& t)CO{RE MO(PO<T
    >(*TH)/= t);}TE <TY T> IN PO<T> PO<T>::OP%(CO T& t)CO{RE MO(PO<T>(*TH)%= t);}TE <TY T> IN CO VE<T>& PO<T>::GetCoefficient()CO NE{RE m_f;}TE <TY T
    > IN CRUI PO<T>::SZ()CO NE{RE m_SZ;}TE <TY T> IN VO PO<T>::resize(CRUI deg_plus)NE{m_f.resize(m_SZ = deg_plus);}TE <TY T> IN VO PO<T>::swap(PO<T
    >& f){m_f.swap(f.m_f);swap(m_SZ,f.m_SZ);}TE <TY T> IN VO PO<T>::swap(VE<T>& f){m_f.swap(f);m_SZ = m_f.SZ();}TE <TY T>VO PO<T>::ReMORedundantZero
    (){CO T& z = c_zero();WH(m_SZ > 0?m_f[m_SZ - 1]== z:false){m_f.pop_back();m_SZ--;}RE;}TE <TY T>string PO<T>::Display()CO NE{string s = "(";if
    (m_SZ > 0){s += to_string(m_f[0]);for(uint i = 1;i < m_SZ;i++){s += "," + to_string(m_f[i]);}}s += ")";RE s;}TE <TY T> IN CO PO<T>& PO<T>::zero
    (){ST CO PO<T> z{};RE z;}TE <TY T> IN CO PO<T>& PO<T>::one(){ST CO PO<T> o{c_one()};RE o;}TE <TY T> IN CO PO<T>& PO<T>::x(){ST CO PO<T> f{1,c_one
    ()};RE f;}TE <TY T> IN CO T& PO<T>::c_zero(){ST CO T z{0};RE z;}TE <TY T> IN CO T& PO<T>::c_one(){ST CO T o{1};RE o;}TE <TY T> IN CO T& PO<T
    >::c_minus_one(){ST CO T m{-1};RE m;}TE <TY T>PO<T> Differential(CRUI n,CO PO<T>& f){CRUI SZ = f.SZ();if(SZ < n){RE PO<T>::zero();}VE<T> df(SZ - 
    n);T coef = T::Factorial(n);uint i = n;WH(i < SZ){df[i - n]= f[i]* coef;i++;(coef *= i)/=(i - n);}RE PO<T>(MO(df));}
TE <TY T> IN PO<T>& PO<T>::OP/=(CO PO<T>& f){RE *TH = Quotient(*TH,f);}TE <TY T>PO<T> PO<T>::Quotient(CO PO<T>& f0,CO PO<T>& f1){AS(f1.m_SZ == 0 || 
    f1[f1.m_SZ-1] != c_zero());if(f0.m_SZ < f1.m_SZ){RE PO<T>::zero();}AS(f1.m_SZ != 0);CO uint f0_TP_SZ = f0.m_SZ - f1.m_SZ + 1;CO uint f1_TP_SZ = 
    f0_TP_SZ < f1.m_SZ?f0_TP_SZ:f1.m_SZ;CO TRPO<T> f1_TP_inverse = Inverse(TRPO<T>(f0_TP_SZ,TP(f1,f1_TP_SZ)));TRPO<T> f0_TP{f0_TP_SZ,TP(f0,f0_TP_SZ)}
    ;f0_TP *= f1_TP_inverse;for(uint d0 =(f0_TP_SZ + 1)/ 2;d0 < f0_TP_SZ;d0++){::swap(f0_TP[d0],f0_TP[f0_TP_SZ - 1 - d0]);}RE f0_TP;}TE <TY T>PO<T>& 
    PO<T>::OP%=(CO PO<T>& f){if(m_SZ >= f.m_SZ){*TH -=(*TH / f)* f;ReMORedundantZero();}RE *TH;}TE <TY T> IN PO<T> PO<T>::OP/(CO PO<T>& f)CO{RE PO<T
    >::Quotient(*TH,f);}TE <TY T> IN PO<T> PO<T>::OP%(CO PO<T>& f)CO{RE MO(PO<T>(*TH)%= f);}
#endif
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Polynomial/LagrangeInterpolation/a_Body.hpp"
#else
// ../Truncate/compress.txtを先に貼る。
TE <TY T,TE <TY...> TY V1,TE <TY...> TY V2>VO SetProductTree(V1<V2<T> >& product_tree){V2<T> *p_node = &(product_tree.back());WH(p_node->SZ()> 1){V2
    <T> node{};for(auto IT = p_node->BE(),EN = p_node->EN();IT != EN;IT++){node.push_back(T{});T& f = *IT;IT++;if(IT == EN){node.back()= f;break
    ;}else{node.back()= f * *IT;}}product_tree.push_front(MO(node));p_node = &(product_tree.front());}RE;}TE <TY T,TE <TY...> TY V1,TE <TY...> TY V2
    ,TE <TY...> TY V3>VO SetPointTree(CO V1<T>& point,V2<V3<PO<T> > >& point_tree){CO PO<T>& x = PO<T>::x();V3<PO<T> > linear{};for(auto& p:point
