#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  /* #define SUBMIT_ONLY */
  #define DEBUG_OUTPUT
#endif
#ifdef INCLUDE_MAIN

VO Solve()
{
  CIN( int , N );
  CIN_A( int , 0 , N , A );
  int sum = Sum( A );
  int Q = 999630629;
  int K = sum - Q;
  MP a{};
  if( K >= 0 ){
    Map<int,MP> imA{};
    FOR( i , 0 , N ){
      imA[A[i]]++;
    }
    FPS<MP> f{ K + 1 };
    RUN( imA , [Ai,ci] ){
      FOREQ( d , Ai , K , Ai ){
        int j = d / Ai;
        f[d] += ( j & 1 ? ci : -ci ) / MP{ j };
      }
    }
    f = Exp( f );
    FOREQ( k , 0 , K ){
      a += f[k];
    }
  }
  RETURN( sum * Power( MP{2} , N - 1 ) - a * Q );
}
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()
{

  CERR( "全ての出力が一致しました。" );
}

/* 圧縮時は中身だけ削除する。*/
IN VO RandomTest( const int& test_case_num )
{
  REPEAT( test_case_num ){

  }
  CERR( "全ての出力が一致しました。" );
}

#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。*/

#define PO Polynomial
#define FPS FormalPowerSeries
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Polynomial/FPS/a_Body.hpp"
#else
TE <TY INT> CE INT Log(INT N){INT AN = 0,pw = 1;WH(N > pw){pw <<= 1;AN++;}RE AN;}

TE <TY T,int EX_lim>CL Power3Power_CE{PU:T m_val[EX_lim];CE Power3Power_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 Power3Power_CE<T,EX_lim>::Power3Power_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& Power3Power_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(&Power3Power_CE<T,EX_lim>::Get()CO)[EX_lim]{RE m_val;}

#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 PO{PU:VE<T> m_f;int 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(CRI 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[](CRI i)CO;IN T& OP[](CRI i);T OP()(CO T& t)CO;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 CRI SZ()CO NE;IN VO resize(CRI deg_plus)NE;int Valuation()CO NE;IN VO swap(PO<T>& f);IN VO swap(VE<T>& f);VO Reduce();VO TP(CRI N_trunc);ST PO<T> NaiveCN(PO<T> f0,CRI valuation0,CO PO<T>& f1,CRI valuation1,CRI N_trunc);ST PO<T> NaiveQuotient(PO<T> f0,CO PO<T>& f1);ST PO<T> NaiveResidue(PO<T> f0,CO PO<T>& f1);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();};

TE <TY T>CL FPS:PU PO<T>{PU:int m_N;IN FPS(CRI N = 0);IN FPS(CO FPS<T>& f);IN FPS(FPS<T>&& f);IN FPS(CRI N,T t);IN FPS(CRI N,CO PO<T>& f);IN FPS(CRI N,PO<T>&& f);IN FPS(CRI N,VE<T>&& f);IN FPS(CRI N,CRI i,T t);IN FPS<T>& OP=(FPS<T> f);IN FPS<T>& OP=(T n);IN FPS<T>& OP=(PO<T> f);IN FPS<T>& OP+=(CO T& t);FPS<T>& OP+=(CO PO<T>& f);IN FPS<T>& OP+=(CO FPS<T>& f);IN FPS<T>& OP-=(CO T& t);FPS<T>& OP-=(CO PO<T>& f);IN FPS<T>& OP-=(CO FPS<T>& f);IN FPS<T>& OP*=(CO T& t);FPS<T>& OP*=(PO<T> f);IN FPS<T>& OP*=(FPS<T> f);IN FPS<T>& OP/=(CO T& t);IN FPS<T>& OP/=(CO FPS<T>& t);TE <TY P> IN FPS<T> OP+(CO P& f)CO;IN FPS<T> OP-()CO;TE <TY P> IN FPS<T> OP-(CO P& f)CO;TE <TY P> IN FPS<T> OP*(CO P& f)CO;TE <TY P> IN FPS<T> OP/(CO P& f)CO;FPS<T> Inverse()CO;IN VO SetTruncation(CRI N)NE;IN CRI GetTruncation()CO NE;IN FPS<T>& TruncateInitial(CRI N)NE;IN FPS<T>& TruncateFinal(CRI N)NE;};
TE <TY T>US FPS = FPS<T>;

