#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  // #define USE_GETLINE
#endif

#ifdef INCLUDE_MAIN

IN VO Solve()
{
  CIN( int , N );
  CIN_A( ll , A , N );
  CIN( int , Q );
  constexpr PrimeEnumeration<int,100> pe{};
  auto& C = pe.length();
  vector t( C , IntervalMultiplyLazySqrtDecomposition{ MultiplicativeMonoid( 1 ) , Module<int,int>() } );
  FOR( c , 0 , C ){
    vector<int> factor( N );
    FOR( i , 0 , N ){
      while( A[i] % pe[c] == 0 ){
	A[i] /= pe[c];
	factor[i]++;
      }
    }
    t[c].Reset( move( factor ) );
  }
  A.clear();
  FOR( q , 0 , Q ){
    CIN( int , type );
    CIN( ll , l , r , x ); l--; r--;
    if( type == 1 ){
      vector<int> e( C );
      FOR( c , 0 , C ){
	while( x % pe[c] == 0 ){
	  x /= pe[c];
	  e[c]++;
	}
      }
      FOR( c , 0 , C ){
	t[c].IntervalSet( l , r , e[c] );
      }
    } else if( type == 2 ){
      vector<int> e( C );
      FOR( c , 0 , C ){
	while( x % pe[c] == 0 ){
	  x /= pe[c];
	  e[c]++;
	}
      }
      FOR( c , 0 , C ){
	t[c].IntervalMultiply( l , r , e[c] );
      }
    } else if( type == 3 ){
      assert( x <= 100 );
      ll answer = 1;
      FOR( c , 0 , C ){
	x % pe[c] == 0 ? answer *= t[c].IntervalProduct( l , r ) + 1 : answer;
      }
      COUT( answer );
    }
  }
}
REPEAT_MAIN(1);

#else // INCLUDE_MAIN

#ifdef INCLUDE_SUB

// COMPAREに使用。圧縮時は削除する。
ll Naive( int N , int M , int K )
{
  ll answer = N + M + K;
  return answer;
}

// COMPAREに使用。圧縮時は削除する。
ll Answer( ll N , ll M , ll K )
{
  // START_WATCH;
  ll answer = N + M + K;

  // // TLに準じる乱択や全探索。デフォルトの猶予は100.0[ms]。
  // CEXPR( double , TL , 2000.0 );
  // while( CHECK_WATCH( TL ) ){

  // }
  return answer;
}

// 圧縮時は中身だけ削除する。
IN VO Experiment()
{
  // CEXPR( int , bound , 10 );
  // FOREQ( N , 0 , bound ){
  //   FOREQ( M , 0 , bound ){
  //     FOREQ( K , 0 , bound ){
  //   	COUT( N , M , K , ":" , Naive( N , M , K ) );
  //     }
  //   }
  //   // cout << Naive( N ) << ",\n"[N==bound];
  // }
}

// 圧縮時は中身だけ削除する。
IN VO SmallTest()
{
  // CEXPR( int , bound , 10 );
  // FOREQ( N , 0 , bound ){
  //   FOREQ( M , 0 , bound ){
  //     FOREQ( K , 0 , bound ){
  //   	COMPARE( N , M , K );
  //     }
  //   }
  //   // COMPARE( N );
  // }
}

#define INCLUDE_MAIN
#include __FILE__

#else // INCLUDE_SUB

#ifdef INCLUDE_LIBRARY

/*

C-x 3 C-x o C-x C-fによるファイル操作用

BFS (5KB)
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/BreadthFirstSearch/compress.txt

CoordinateCompress (3KB)
c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/CoordinateCompress/compress.txt

DFSOnTree (11KB)
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/DepthFirstSearch/Tree/a.hpp

Divisor (4KB)
c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Prime/Divisor/compress.txt

IntervalAddBIT (9KB)
c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/AffineSpace/BIT/IntervalAdd/compress.txt

Polynomial (21KB)
c:/Users/user/Documents/Programming/Mathematics/Polynomial/compress.txt

UnionFind (3KB)
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/UnionFindForest/compress.txt