    ){linear.push_back(x - PO<T>(p));}point_tree.push_front(MO(linear));SetProductTree(point_tree);RE;}TE <TY T,TE <TY...> TY V1,TE <TY...> TY V2,TE 
    <TY...> TY V3>VO SetPointTreeEvaluation(CO PO<T>& f,CO V1<V2<PO<T> > >& point_tree,V3<T>& AN){AS(!point_tree.empty());CO V2<PO<T> >& prod = 
    point_tree.front();if(prod.empty()){RE;}CO PO<T>& zero = PO<T>::zero();auto IT_tree = point_tree.BE(),EN_tree = point_tree.EN();LI<PO<T> > RS ={f 
    % IT_tree->front()};IT_tree++;WH(IT_tree != EN_tree){auto IT_RS = RS.BE(),EN_RS = RS.EN();auto IT_node = IT_tree->BE(),EN_node = IT_tree->EN();WH
    (IT_RS != EN_RS){CO PO<T>& f = *IT_node;IT_node++;if(IT_node != EN_node){*(RS.insert(IT_RS,zero))= *IT_RS % f;*IT_RS %= *IT_node;IT_node++;}IT_RS
    ++;}IT_tree++;}for(auto& f:RS){AN.push_back(f[0]);}RE;}TE <TY T,TE <TY...> TY V1,TE <TY...> TY V2> IN VO SetMultipointEvaluation(CO PO<T>& f,CO 
    V1<T>& point,V2<T>& AN){LI<LI<PO<T> > > pt{};SetPointTree(point,pt);SetPointTreeEvaluation(f,pt,AN);}
TE <TY T>PO<T> LagrangeInterpolation(CO VE<T>& arg,CO VE<T>& val){CO int SZ = arg.SZ();AS(SZ > 0 && int(val.SZ())== SZ);LI<VE<PO<T> > > pt{}
    ;SetPointTree(arg,pt);CO VE<PO<T> >& top = pt.front();auto& f = top.front();PO<T> g{Differential(1,f)};VE<T> coef{};SetPointTreeEvaluation(g,pt
    ,coef);PO<T> AN{};auto IT = pt.back().BE();for(int i = 0;i < SZ;i++){AN += f / *(IT++)*(val[i]/ coef[i]);}AN.ReMORedundantZero();RE AN;}
#endif
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/LinearAlgebra/Rank/Mod/Debug/a_Body.hpp"
#else
#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)};}
#endif
/* AAA 常設でないライブラリは以上に挿入する。*/
#define INCLUDE_SUB
#include __FILE__
#else /* INCLUDE_LIBRARY */
#ifdef DEBUG
  #define _GLIBCXX_DEBUG
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if CE( bound_test_case_num > 1 ){ 
      SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } FINISH_MAIN
  #define FINISH_MAIN REPEAT( test_case_num ){ if CE( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } 
      Solve(); CERR( "" ); } }
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #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__ )
  #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] ); }
  #endif
  #define SET_ASSERT( A , MIN , MAX ) SET( A ); ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ )
  #define CERR( ... ) 
  #define CERRNS( ... ) 
  #define COUT_A( I , N , A ) CoutArray( cout , I , N , A ) << ENDL
  #define CERR_A( I , N , A ) 
  #define TLE( CONDITION ) if( !( CONDITION ) ){ ll TLE_VAR = 1; while( TLE_VAR != 0 ){ ( TLE_VAR += 2 ) %= int( 1e9 ); } COUT( TLE_VAR ); }
  #define MLE( CONDITION ) if( !( CONDITION ) ){ vector<vector<ll>> MLE_VAR{}; REPEAT( 1e6 ){ MLE_VAR.push_back( vector<ll>( 1e6 ) ); } COUT( MLE_VAR 
      ); }
  #define OLE( CONDITION ) if( !( CONDITION ) ){ REPEAT( 1e8 ){ COUT( "OLE" ); } }
#endif
#ifdef REACTIVE