#define PS_FOR_FFT_BODY(MOD,LE,PR,IPR,TYPE)ST_AS((TYPE::DeRP(PR)*= TYPE::DeRP(IPR))== TYPE::DeRP(1));TE <> CE CO uint LimitOfPowerForFFT<TYPE> = LE - 1;TE <> IN CO TYPE(&PrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<TYPE>]{ST CE Power3Power_CE<TYPE,LimitOfPowerForFFT<TYPE>> PRT{PR};ST_AS(PRT.m_val[0]== TYPE::DeRP(1));RE PRT.Get();}TE <> IN CO TYPE(&InversePrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<TYPE>]{ST CE Power3Power_CE<TYPE,LimitOfPowerForFFT<TYPE>> IPRT{IPR};ST_AS(IPRT.m_val[0]== TYPE::DeRP(1)&&(TYPE::DeRP(PR)*= TYPE::DeRP(IPR))== TYPE::DeRP(1));RE IPRT.Get();}TE <> IN PO<TYPE>& PO<TYPE>::OP*=(PO<TYPE> f){CO int SZ = m_SZ + f.m_SZ - 1;RE *TH = FFTCN(MO(*TH),MO(f),SZ);}TE <> IN PO<TYPE>& PO<TYPE>::OP/=(CO PO<TYPE>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();if(m_SZ < f.m_SZ){RE *TH = zero();}RE *TH = FFTQuotient(MO(*TH),f);}TE <> IN PO<TYPE>& PO<TYPE>::OP%=(CO PO<TYPE>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();RE *TH = FFTResidue(MO(*TH),f);}
#define PS_FOR_FFT(MOD,LE,PR,IPR,MINT)PS_FOR_FFT_BODY(MOD,LE,PR,IPR,MINT<MOD>)
TE <TY T> CE CO int LimitOfPowerForFFT;TE <TY T> IN CO T(&PrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<T>];TE <TY T> IN CO T(&InversePrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<T>];
TE <TY T>VO CooleyTukey(VE<T>& f,CRI N_shift,CRI N_input_start,CRI N_input_lim,CRI N_trunc,CRI two_pw,CRI EX,CO T(&PRT)[LimitOfPowerForFFT<T>]){AS(N_input_lim - N_input_start <= two_pw);CO int N_zero = N_shift + N_input_start,le = N_zero + two_pw;AS(N_zero <= N_trunc);CO int N_input_final = min(N_input_start + two_pw,int(f.SZ()));for(int i = N_input_lim;i < N_input_final;i++){f[i]= T{};}WH(int(f.SZ())< le){f.push_back(T{});}ST VE<int> bit_reverse[32]={VE<int>(1)};ST int e_next = 1;ST int two_pw_next = 1;ST int two_pw_next2 = 2;ST VE<int>* p_bit_reverse_prev = bit_reverse;ST VE<int>* p_bit_reverse_curr = p_bit_reverse_prev + 1;WH(e_next <= EX){*p_bit_reverse_curr = VE<int>(two_pw_next2);int* p_bit_reverse_curr_i = &((*p_bit_reverse_curr)[0]);int* p_bit_reverse_curr_i_plus = p_bit_reverse_curr_i + two_pw_next;int* p_bit_reverse_prev_i = &((*p_bit_reverse_prev)[0]);for(int 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<int>& bit_reverse_EX = bit_reverse[EX];int bit_num = 0;CO int* 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;int i,j,j_lim,two_pw_curr = 1,two_pw_curr_2 = 2;WH(two_pw_curr < two_pw){CO int N_input_final_curr = N_input_start + two_pw_curr;bit_num = 0;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_final_curr];f[j + N_input_start]+= f[j + N_input_final_curr];f[j + N_input_final_curr]= 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;}if(N_trunc < le){f.resize(N_trunc);}if(N_shift > 0){for(int i = N_trunc - 1;i >= N_zero;i--){f[i]= MO(f[i - N_shift]);}for(int i = N_zero - 1;i >= N_input_start;i--){f[i]= T{};}}RE;}TE <TY T> IN VO FFT(VE<T>& f,CRI N_input_start,CRI N_input_lim,CRI two_pw,CRI EX){CooleyTukey<T>(f,0,N_input_start,N_input_lim,N_input_start + two_pw,two_pw,EX,PrimitiveRootOfTwoForFFT<T>());}TE <TY T> IN VO IFFT(VE<T>& f,CRI N_shift,CRI N_input_start,CRI N_input_lim,CRI N_trunc,CRI two_pw,CO T& two_pw_inv,CRI EX){CooleyTukey<T>(f,N_shift,N_input_start,N_input_lim,N_trunc,two_pw,EX,InversePrimitiveRootOfTwoForFFT<T>());CO int SZ = f.SZ();for(int i = N_shift + N_input_start;i < SZ;i++){f[i]*= two_pw_inv;}}TE <TY T>PO<T> FFTCN(PO<T> f0,PO<T> f1,CRI N_trunc){f0.Reduce();if(f0.m_SZ == 0){RE MO(f0);}f1.Reduce();if(f1.m_SZ == 0){RE MO(f1);}AS(f0.m_SZ <= N_trunc);CO int valuation0 = f0.Valuation();CO int valuation1 = f1.Valuation();if(N_trunc <= valuation0 + valuation1){RE f0.zero();}CO int le0 = f0.m_SZ - valuation0;CO int le1 = min(f1.m_SZ,N_trunc)- valuation1;CO int le = le0 + le1 - 1;CO int EX = Log(le);if(min(le0,le1)<= EX){RE f0.NaiveCN(MO(f0),valuation0,MO(f1),valuation1,min(f0.m_SZ + f1.m_SZ - 1,N_trunc));}CO int two_pw = 1 << EX;FFT(f0.m_f,valuation0,f0.m_SZ,two_pw,EX);FFT(f1.m_f,valuation1,valuation1 + le1,two_pw,EX);for(int i = 0;i < two_pw;i++){f0.m_f[i + valuation0]*= f1.m_f[i + valuation1];}IFFT(f0.m_f,valuation1,valuation0,valuation0 + two_pw,N_trunc,two_pw,f0.c_one()/ two_pw,EX);f0.m_SZ = f0.m_f.SZ();RE MO(f0);}TE <TY T>PO<T> FFTQuotient(PO<T> f0,PO<T> f1){AS(f1.m_SZ > 0 && f1[f1.m_SZ-1]!= f0.c_zero());if(f0.m_SZ < f1.m_SZ){RE PO<T>::zero();}CO int f0_TP_SZ = f0.m_SZ - f1.m_SZ + 1;CO int f1_TP_SZ = min(f0_TP_SZ,f1.m_SZ);f1.TP(f1_TP_SZ);CO FPS<T> f1_TP_inverse = FPS<T>(f0_TP_SZ,MO(f1)).Inverse();f0.TP(f0_TP_SZ);FPS<T> f0_TP{f0_TP_SZ,MO(f0)};f0_TP *= f1_TP_inverse;f0_TP.TP(f0_TP_SZ);RE f0_TP;}TE <TY T>PO<T> FFTResidue(PO<T> f0,CO PO<T>& f1){if(f0.m_SZ >= f1.m_SZ){f0 -=(f0 / f1)* f1;f0.Reduce();}RE MO(f0);}
PS_FOR_FFT(P,24,31,128805723,Mod);
PS_FOR_FFT(167772161,26,17,29606852,Mod);
PS_FOR_FFT(469762049,27,30,15658735,Mod);
PS_FOR_FFT(754974721,25,362,415027540,Mod);