  #ifdef DEBUG
    #define RSET( A , ... ) A = __VA_ARGS__
  #else
    #define RSET( A , ... ) SET( A )
  #endif
  #define RCIN( LL , A , ... ) LL A; RSET( A , __VA_ARGS__ )
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now(); double loop_average_time = 0.0 , loop_start_time = 0.0 , 
    current_time = 0.0; int 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_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 ); cerr << fixed << setprecision( DECIMAL_DIGITS )
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ , false ); auto answer = Answer( __VA_ARGS__ , false ); bool match = naive == answer; CERR( "
    (" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive == " , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ CERR( 
    "出力の不一致が検出されました。" ); RE; }
/* 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
/* 型のエイリアス */
#define decldecay_t(VAR)decay_t<decltype(VAR)>
TE <TY F,TY...Args> US ret_t = decltype(declval<F>()(declval<Args>()...));
TE <TY T> US inner_t = TY T::type;
US uint = unsigned int;
US ll = long long;
US ull = unsigned long long;
US ld = long double;
US lld = __float128;
/* VVV 常設ライブラリは以下に挿入する。*/
#ifdef DEBUG
  #include "C:/Users/user/Documents/Programming/Contest/Template/Local/a_Body.hpp"
#else
/* Random (1KB)*/
ll GetRand(CRI Rand_min,CRI Rand_max){AS(Rand_min <= Rand_max);ll AN = time(NULL);RE AN * rand()%(Rand_max + 1 - Rand_min)+ Rand_min;}
/* Set (2KB)*/
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Mathematics/Utility/Set/a_Body.hpp"
#else
#define DC_OF_HASH(...)struct hash<__VA_ARGS__>{IN size_t OP()(CO __VA_ARGS__& n)CO;};
CL is_ordered{PU:is_ordered()= delete;TE <TY T> ST CE auto Check(CO T& t)-> decltype(t < t,true_type());ST CE false_type Check(...);TE <TY T> ST CE 
    CO bool value = is_same_v< decltype(Check(declval<T>())),true_type >;};
TE <TY T>US Set = conditional_t<is_COructible_v<unordered_set<T>>,unordered_set<T>,conditional_t<is_ordered::value<T>,set<T>,VO>>;
#define DF_OF_POP_FOR_SET(SET)TE <TY T> IN T pop_max(SET& S){AS(!S.empty());auto IT = --S.EN();CO T AN = MO(*IT);S.erase(IT);RE AN;}TE <TY T> IN T 
    pop_min(SET& S){AS(!S.empty());auto IT = S.BE();CO T AN = MO(*IT);S.erase(IT);RE AN;}TE <TY T> IN SET& OP+=(SET& S,T t){S.insert(MO(t));RE S;}TE 
    <TY T> IN SET& OP-=(SET& S,CO T& t){S.erase(t);RE S;}TE <TY T> IN CO T& Get(CO SET& S,int i){auto BE = S.BE(),EN = S.EN();auto& IT = i < 0?(++i
    ,--EN):BE;WH(i > 0 && IT != EN){--i;++IT;}WH(i < 0 && IT != BE){++i;--IT;}AS(i == 0);RE *IT;}
#define DF_OF_UNION_FOR_SET(SET)TE <TY T> IN SET& OP|=(SET& a0,CO SET& a1){for(auto& t:a1){a0 += t;}RE a0;}TE <TY T> IN SET OP|(SET a0,CO SET& a1){RE 
    MO(a0 |= a1);}
TE <TY SET,TY T> IN TY SET::const_iterator MaximumLeq(CO SET& S,CO T& t){auto IT = S.upper_bound(t);RE IT == S.BE()?S.EN():--IT;}TE <TY SET,TY T> IN 
    TY SET::const_iterator MaximumLt(CO SET& S,CO T& t){auto IT = S.lower_bound(t);RE IT == S.BE()?S.EN():--IT;}TE <TY SET,TY T> IN TY SET
    ::const_iterator MinimumGeq(CO SET& S,CO T& t){RE S.lower_bound(t);}TE <TY SET,TY T> IN TY SET::const_iterator MinimumGt(CO SET& S,CO T& t){RE S
    .upper_bound(t);}TE <TY SET,TY ITERATOR> IN VO EraseBack(SET& S,ITERATOR& IT){IT = S.erase(IT);}TE <TY SET,TY ITERATOR> IN VO EraseFront(SET& S
    ,ITERATOR& IT){IT = S.erase(IT);IT == S.BE()?IT = S.EN():--IT;}TE <TE <TY...> TY SET,TY T,TY...Args> IN bool In(CO SET<T,Args...>& S,CO T& t){RE 
    S.count(t)== 1;}DF_OF_POP_FOR_SET(set<T>);DF_OF_POP_FOR_SET(unordered_set<T>);DF_OF_POP_FOR_SET(multiset<T>);DF_OF_POP_FOR_SET(unordered_multiset
    <T>);DF_OF_UNION_FOR_SET(set<T>);DF_OF_UNION_FOR_SET(unordered_set<T>);DF_OF_UNION_FOR_SET(multiset<T>);DF_OF_UNION_FOR_SET(unordered_multiset<T
    >);DF_OF_UNION_FOR_SET(VE<T>);DF_OF_UNION_FOR_SET(LI<T>);
#endif
/* Tuple (6KB)*/
#define DF_OF_AR_FOR_TUPLE(OPR)TE <TY T,TY U,TE <TY...> TY PAIR> IN auto OP OPR ## =(PAIR<T,U>& t0,CO PAIR<T,U>& t1)-> decltype((get<0>(t0),t0))&{get
    <0>(t0)OPR ## = get<0>(t1);get<1>(t0)OPR ## = get<1>(t1);RE t0;}TE <TY T,TY U,TY V,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V>& t0,CO 
    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);RE t0
    ;}TE <TY T,TY U,TY V,TY W,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V,W>& t0,CO 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);RE t0;}TE <TY ARG,TY T,TY U,TE 
    <TY...> TY PAIR> IN auto OP OPR ## =(PAIR<T,U>& t0,CO ARG& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = t1;get<1>(t0)OPR ## = t1;RE t0;}TE 
    <TY ARG,TY T,TY U,TY V,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V>& t0,CO ARG& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = t1
    ;get<1>(t0)OPR ## = t1;get<2>(t0)OPR ## = t1;RE t0;}TE <TY ARG,TY T,TY U,TY V,TY W,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V,W>& t0,CO 
    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;RE t0;}TE <TE <TY
    ...> TY TUPLE,TY...ARGS,TY ARG> IN auto OP OPR(CO TUPLE<ARGS...>& t0,CO ARG& t1)-> decldecay_t((get<0>(t0),t0)){auto t = t0;RE MO(t OPR ## = t1
    );}
#define DF_OF_INCREMENT_FOR_TUPLE(INCR)TE <TY T,TY U,TE <TY...> TY PAIR> IN auto OP INCR(PAIR<T,U>& t)-> decltype((get<0>(t),t))&{INCR get<0>(t);INCR 
    get<1>(t);RE t;}TE <TY T,TY U,TY V,TE <TY...> TY TUPLE> IN auto OP INCR(TUPLE<T,U,V>& t)-> decltype((get<0>(t),t))&{INCR get<0>(t);INCR get<1>(t
    );INCR get<2>(t);RE t;}TE <TY T,TY U,TY V,TY W,TE <TY...> TY TUPLE> IN auto OP 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);RE t;}
TE <CL Traits,TY T> IN IS& OP>>(IS& is,tuple<T>& arg){RE is >> get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY...> TY V> IN auto OP>>(IS& is,V<T,U>& arg
    )-> decltype((get<0>(arg),is))&{RE is >> get<0>(arg)>> get<1>(arg);}TE <CL Traits,TY T,TY U,TY V> IN IS& OP>>(IS& is,tuple<T,U,V>& arg){RE is >> 