TE <TY T> IN FPS<T>::FPS(CRI N):PO<T>(),m_N(N){AS(m_N > 0);}TE <TY T> IN FPS<T>::FPS(CO FPS<T>& f):PO<T>(f),m_N(f.m_N){}TE <TY T> IN FPS<T>::FPS(FPS<T>&& f):PO<T>(MO(f.m_f)),m_N(f.m_N){}TE <TY T> IN FPS<T>::FPS(CRI N,T t):PO<T>(MO(t)),m_N(N){AS(m_N > 0);}TE <TY T> IN FPS<T>::FPS(CRI N,CO PO<T>& f):PO<T>(),m_N(N){AS(m_N > 0);TH->m_SZ = f.m_SZ < m_N?f.m_SZ:m_N;TH->m_f = VE<T>(TH->m_SZ);for(int i = 0;i < TH->m_SZ;i++){TH->m_f[i]= f.m_f[i];}}TE <TY T> IN FPS<T>::FPS(CRI N,PO<T>&& f):PO<T>(),m_N(N){if(f.m_SZ < m_N * 2){PO<T>::OP=(MO(f));if(f.m_SZ > m_N){TruncateFinal(m_N);}}else{TH->m_f = VE<T>(m_N);for(int i = 0;i < m_N;i++){TH->m_f[i]= MO(f.m_f[i]);}TH->m_SZ = m_N;}}TE <TY T> IN FPS<T>::FPS(CRI N,VE<T>&& f):PO<T>(),m_N(N){AS(m_N > 0);CO int f_SZ = f.SZ();if(f_SZ < m_N * 2){PO<T>::OP=(MO(f));if(f_SZ > m_N){TruncateFinal(m_N);}}else{TH->m_f = VE<T>(m_N);for(int i = 0;i < m_N;i++){TH->m_f[i]= MO(f[i]);}}}TE <TY T> IN FPS<T>::FPS(CRI N,CRI i,T t):PO<T>(),m_N(N){AS(m_N > 0);if(i < m_N?t != TH->c_zero():false){(*TH)[i]= MO(t);}}TE <TY T> IN FPS<T>& FPS<T>::OP=(FPS<T> f){PO<T>::OP=(MO(f.m_f));m_N = f.m_N;RE *TH;}TE <TY T> IN FPS<T>& FPS<T>::OP=(T n){PO<T>::OP=(MO(n));RE *TH;}TE <TY T> IN FPS<T>& FPS<T>::OP=(PO<T> f){RE OP=(FPS<T>(m_N,MO(f)));}TE <TY T> IN FPS<T>& FPS<T>::OP+=(CO T& t){PO<T>::OP+=(t);RE *TH;}TE <TY T>FPS<T>& FPS<T>::OP+=(CO PO<T>& f){CRI SZ_f = m_N < f.m_SZ?m_N:f.m_SZ;CRI SZ = TH->m_SZ < SZ_f?TH->m_SZ:SZ_f;for(int i = 0;i < SZ;i++){TH->m_f[i]+= f.m_f[i];}for(int i = SZ;i < SZ_f;i++){TH->m_f.push_back(f.m_f[i]);}TH->m_SZ = TH->m_f.SZ();RE *TH;}TE <TY T> IN FPS<T>& FPS<T>::OP+=(CO FPS<T>& f){AS(m_N <= f.m_N);CO PO<T>& f_ref = f;RE OP+=(f_ref);}TE <TY T> IN FPS<T>& FPS<T>::OP-=(CO T& t){PO<T>::OP-=(t);RE *TH;}TE <TY T>FPS<T>& FPS<T>::OP-=(CO PO<T>& f){CRI SZ_f = m_N < f.m_SZ?m_N:f.m_SZ;CRI SZ = TH->m_SZ < SZ_f?TH->m_SZ:SZ_f;for(int i = 0;i < SZ;i++){TH->m_f[i]-= f.m_f[i];}for(int i = SZ;i < SZ_f;i++){TH->m_f.push_back(-f.m_f[i]);}TH->m_SZ = TH->m_f.SZ();RE *TH;}TE <TY T> IN FPS<T>& FPS<T>::OP-=(CO FPS<T>& f){AS(m_N <= f.m_N);CO PO<T>& f_ref = f;RE OP-=(f_ref);}TE <TY T> IN FPS<T>& FPS<T>::OP*=(CO T& t){PO<T>::OP*=(t);RE *TH;}TE <TY T>FPS<T>& FPS<T>::OP*=(PO<T> f){*TH = FFTCN(forward<PO<T>>(*TH),MO(f),m_N);RE *TH;}TE <TY T> IN FPS<T>& FPS<T>::OP*=(FPS<T> f){AS(m_N <= f.m_N);RE OP*=(forward<PO<T>>(f));}TE <TY T> IN FPS<T>& FPS<T>::OP/=(CO T& t){PO<T>::OP/=(t);RE *TH;}TE <TY T> IN FPS<T>& FPS<T>::OP/=(CO FPS<T>& f){AS(m_N <= f.m_N);RE OP*=(m_N == f.m_N?f.Inverse():FPS<T>(m_N,f).Inverse());}TE <TY T> TE <TY P> IN FPS<T> FPS<T>::OP+(CO P& f)CO{RE MO(FPS<T>(*TH)+= f);}TE <TY T> IN FPS<T> FPS<T>::OP-()CO{RE MO(FPS<T>(m_N)-= *TH);}TE <TY T> TE <TY P> IN FPS<T> FPS<T>::OP-(CO P& f)CO{RE MO(FPS<T>(*TH)-= f);}TE <TY T> TE <TY P> IN FPS<T> FPS<T>::OP*(CO P& f)CO{RE MO(FPS<T>(*TH)*= f);}TE <TY T> TE <TY P> IN FPS<T> FPS<T>::OP/(CO P& f)CO{RE MO(FPS<T>(*TH)/= f);}TE <TY T>FPS<T> FPS<T>::Inverse()CO{AS(TH->m_SZ > 0 && TH->m_f[0]!