    get<0>(arg)>> get<1>(arg)>> get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W> IN IS& OP>>(IS& is,tuple<T,U,V,W>& arg){RE is >> get<0>(arg)>> get<1
    >(arg)>> get<2>(arg)>> get<3>(arg);}TE <CL Traits,TY T> IN OS& OP<<(OS& os,CO tuple<T>& arg){RE os << get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY
    ...> TY V> IN auto OP<<(OS& os,CO V<T,U>& arg)-> decltype((get<0>(arg),os))&{RE os << get<0>(arg)<< " " << get<1>(arg);}TE <CL Traits,TY T,TY U
    ,TY V> IN OS& OP<<(OS& os,CO tuple<T,U,V>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W
    > IN OS& OP<<(OS& os,CO tuple<T,U,V,W>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg)<< " " << get<3>(arg
    );}DF_OF_AR_FOR_TUPLE(+);TE <TY T,TY U,TE <TY...> TY V> IN auto OP-(CO V<T,U>& t)-> decltype(get<0>(t),t){RE{-get<0>(t),-get<1>(t)};}TE <TY T,TY 
    U,TY V> IN tuple<T,U,V> OP-(CO tuple<T,U,V>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t)};}TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W> OP-(CO tuple<T,U
    ,V,W>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t),-get<3>(t)};}DF_OF_AR_FOR_TUPLE(-);DF_OF_AR_FOR_TUPLE(*);DF_OF_AR_FOR_TUPLE(/);DF_OF_AR_FOR_TUPLE(%
    );DF_OF_INCREMENT_FOR_TUPLE(++);DF_OF_INCREMENT_FOR_TUPLE(--);
TE <int n>CL TupleAccessIndex{};TE <TY...Types>CL Tuple:PU tuple<Types...>{PU:IN Tuple(Types&&... args);TE <TY...Args> IN Tuple(Args&&... args);TE 
    <int n> IN auto& OP[](CO TupleAccessIndex<n>& i)NE;TE <int n> IN CO auto& OP[](CO TupleAccessIndex<n>& i)CO NE;};TE <TY...Types>CL tuple_size
    <Tuple<Types...>>:PU tuple_size<tuple<Types...>>{};TE <size_t n,TY...Types>CL tuple_element<n,Tuple<Types...>>:PU tuple_element<n,tuple<Types
    ...>>{};
TE <TY T,TY U> US Pair = Tuple<T,U>;TE <TY INT> US T2 = Pair<INT,INT>;TE <TY INT> US T3 = Tuple<INT,INT,INT>;TE <TY INT> US T4 = Tuple<INT,INT,INT
    ,INT>;
CE TupleAccessIndex<0> O{};CE TupleAccessIndex<1> I{};CE TupleAccessIndex<2> II{};CE TupleAccessIndex<3> III{};
TE <TY...Types> IN Tuple<Types...>::Tuple(Types&&... args):tuple<Types...>(MO(args)...){}TE <TY...Types> TE <TY...Args> IN Tuple<Types...>::Tuple
    (Args&&... args):tuple<Types...>(forward<Args>(args)...){}TE <TY...Types> TE <int n> IN auto& Tuple<Types...>::OP[](CO TupleAccessIndex<n>& i
    )NE{RE get<n>(*TH);}TE <TY...Types> TE <int n> IN CO auto& Tuple<Types...>::OP[](CO TupleAccessIndex<n>& i)CO NE{RE get<n>(*TH);}
#define DF_OF_HASH_FOR_TUPLE(PAIR)TE <TY T,TY U> IN size_t hash<PAIR<T,U>>::OP()(CO PAIR<T,U>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST 
    CO hash<T> h0;ST CO hash<U> h1;RE(h0(get<0>(n))* seed)^ h1(get<1>(n));}
TE <TY T> DC_OF_HASH(tuple<T>);TE <TY T,TY U> DC_OF_HASH(pair<T,U>);TE <TY T,TY U> DC_OF_HASH(tuple<T,U>);TE <TY T,TY U,TY V> DC_OF_HASH(tuple<T,U,V
    >);TE <TY T,TY U,TY V,TY W> DC_OF_HASH(tuple<T,U,V,W>);