= TH->c_zero());CO PO<T>& TH_ref = *TH;int pw;int pw_2 = 1;FPS<T> f_inv{pw_2,TH->c_one()/ TH->m_f[0]};WH(pw_2 < m_N){pw = pw_2;pw_2 <<= 1;f_inv.SetTruncation(pw_2);auto temp = f_inv * TH_ref;temp[0]--;temp *= f_inv;for(int i = pw;i < pw_2;i++){f_inv[i]-= temp[i];}}f_inv.SetTruncation(m_N);RE f_inv;}TE <TY T> IN VO FPS<T>::SetTruncation(CRI N)NE{if(N < m_N){TruncateFinal(N);}m_N = N;}TE <TY T> IN CRI FPS<T>::GetTruncation()CO NE{RE m_N;}TE <TY T> IN FPS<T>& FPS<T>::TruncateInitial(CRI N)NE{CRI SZ = N < TH->m_SZ?N:TH->m_SZ;for(int i = 0;i < SZ;i++){TH->m_f[i]= 0;}RE *TH;}TE <TY T> IN FPS<T>& FPS<T>::TruncateFinal(CRI N)NE{WH(TH->m_SZ > N){TH->m_f.pop_back();TH->m_SZ--;}RE *TH;}TE <TY T>FPS<T> Differential(CO FPS<T>& f){auto& SZ = f.SZ();auto& N = f.GetTruncation();if(SZ < 1){RE FPS<T>(1 < N?N - 1:1);}VE<T> df(SZ - 1);for(int i = 1;i < SZ;i++){df[i - 1]= f[i]* i;}RE FPS<T>(1 < N?N - 1:1,MO(df));}TE <TY T>FPS<T> Differential(CRI n,CO FPS<T>& f){if(n == 0){RE f;}if(n == 1){RE Differential<T>(f);}auto& SZ = f.SZ();auto& N = f.GetTruncation();if(SZ < n){RE FPS<T>(n < N?N - n:1);}VE<T> df(SZ - n);T coef = T::Factorial(n),numer = n,denom = 0;for(int i = n;i < SZ;i++){df[i - n]= f[i]* coef;(coef *= ++numer)/= ++denom;}RE FPS<T>(n < N?N - n:1,MO(df));}TE <TY T>FPS<T> ShiftedIntegral(CRI n,CO FPS<T>& f,CRI shift){auto& SZ = f.SZ();auto& N = f.GetTruncation();if(SZ + n < shift){RE FPS<T>{N + n > shift?N + n - shift:1};}VE<T> F(SZ + n - shift);if(n == 0){for(int i = shift;i < SZ;i++){F[i - shift]= f[i];}}else if(n == 1){CO int i_min = max(shift - 1,0);T denom = i_min;for(int i = i_min;i < SZ;i++){F[i + 1 - shift]= f[i]/ ++denom;}}else{CO int i_min = max(shift - n,0);for(int i = i_min;i < SZ;i++){F[i + n - shift]= f[i]* T::Factorial(i)* T::FactorialInverse(n + i);}}RE FPS<T>(N + n - shift,MO(F));}TE <TY T> IN FPS<T> Integral(CO FPS<T>& f){RE ShiftedIntegral<T>(1,f,0);}TE <TY T> IN FPS<T> Integral(CRI n,CO FPS<T>& f){RE ShiftedIntegral<T>(n,f,0);}TE <TY T>FPS<T> Exp(CO FPS<T>& f){AS(f[0]== f.c_zero());CRI N = f.GetTruncation();int pw;int pw_2 = 1;FPS<T> f_exp{pw_2,f.c_one()};WH(pw_2 < N){pw = pw_2;pw_2 *= 2;f_exp.SetTruncation(pw_2);auto temp = Differential<T>(f_exp);temp /= f_exp;temp = ShiftedIntegral(1,temp,pw);for(int i = 0;i < pw;i++){temp[i]-= f[i | pw];}temp *= f_exp;for(int i = 0;i < pw;i++){f_exp[i|pw]-= temp[i];}}f_exp.SetTruncation(N);RE f_exp;}TE <TY T> IN FPS<T> Log(CO FPS<T>& f){AS(f[0]== f.c_one());RE Integral<T>(Differential<T>(f)/= f);}TE <TY T,CL Traits> IN OS& OP<<(OS& os,CO FPS<T>& f){CO int N = f.GetTruncation();for(int i = 0;i < N;i++){(i > 0?os << " ":os)<< f[i];}RE os;}