TE <TY T> IN size_t hash<tuple<T>>::OP()(CO tuple<T>& n)CO{ST CO hash<T> h;RE h(get<0>(n));}DF_OF_HASH_FOR_TUPLE(pair);DF_OF_HASH_FOR_TUPLE(tuple);TE 
    <TY T,TY U,TY V> IN size_t hash<tuple<T,U,V>>::OP()(CO tuple<T,U,V>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<pair<T,U>> h01
    ;ST CO hash<V> h2;RE(h01({get<0>(n),get<1>(n)})* seed)^ h2(get<2>(n));}TE <TY T,TY U,TY V,TY W> IN size_t hash<tuple<T,U,V,W>>::OP()(CO tuple<T,U
    ,V,W>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<pair<T,U>> h01;ST CO hash<pair<V,W>> h23;RE(h01({get<0>(n),get<1>(n)})* seed
    )^ h23({get<2>(n),get<3>(n)});}
/* Vector (3KB)*/
#define DC_OF_COUT_FOR_VE(V)TE <CL Traits,TY Arg> IN OS& OP<<(OS& os,CO V<Arg>& arg)
#define DF_OF_COUT_FOR_VE(V)TE <CL Traits,TY Arg> IN OS& OP<<(OS& os,CO V<Arg>& arg){auto BE = arg.BE(),EN = arg.EN();auto IT = BE;WH(IT != EN){(IT 
    == BE?os:os << " ")<< *IT;IT++;}RE os;}
DF_OF_COUT_FOR_VE(VE);DF_OF_COUT_FOR_VE(LI);DF_OF_COUT_FOR_VE(set);DF_OF_COUT_FOR_VE(unordered_set);DF_OF_COUT_FOR_VE(multiset);IN VO VariadicResize
    (CRI SZ){}TE <TY Arg,TY... ARGS> IN VO VariadicResize(CRI SZ,Arg& arg,ARGS&... args){arg.resize(SZ);VariadicResize(SZ,args...);}
#define DF_OF_SCALAR_ACTION_FOR_VE(V,OPR)TE <TY T> IN V<T>& OP OPR ## =(V<T>& a,CO T& t){for(auto& s:a){a OPR ## = t;}RE a;}
#define DF_OF_AR_FOR_VE(V,OPR)TE <TY T> IN V<T>& OP OPR ## =(V<T>& a0,CO V<T>& a1){AS(a0.SZ()<= a1.SZ());auto IT0 = a0.BE(),EN0 = a0.EN();auto IT1 = 
    a1.BE();WH(IT0 != EN0){*(IT0++)OPR ## = *(IT1++);}RE a0;}TE <TY T,TY U> IN V<T> OP OPR(V<T> a,CO U& u){RE MO(a OPR ## = u);}
#define DF_OF_INCREMENT_FOR_VE(V,INCR)TE <TY T> IN V<T>& OP INCR(V<T>& a){for(auto& i:a){INCR i;}RE a;}
#define DF_OF_ARS_FOR_VE(V)TE <TY T> IN V<T>& OP+=(V<T>& a,CO T& t){a.push_back(t);RE a;}DF_OF_SCALAR_ACTION_FOR_VE(V,*);DF_OF_SCALAR_ACTION_FOR_VE(V
    ,/);DF_OF_SCALAR_ACTION_FOR_VE(V,%);DF_OF_AR_FOR_VE(V,+);DF_OF_AR_FOR_VE(V,-);DF_OF_AR_FOR_VE(V,*);DF_OF_AR_FOR_VE(V,/);DF_OF_AR_FOR_VE(V,%
    );DF_OF_INCREMENT_FOR_VE(V,++);DF_OF_INCREMENT_FOR_VE(V,--);TE <TY T> IN V<T> OP*(CO T& scalar,V<T> v){for(auto& t:v){t *= scalar;}RE MO(v);}TE 
    <TY T> IN T pop(V<T>& a){AS(!a.empty());T AN = MO(a.back());a.pop_back();RE AN;}
DF_OF_ARS_FOR_VE(VE);DF_OF_ARS_FOR_VE(LI);TE <TY V> IN auto Get(V& a){RE[&](CRI i = 0)-> CO decldecay_t(a[0])&{RE a[i];};}TE <TY T> IN VE<T> id(CRI 
    SZ){VE<T> AN(SZ);for(int i = 0;i < SZ;i++){AN[i]= i;}RE AN;}TE <TY T> IN VO Sort(VE<T>& a,CO bool& reversed = false){if(reversed){ST auto comp 
    =[](CO T& t0,CO T& t1){RE t1 < t0;};sort(a.BE(),a.EN(),comp);}else{sort(a.BE(),a.EN());}}TE <TY T0,TY T1> IN VO Sort(VE<T0>& a,VE<T1>& b,CO bool& 
    reversed = false){CO int SZ = a.SZ();AS(SZ == int(b.SZ()));VE<pair<T0,T1>> v(SZ);for(int i = 0;i < SZ;i++){v[i]={MO(a[i]),MO(b[i])};}Sort(v
    ,reversed);for(int i = 0;i < SZ;i++){a[i]= MO(v[i].first);b[i]= MO(v[i].second);}}TE <TY T> IN VE<int> IndexSort(CO VE<T>& a,CO bool& reversed = 