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(CRI 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[](CRI i)CO{RE m_SZ <= i?c_zero():m_f[i];}TE <TY T> IN T& PO<T>::OP[](CRI 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(int 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){for(int i = 0;i < m_SZ;i++){m_f[i]+= f.m_f[i];}for(int i = m_SZ;i < f.m_SZ;i++){m_f.push_back(f.m_f[i]);}m_SZ = f.m_SZ;}else{for(int 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){for(int i = 0;i < m_SZ;i++){m_f[i]-= f.m_f[i];}for(int i = m_SZ;i < f.m_SZ;i++){m_f.push_back(- f.m_f[i]);}m_SZ = f.m_SZ;}else{for(int 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 T& t){if(t == c_one()){RE *TH;}CO T t_inv{c_one()/ t};for(int i = 0;i < m_SZ;i++){OP[](i)*= t_inv;}RE *TH;}TE <TY T>PO<T>& PO<T>::OP%=(CO T& t){if(t == c_one()){RE *TH = zero();}for(int i = 0;i < m_SZ;i++){m_f[i]%= t;}RE *TH;}TE <TY T>bool PO<T>::OP==(CO PO<T>& f)CO{CRI SZ0 = SZ();CRI SZ1 = f.SZ();CRI SZ_max = SZ0 < SZ1?SZ1:SZ0;for(int 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{CRI SZ_max = SZ();CO T& zero = PO<T>::c_zero();for(int 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(){Reduce();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 PO<T>& f)CO{RE MO(PO<T>(*TH)/= f);}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 PO<T>& f)CO{RE MO(PO<T>(*TH)%= f);}TE <TY T> IN CO VE<T>& PO<T>::GetCoefficient()CO NE{RE m_f;}TE <TY T> IN CRI PO<T>::SZ()CO NE{RE m_SZ;}TE <TY T> IN VO PO<T>::resize(CRI deg_plus)NE{m_f.resize(m_SZ = deg_plus);}TE <TY T>int PO<T>::Valuation()CO NE{for(int i = 0;i < m_SZ;i++){if(m_f[i]!= c_zero()){RE i;}}RE -1;}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>::Reduce(){CO T& z = c_zero();WH(m_SZ > 0 && m_f[m_SZ - 1]== z){m_f.pop_back();m_SZ--;}RE;}TE <TY T>VO PO<T>::TP(CRI N_trunc){WH(N_trunc > m_SZ){m_f.push_back(c_zero());m_SZ++;}CO int N_half = min(N_trunc,(m_SZ + 1)/ 2);for(int d = 0;d < N_half;d++){::swap(m_f[d],m_f[m_SZ - 1 - d]);}m_f.resize(N_trunc);m_SZ = N_trunc;RE;}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(CRI n,CO PO<T>& f){CRI SZ = f.SZ();if(SZ < n){RE PO<T>::zero();}VE<T> df(SZ - n);T coef = T::Factorial(n);int i = n;WH(i < SZ){df[i - n]= f[i]* coef;i++;(coef *= i)/=(i - n);}RE PO<T>(MO(df));}
TE <TY T>PO<T> PO<T>::NaiveCN(PO<T> f0,CRI valuation0,CO PO<T>& f1,CRI valuation1,CRI N_trunc){CO int SZ0 = f0.SZ();AS(0 <= valuation0 && valuation0 < SZ0);CO int SZ1 = f1.SZ();AS(0 <= valuation1 && valuation1 < SZ1);CO int i_ulim = min(SZ0,N_trunc - valuation1);for(int i = i_ulim - 1;i >= valuation0;i--){CO T f0i = f0[i];f0[i]*= f1[0];CO int j_ulim = min(SZ1,N_trunc - i),j_min = max(valuation1,1);for(int j = j_ulim - 1;j >= j_min;j--){f0[i+j]+= f0i * f1[j];}}RE MO(f0);}TE <TY T,CL Traits> IN OS& OP<<(OS& os,CO PO<T>& f){auto& SZ = f.SZ();for(int i = 0;i < SZ;i++){(i == 0?os:os << " ")<< f[i];}RE os;}
TE <TY T>PO<T> PO<T>::NaiveQuotient(PO<T> f0,CO PO<T>& f1){CO int diff = f0.m_SZ - f1.m_SZ;if(diff < 0){RE MO(f0);}CO T r = f0[f0.m_SZ-1]/ f1[f1.m_SZ-1];f0.m_f.pop_back();f0.m_SZ--;f0.Reduce();for(int i = diff;i < f0.m_SZ;i++){f0[i]-= r * f1[i - diff];}f0.Reduce();f0 = NaiveQuotient(MO(f0),f1);f0[diff]= r;RE MO(f0);}TE <TY T>PO<T> PO<T>::NaiveResidue(PO<T> f0,CO PO<T>& f1){CO int diff = f0.m_SZ - f1.m_SZ;if(diff < 0){RE MO(f0);}CO T r = f0[f0.m_SZ-1]/ f1[f1.m_SZ-1];f0.m_f.pop_back();f0.m_SZ--;f0.Reduce();for(int i = diff;i < f0.m_SZ;i++){f0[i]-= r * f1[i - diff];}f0.Reduce();RE NaiveResidue(MO(f0),f1);}TE <TY T>PO<T>& PO<T>::OP*=(PO<T> f){Reduce();if(m_SZ == 0){RE *TH;}f.Reduce();if(f.m_SZ == 0){RE *TH = MO(f);}CO int valuation0 = TH->Valuation();CO int valuation1 = f.Valuation();CO int le0 = m_SZ - valuation0;CO int le1 = f.m_SZ - valuation1;CO int SZ = m_SZ + f.m_SZ - 1;m_f = NaiveCN(MO(*TH),valuation0,f,valuation1,SZ);m_SZ = m_f.SZ();RE *TH;}TE <TY T>PO<T>& PO<T>::OP/=(CO PO<T>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();if(m_SZ < f.m_SZ){RE *TH = zero();}RE *TH = NaiveQuotient(MO(*TH),f);}TE <TY T>PO<T>& PO<T>::OP%=(CO PO<T>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();RE *TH = NaiveResidue(MO(*TH),f);}
#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 DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , test_case_num_bound , BOUND ); int test_case_num = 1; if CE( test_case_num_bound > 1 ){ FINISH_MAIN
  #ifdef USE_GETLINE
    #define SET_SEPARATE( SEPARATOR , ... ) VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
    #define SET( ... ) SET_SEPARATE( '\n' , __VA_ARGS__ )
    #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; SET_SEPARATE( SEPARATOR , __VA_ARGS__ )
    #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
    #define FINISH_MAIN GETLINE( test_case_num_str ); test_case_num = stoi( test_case_num_str ); ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } }
  #else
    #define SET( ... ) VariadicCin( cin , __VA_ARGS__ )
    #define CIN( LL , ... ) LL __VA_ARGS__; SET( __VA_ARGS__ )
    #define SET_A( I , N , ... ) VariadicResize( N + I , __VA_ARGS__ ); FOR( VARIABLE_FOR_SET_A , 0 , N ){ VariadicSet( cin , VARIABLE_FOR_SET_A + I , __VA_ARGS__ ); }
    #define CIN_A( LL , I , N , ... ) VE<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
    #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) VE<VE<LL>> VAR( N0 + I0 ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , VAR[VARIABLE_FOR_CIN_AA + I0] ); }
    #define FINISH_MAIN SET_ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } }
  #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 ); } cerr << TLE_VAR << endl; }
  #define MLE( CONDITION ) if( !( CONDITION ) ){ vector<vector<ll>> MLE_VAR{}; REPEAT( 1e6 ){ MLE_VAR.push_back( vector<ll>( 1e6 ) ); } cerr << MLE_VAR << endl; }
  #define OLE( CONDITION ) if( !( CONDITION ) ){ REPEAT( 1e8 ){ cerr << "OLE\n"; } }
#endif
#ifdef REACTIVE
  #ifndef DEBUG
    #define LOCAL( ... ) 
    #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 = loop_average_time , current_time = loop_start_time; int loop_count = current_time; assert( loop_count == 0 )
#define CURRENT_TIME ( current_time = static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 ) )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME , loop_count == 0 ? loop_start_time = current_time : loop_average_time = ( current_time - loop_start_time ) / loop_count , ++loop_count , current_time < TL_MS - loop_average_time * 2 - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( I , N , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A + I] , MIN , MAX ); }
#define SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_AA0 , 0 , N0 ){ FOR( VARIABLE_FOR_SET_AA1 , 0 , N1 ){ SET_ASSERT( A[VARIABLE_FOR_SET_AA0 + I0][VARIABLE_FOR_SET_AA1 + I1] , MIN , MAX ); } }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); SET_A_ASSERT( I , N , A , MIN , MAX )
#define CIN_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) vector A( N0 + I0 , vector<decldecay_t( MAX )>( N1 + I1 ) ); SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX )
#define PR1( A1 , ... ) A1
#define PR2( A1 , A2 , ... ) A2
#define PR3( A1 , A2 , A3 , ... ) A3
#define FOR_( VAR , INITIAL , FINAL , UPPER , COMP , INCR ) for( decldecay_t( UPPER ) VAR = INITIAL ; VAR COMP FINAL ; VAR INCR )
#define FOR( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , < , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQ( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , <= , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQINV( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , INITIAL , + 1 > , PR3( __VA_ARGS__ , -= PR2( __VA_ARGS__ , ? ) , -- ) )
#define ITR( ARRAY ) auto begin_ ## ARRAY = ARRAY .BE() , itr_ ## ARRAY = begin_ ## ARRAY , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define RUN( ARRAY , ... ) for( auto&& __VA_ARGS__ : ARRAY )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS ); cerr << fixed << setprecision( DECIMAL_DIGITS )
#define 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; }
#define CHECK( ... ) auto answer = Answer( __VA_ARGS__ , false ); CERR( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Answer == " , answer )