    false){auto index = id<int>(a.SZ());if(reversed){sort(index.BE(),index.EN(),[&](CRI i,CRI j){RE a[j]< a[i];});}else{sort(index.BE(),index.EN
    (),[&](CRI i,CRI j){RE a[i]< a[j];});}RE index;}TE <TY V> IN int len(CO V& a){RE a.SZ();}
/* Map (1KB)*/
#define DF_OF_AR_FOR_MAP(MAP,OPR)TE <TY T,TY U> IN MAP<T,U>& OP OPR ## =(MAP<T,U>& a,CO pair<T,U>& v){a[v.first]OPR ## = v.second;RE a;}TE <TY T,TY U
    > IN MAP<T,U>& OP OPR ## =(MAP<T,U>& a0,CO MAP<T,U>& a1){for(auto&[t,u]:a1){a0[t]OPR ## = u;}RE a0;}TE <TY T,TY U,TY ARG> IN MAP<T,U> OP OPR(MAP
    <T,U> a,CO ARG& arg){RE MO(a OPR ## = arg);}
#define DF_OF_ARS_FOR_MAP(MAP)DF_OF_AR_FOR_MAP(MAP,+);DF_OF_AR_FOR_MAP(MAP,-);DF_OF_AR_FOR_MAP(MAP,*);DF_OF_AR_FOR_MAP(MAP,/);DF_OF_AR_FOR_MAP(MAP,%
    );
TE <TY T,TY U>US Map = conditional_t<is_COructible_v<unordered_map<T,int>>,unordered_map<T,U>,conditional_t<is_ordered::value<T>,map<T,U>,VO>>;
DF_OF_ARS_FOR_MAP(map);DF_OF_ARS_FOR_MAP(unordered_map);
/* StdStream (2KB)*/
TE <CL Traits> IN IS& VariadicCin(IS& is){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicCin(IS& is,Arg& arg,ARGS&... args){RE VariadicCin(is 
    >> arg,args...);}TE <CL Traits> IN IS& VariadicSet(IS& is,CRI i){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicSet(IS& is,CRI i,Arg& arg
    ,ARGS&... args){RE VariadicSet(is >> arg[i],i,args...);}TE <CL Traits> IN IS& VariadicGetline(IS& is,CO char& separator){RE is;}TE <CL Traits,TY 
    Arg,TY... ARGS> IN IS& VariadicGetline(IS& is,CO char& separator,Arg& arg,ARGS&... args){RE VariadicGetline(getline(is,arg,separator),separator
    ,args...);}TE <CL Traits,TY Arg> IN OS& VariadicCout(OS& os,Arg&& arg){RE os << forward<Arg>(arg);}TE <CL Traits,TY Arg1,TY Arg2,TY... ARGS> IN 
    OS& VariadicCout(OS& os,Arg1&& arg1,Arg2&& arg2,ARGS&&... args){RE VariadicCout(os << forward<Arg1>(arg1)<< " ",forward<Arg2>(arg2),forward<ARGS
    >(args)...);}TE <CL Traits,TY Arg> IN OS& VariadicCoutNonSep(OS& os,Arg&& arg){RE os << forward<Arg>(arg);}TE <CL Traits,TY Arg1,TY Arg2,TY... 
    ARGS> IN OS& VariadicCoutNonSep(OS& os,Arg1&& arg1,Arg2&& arg2,ARGS&&... args){RE VariadicCoutNonSep(os << forward<Arg1>(arg1),forward<Arg2>(arg2
    ),forward<ARGS>(args)...);}TE <CL Traits,TY ARRAY> IN OS& CoutArray(OS& os,CRI i_start,CRI i_ulim,ARRAY&& a){for(int i = i_start;i < i_ulim;i++){
    (i == i_start?os:(os << " "))<< a[i];}RE os;}
/* 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_HASH_FOR_MOD(MOD)IN size_t hash<MOD>::OP()(CO MOD& n)CO{ST CO hash<decldecay_t(n.RP())> h;RE h(n.RP());}
TE <uint M> DC_OF_HASH(Mod<M>); TE <uint M> DF_OF_HASH_FOR_MOD(Mod<M>);
/* Loop (1KB)*/
TE <TY INT> bool NextLoop(CRI SZ,CO VE<INT>& lower_bound,CO VE<INT>& upper_limit,VE<INT>& index){int depth = 0;WH(depth < SZ){if(++index[depth]< 