/* 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(CRL Rand_min,CRL Rand_max){AS(Rand_min <= Rand_max);ll AN = time(NULL);RE AN * rand()%(Rand_max + 1 - Rand_min)+ Rand_min;}

/* Set (2KB)*/
#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();T AN = *IT;S.erase(IT);RE AN;}TE <TY T> IN T pop_min(SET& S){AS(!S.empty());auto IT = S.BE();T AN = *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,TY U> IN SET& OP<<=(SET& S,U&& u){S.insert(T{forward<U>(u)});RE S;}TE <TY T> IN SET& OP>>=(SET& S,CO T& t){S.erase(t);RE S;}TE <TY T,TY U> IN SET& OP>>=(SET& S,CO U& u){RE S >>= T{u};}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& S0,SET S1){S0.merge(MO(S1));RE S0;}TE <TY T> IN SET OP|(SET S0,SET S1){RE MO(S0.SZ()< S1.SZ()?S1 |= MO(S0):S0 |= MO(S1));}
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 T& t,CO SET<T,Args...>& S){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>);

/* 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)-> decldecay_t((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 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_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> IN V<T>& OP OPR ## =(V<T>& a,CO T& t){for(auto& x:a){x OPR## = t;}RE a;}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_SHIFT_FOR_VE(V)TE <TY T> IN V<T>& OP<<=(V<T>& a,T t){a.push_back(MO(t));RE a;}TE <TY T,TY U> IN V<T>& OP<<=(V<T>& a,U&& u){RE a <<= T{forward<U>(u)};}TE <TY T> IN T pop(V<T>& a){AS(!a.empty());T AN = MO(a.back());a.pop_back();RE AN;}
#define DF_OF_ARS_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-(V<T> a){RE MO(a *= T(-1));}TE <TY T> IN V<T> OP*(CO T& t,V<T> v){RE MO(v *= t);}DF_OF_SHIFT_FOR_VE(V);
DF_OF_ARS_FOR_VE(VE);DF_OF_ARS_FOR_VE(LI);DF_OF_SHIFT_FOR_VE(basic_string);
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 V> IN VO Sort(V& a,CO bool& reversed = false){US T = decltype(a[0]);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 V0,TY V1> IN VO Sort(V0& a,V1& b,CO bool& reversed = false){CO int SZ = a.SZ();AS(SZ == int(b.SZ()));VE<pair<decltype(a[0]),decltype(b[0])>> 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 V> IN pair<VE<int>,VE<int>> IndexSort(CO V& a,CO bool& reversed = false){CO int SZ = a.SZ();auto index = id<int>(SZ),ord = index;sort(index.BE(),index.EN(),[&](CRI i,CRI j){CO pair ti{a[i],i},tj{a[j],j};RE reversed?tj < ti:ti < tj;});for(int i = 0;i < SZ;i++){ord[index[i]]= i;}RE{MO(index),MO(ord)};}TE <TY V> IN int len(CO V& a){RE a.SZ();}TE <TY V> IN VO Reverse(V& a){CO int SZ = len(a),half = SZ / 2;for(int i = 0;i < half;i++){swap(a[i],a[SZ-1-i]);}};TE <TY V> IN V Reversed(V a){Reverse(a);RE MO(a);}

/* 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 = 2e6;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);CE Mod<M>& OP^=(ll EX);CE Mod<M>& OP<<=(ll n);CE Mod<M>& OP>>=(ll 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(/,);CE Mod<M> OP^(ll EX)CO;CE Mod<M> OP<<(ll n)CO;CE Mod<M> OP>>(ll n)CO;CE Mod<M> OP-()CO NE;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>& Factorial(CRL n);ST IN CO Mod<M>& FactorialInverse(CRL n);ST IN Mod<M> Combination(CRL n,CRL i);ST IN CO Mod<M>& zero()NE;ST IN CO Mod<M>& one()NE;ST CE uint GetModulo()NE;CE Mod<M>& SignInvert()NE;IN Mod<M>& Invert();CE Mod<M>& PPW(ll EX)NE;CE Mod<M>& NNPW(ll EX)NE;ST IN CO Mod<M>& Inverse(CRI n);ST IN CO Mod<M>& TwoPower(CRI n);ST IN CO Mod<M>& TwoPowerInverse(CRI n);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> CE Mod<M>& Mod<M>::PPW(ll 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> CE Mod<M>& Mod<M>::NNPW(ll EX)NE{RE EX == 0?(m_n = 1,*TH):PPW(MO(EX));}TE <uint M> CE Mod<M>& Mod<M>::OP^=(ll EX){bool neg = EX < 0;AS(!(neg && m_n == 0));RE NNPW(MO(neg?(EX %= COants::g_order)== 0?EX:EX += COants::g_order:EX));}TE <uint M> CE Mod<M>& Mod<M>::OP<<=(ll n){RE *TH *=(n < 0 && -n < int(COants::g_memory_le))?TwoPowerInverse(- int(n)):(n >= 0 && n < int(COants::g_memory_le))?TwoPower(int(n)):Mod<M>(2)^= MO(n);}TE <uint M> CE Mod<M>& Mod<M>::OP>>=(ll n){RE *TH <<= MO(n *= -1);}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> CE Mod<M> Mod<M>::OP^(ll EX)CO{RE MO(Mod<M>(*TH)^= MO(EX));}TE <uint M> CE Mod<M> Mod<M>::OP<<(ll n)CO{RE MO(Mod<M>(*TH)<<= MO(n));}TE <uint M> CE Mod<M> Mod<M>::OP>>(ll 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):NNPW(COants::g_order_minus);}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(CRI n){AS(0 < n && n < int(COants::g_memory_le));ST VE<Mod<M>> memory ={zero(),one()};ST int 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>::TwoPower(CRI n){AS(0 <= n && n < int(COants::g_memory_le));ST VE<Mod<M>> memory ={one()};ST int le_curr = 1;WH(le_curr <= n){memory.push_back(memory.back()+ memory.back());le_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::TwoPowerInverse(CRI n){AS(0 <= n && n < int(COants::g_memory_le));ST VE<Mod<M>> memory ={one()};ST int le_curr = 1;WH(le_curr <= n){auto& m = memory.back().m_n;memory.push_back(DeRP(((m & 1)== 0?m:m + M)>> 1));le_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::Factorial(CRL n){AS(n >= 0);if(ll(M)<= n){RE zero();}ST VE<Mod<M>> memory ={one(),one()};ST int 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(CRL n){AS(0 <= n && n < ll(M));ST VE<Mod<M>> memory ={one(),one()};ST int 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(CRL n,CRL i){RE 0 <= i && 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> 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> CE Mod<M> Power(Mod<M> n,ll EX){RE MO(n ^= 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>);