    upper_limit[depth]){break;}index[depth]= lower_bound[depth];depth++;}RE depth < SZ;}TE <TY INT> bool NextLoop(CO VE<INT>& lower_bound,CO VE<INT>& 
    upper_limit,VE<INT>& index){RE NextLoop(index.SZ(),lower_bound,upper_limit,index);}TE <TY INT> bool NextLoopEq(CRI SZ,CO VE<INT>& lower_bound,CO 
    VE<INT>& upper_bound,VE<INT>& index){int depth = 0;WH(depth < SZ){if(++index[depth]<= upper_bound[depth]){break;}index[depth]= lower_bound[depth]
    ;depth++;}RE depth < SZ;}TE <TY INT> bool NextLoopEq(CO VE<INT>& lower_bound,CO VE<INT>& upper_bound,VE<INT>& index){RE NextLoopEq(index.SZ
    (),lower_bound,upper_bound,index);}
/* string (1KB)*/
TE <TY INT> IN char IntToChar(CO INT& i,CO char& c = 'a'){RE c + i;}TE <TY INT> IN INT CharToInt(CO char& i){RE i -(i < 'a'?'A':'a');}TE <TY INT
    >string ArrayToString(CO VE<INT>& A,CO char& c = 'a'){CO int N = A.SZ();string S(N,c);for(int i = 0;i < N;i++){S[i]= IntToChar<INT>(A[i],c);}RE S
    ;}TE <TY INT>VE<INT> StringToArray(CO string& S){CO int N = S.SZ();VE<int> A(N);for(int i = 0;i < N;i++){A[i]= CharToInt<INT>(S[i]);}RE A;}
/* Sum (2KB) */
TE <TY T,TE <TY...> TY V,TY OPR> T LeftConnectiveProd(CO V<T>& f,OPR opr){AS(!f.empty());auto IT = f.BE(),EN = f.EN();T AN = *(IT++);WH(IT != EN){AN 
    = opr(MO(AN),*(IT++));}RE AN;}TE <TY T,TE <TY...> TY V> IN T Sum(CO V<T>& f){RE LeftConnectiveProd(f,[](T t0,CO T& t1){RE MO(t0 += t1);});}TE <TY 
    T,TE <TY...> TY V> IN T Prod(CO V<T>& f){RE LeftConnectiveProd(f,[](T t0,CO T& t1){RE MO(t0 *= t1);});}TE <TY T,TE <TY...> TY V> IN T Max(CO V<T
    >& f){RE *max_element(f.BE(),f.EN());}TE <TY T,TE <TY...> TY V> IN T Min(CO V<T>& f){RE *min_element(f.BE(),f.EN());}TE <TY T,TY U> IN T SetMax
    (T& n,CO U& m){RE n < m?n = m:n;}TE <TY T,TY U> IN T SetMin(T& n,CO U& m){RE n > m?n = m:n;}TE <TY T,TY UINT>T Power(T t,UINT EX,T init = 1){(EX 
    & 1)== 1?init *= t:init;EX >>= 1;WH(EX > 0){t = Square(t);(EX & 1)== 1?init *= t:init;EX >>= 1;}RE MO(init);}TE <TY INT> IN INT 
    ArithmeticProgressionSum(CO INT& l,INT r,CO INT& d = 1){AS(l <= r);CO INT c =(r - l)/ d;RE(c & 1)== 0?(c + 1)*(l + d *(c >> 1)):((c + 1)>> 1)*((l 
    << 1)+ d * c);}TE <TY INT> IN INT ArithmeticProgressionSum(CO INT& r){RE ArithmeticProgressionSum(INT{},r);}TE <TY T,TY UINT> IN T 
    GeometricProgressionSum(T rate,UINT EX_max,CO T& init = 1){T rate_minus = rate - 1;RE rate_minus == 0?init * ++EX_max:(Power(MO(rate),MO(++EX_max
    ))- 1)/ MO(rate_minus)* init;}TE <TY T,TY UINT>T GeometricProgressionLinearCombinationSum(VE<T> rate,VE<UINT> EX_max,CO VE<T>& init){CO int SZ = 
    init.SZ();AS(int(rate.SZ())== SZ && int(EX_max.SZ())== SZ);T AN{};for(int i = 0;i < SZ;i++){AN += GeometricProgressionSum(MO(rate[i]),MO
    (EX_max[i]),init[i]);}RE AN;}
#endif
/* AAA 常設ライブラリは以上に挿入する。*/
#define INCLUDE_LIBRARY
#include __FILE__
#endif /* INCLUDE_LIBRARY */
#endif /* INCLUDE_SUB */
#endif /* INCLUDE_MAIN */
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