/* Iteration (3KB) */
#define SPECIALSATION_OF_AR_PROGRESSION_SUM(TYPE)TE <> IN TYPE ArithmeticProgressionSum(CO TYPE& l,CO TYPE& r,CO TYPE& d){RE SpecialisedArithmeticProgressionSum(l,r,d);}
TE <TY T,TY U,TE <TY...> TY V,TY OPR> T LeftConnectiveProd(T t,CO V<U>& f,OPR opr){for(auto& u:f){t = opr(MO(t),u);}RE MO(t);}TE <TY T,TY U,TE <TY...> TY V> IN T Sum(T t,CO V<U>& f){RE LeftConnectiveProd(MO(t),f,[](T t0,CO U& u1){RE MO(t0 += u1);});}TE <TY T,TE <TY...> TY V> IN T Sum(CO V<T>& f){RE Sum(T{},f);}TE <TY T,TY U,TE <TY...> TY V> IN T Prod(T t,CO V<U>& f){RE LeftConnectiveProd(MO(t),f,[](T t0,CO U& u1){RE MO(t0 *= u1);});}TE <TY T,TE <TY...> TY V> IN T Prod(CO V<T>& f){RE Prod(T{1},f);}TE <TY T> IN T& SetMax(T& t){RE t;}TE <TY T,TY U,TY... Args> IN T& SetMax(T& t0,CO U& u1,CO Args&... args){RE SetMax(t0 < u1?t0 = u1:t0,args...);}TE <TY T> IN T& SetMin(T& t){RE t;}TE <TY T,TY U,TY... Args> IN T& SetMin(T& t0,CO U& u1,CO Args&... args){RE SetMin(u1 < t0?t0 = u1:t0,args...);}TE <TY T,TE <TY...> TY V> IN CO T& Max(CO V<T>& f){RE *max_element(f.BE(),f.EN());}TE <TY T,TY U,TY...Args> IN T Max(T t0,CO U& t1,CO Args&... args){RE MO(SetMax(t0,t1,args...));}TE <TY T,TE <TY...> TY V> IN CO T& Min(CO V<T>& f){RE *min_element(f.BE(),f.EN());}TE <TY T,TY U,TY...Args> IN T Min(T t0,CO U& t1,CO Args&... args){RE MO(SetMin(t0,t1,args...));}TE <TY T,TY UINT>T Power(CO T& t,CO UINT& EX,T init = 1){RE EX > 1?Power(t * t,EX >> 1,MO(EX & 1?init *= t:init)):MO(EX > 0?init *= t:(AS(EX == 0),init));}TE <TY T> IN T PowerMemorisation(CO T& t,CRI EX){AS(EX >= 0);ST Map<T,VE<T>> memory{};auto& AN = memory[t];if(AN.empty()){AN.push_back(1);}WH(int(AN.SZ())<= EX){AN.push_back(AN.back()* t);}RE AN[EX];}TE <TY INT> IN INT ArithmeticProgressionSum(CO INT& l,CO INT& r,CO INT& d = 1){RE(l + r)*(r - l + 1)/ 2;}TE <TY INT> IN INT SpecialisedArithmeticProgressionSum(CO INT& l,CO INT& r,CO INT& d){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);}
SPECIALSATION_OF_AR_PROGRESSION_SUM(int);
SPECIALSATION_OF_AR_PROGRESSION_SUM(uint);
SPECIALSATION_OF_AR_PROGRESSION_SUM(ll);
SPECIALSATION_OF_AR_PROGRESSION_SUM(ull);
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;}

/* Sqrt (1KB) */
TE <TY INT>INT RoundDownSqrt(CO INT& n){ST_AS(is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull>);AS(n >= 0);if(n <= 1){RE n;}CE INT r_max = is_same_v<INT,int>?46341:is_same_v<INT,uint>?65536:is_same_v<INT,ll>?3037000500:4294967296;INT l = 1,r = min(r_max,n);WH(l < r - 1){CO INT m =(l + r)>> 1;(m <= n / m?l:r)= m;}RE l;}TE <TY INT>INT RoundUpSqrt(CO INT& n){ST_AS(is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull>);AS(n >= 0);if(n <= 2){RE n;}CE INT r_max = is_same_v<INT,int>?46341:is_same_v<INT,uint>?65536:is_same_v<INT,ll>?3037000500:4294967296;CO INT n_minus = n - 1;INT l = 1,r = min(r_max,n);WH(l + 1 < r){CO INT m =(l + r)>> 1;(m <= n_minus / m?l:r)= m;}RE r;}TE <TY INT> bool IsSquare(CO INT& n){CO INT r = RoundDownSqrt(n);RE n == r * r;}

/* 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 = 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 = 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;}TE <TY INT>string ArrayToParenthesisString(CO VE<INT>& A){CO int N = A.SZ();string S(N,'(');for(int i = 0;i < N;i++){AS(0 <= A[i]&& A[i]<= 1);S[i]= "()"[A[i]];}RE S;}TE <TY INT = int>VE<INT> ParenthesisStringToArray(CO string& S){CO int N = S.SZ();VE<int> A(N);for(int i = 0;i < N;i++){A[i]= S[i]- '(';}RE A;}
#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 */