*/

// VVV 常設でないライブラリは以下に挿入する。

TE <TY INT,INT val_limit,int LE_max = val_limit>CL PrimeEnumeration{PU:bool m_is_composite[val_limit];INT m_val[LE_max];int m_LE;CE PrimeEnumeration();CE CO INT& OP[](CRI n) CO;CE CO INT& Get(CRI n) CO;CE CO bool& IsComposite(CRI i) CO;CE CRI LE() CO NE;};
TE <TY INT,INT val_limit,int LE_max>CE PrimeEnumeration<INT,val_limit,LE_max>::PrimeEnumeration():m_is_composite(),m_val(),m_LE(0){for(INT i = 2;i < val_limit;i++){if(! m_is_composite[i]){INT j = i;WH((j += i)< val_limit){m_is_composite[j] = true;}m_val[m_LE++] = i;if(m_LE >= LE_max){break;}}}}TE <TY INT,INT val_limit,int LE_max> CE CO INT& PrimeEnumeration<INT,val_limit,LE_max>::OP[](CRI n)CO{assert(n < m_LE);RE m_val[n];}TE <TY INT,INT val_limit,int LE_max> CE CO INT& PrimeEnumeration<INT,val_limit,LE_max>::Get(CRI n)CO{RE OP[](n);}TE <TY INT,INT val_limit,int LE_max> CE CO bool& PrimeEnumeration<INT,val_limit,LE_max>::IsComposite(CRI i)CO{assert(i < val_limit);RE m_is_composite[i];}TE <TY INT,INT val_limit,int LE_max> CE CRI PrimeEnumeration<INT,val_limit,LE_max>::LE()CO NE{RE m_LE;}
TE <TY INT,INT val_limit,int LE_max,TY INT1,TY INT2,TY INT3>VO SetPrimeFactorisation(CO PrimeEnumeration<INT,val_limit,LE_max>& prime,CO INT1& n,VE<INT2>& P,VE<INT3>& EX){INT1 n_copy = n;int i = 0;WH(i < prime.m_LE){CO INT2& p = prime[i];if(p * p > n_copy){break;}if(n_copy % p == 0){P.push_back(p);EX.push_back(1);INT3& EX_back = EX.back();n_copy /= p;WH(n_copy % p == 0){EX_back++;n_copy /= p;}}i++;}if(n_copy != 1){P.push_back(n_copy);EX.push_back(1);}RE;}


TE <TY R,TY U>CL VirtualModule{PU:VI U Action(CO R& r,CO U& u)= 0;IN U PW(CO U& u,CO R& r);IN U ScalarProduct(CO R& r,CO U& u);};TE <TY R,TY U,TY O_U,TY GROUP>CL AbstractModule:VI PU VirtualModule<R,U>,PU GROUP{PU:O_U m_o_U;IN AbstractModule(CO R& dummy,O_U o_U,GROUP M);IN U Action(CO R& r,CO U& u);};TE <TY R,TY O_U,TY GROUP> AbstractModule(CO R& dummy,O_U o_U,GROUP M)-> AbstractModule<R,inner_t<GROUP>,O_U,GROUP>;TE <TY R,TY U>CL Module:VI PU VirtualModule<R,U>,PU AdditiveGroup<U>{PU:IN U Action(CO R& r,CO U& u);};
TE <TY R,TY U,TY O_U,TY GROUP> IN AbstractModule<R,U,O_U,GROUP>::AbstractModule(CO R& dummy,O_U o_U,GROUP M):GROUP(MO(M)),m_o_U(MO(o_U)){ST_AS(is_same_v<U,inner_t<GROUP>> && is_invocable_r_v<U,O_U,R,U>);}TE <TY R,TY U,TY O_U,TY GROUP> IN U AbstractModule<R,U,O_U,GROUP>::Action(CO R& r,CO U& u){RE m_o_U(r,u);}TE <TY R,TY U> IN U Module<R,U>::Action(CO R& r,CO U& u){RE r * u;}TE <TY R,TY U> IN U VirtualModule<R,U>::PW(CO U& u,CO R& r){RE Action(r,u);}TE <TY R,TY U> IN U VirtualModule<R,U>::ScalarProduct(CO R& r,CO U& u){RE Action(r,u);}

IN CE int Sqrt(CRI N)NE{if(N <= 1){RE 1;}int left = 0;int right = N;WH(left + 1 < right){int m =(left + right)/ 2;(m <=(N - 1)/ m?left:right)= m;}RE right;}

TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE>CL IntervalMultiplyLazySqrtDecomposition{PU:PT_MAGMA m_L;R_MODULE m_M;int m_N;int m_N_sqrt;int m_N_d;int m_N_m;VE<U> m_a;VE<U> m_b;VE<U> m_lazy_substitution;VE<bool> m_suspENed;VE<R> m_lazy_action;VE<U> m_lazy_MU;IN IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,CRI N = 0);IN IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,CRI N,CRI N_sqrt);IN IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,VE<U> a);IN IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,VE<U> a,CRI N_sqrt);TE <TY...Args> IN VO Reset(Args&&... args);IN VO Set(CRI i,CO U& u);IN VO IntervalSet(CRI i_start,CRI i_final,CO U& u);IN VO IntervalAct(CRI i_start,CRI i_final,CO R& r);IN VO IntervalMultiply(CRI i_start,CRI i_final,CO U& u);IN U OP[](CRI i);IN U Get(CRI i);IN U IntervalProduct(CRI i_start,CRI i_final);IN VO Initialise();IN VO SetProduct(CRI i);IN VO SolveSuspENedSubstitution(CRI d,CO U& u);IN VO IntervalSet_Body(CRI i_min,CRI i_ulim,CO U& u);IN VO SolveSuspENedAction(CRI d);IN VO IntervalAct_Body(CRI i_min,CRI i_ulim,CO R& r);IN VO IntervalMultiply_Body(CRI i_min,CRI i_ulim,CO U& u);IN U IntervalProduct_Body(CRI i_min,CRI i_ulim);};TE <TY PT_MAGMA,TY R_MODULE,TY...Args> IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,CO Args&... args)-> IntervalMultiplyLazySqrtDecomposition<inner_t<PT_MAGMA>,PT_MAGMA,inner_t<R_MODULE>,R_MODULE>;
TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,CRI N):IntervalMultiplyLazySqrtDecomposition(MO(L),MO(M),N,Sqrt(N)){}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,CRI N,CRI N_sqrt):m_L(MO(L)),m_M(MO(M)),m_N(N),m_N_sqrt(N_sqrt),m_N_d((m_N + m_N_sqrt - 1)/ m_N_sqrt),m_N_m(m_N_d * m_N_sqrt),m_a(N,m_M.One()),m_b(m_N_d,m_M.One()),m_lazy_substitution(m_b),m_suspENed(m_N_d),m_lazy_action(m_N_d,m_L.Point()),m_lazy_MU(m_b){Initialise();}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,VE<U> a):m_L(MO(L)),m_M(MO(M)),m_N(a.SZ()),m_N_sqrt(Sqrt(m_N)),m_N_d((m_N + m_N_sqrt - 1)/ m_N_sqrt),m_N_m(m_N_d * m_N_sqrt),m_a(MO(a)),m_b(m_N_d,m_M.One()),m_lazy_substitution(m_b),m_suspENed(m_N_d),m_lazy_action(m_N_d,m_L.Point()),m_lazy_MU(m_b){Initialise();}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,R_MODULE M,VE<U> a,CRI N_sqrt):m_L(MO(L)),m_M(MO(M)),m_N(a.SZ()),m_N_sqrt(N_sqrt),m_N_d((m_N + m_N_sqrt - 1)/ m_N_sqrt),m_N_m(m_N_d * m_N_sqrt),m_a(MO(a)),m_b(m_N_d,m_M.One()),m_lazy_substitution(m_b),m_suspENed(m_N_d),m_lazy_action(m_N_d,m_L.Point()),m_lazy_MU(m_b){Initialise();}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::Initialise(){ST_AS(is_same_v<R,inner_t<PT_MAGMA>> && is_same_v<U,inner_t<R_MODULE>>);m_a.reSZ(m_N_m,m_M.One());int i_min = 0;int i_ulim = m_N_sqrt;for(int d = 0;d < m_N_d;d++){U& m_bd = m_b[d];for(int i = i_min;i < i_ulim;i++){m_bd = m_M.Product(m_bd,m_a[i]);}i_min = i_ulim;i_ulim += m_N_sqrt;}}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> TE <TY...Args> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::Reset(Args&&...args){*TH = IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>(MO(m_L),MO(m_M),forward<Args>(args)...);}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::Set(CRI i,CO U& u){CO int d = i / m_N_sqrt;CO int i_min = d * m_N_sqrt;CO int i_ulim = i_min + m_N_sqrt;U& m_ai = m_a[i];U& m_bd = m_b[d];if(m_suspENed[d]){U& m_lazy_substitution_d = m_lazy_substitution[d];if(m_lazy_substitution_d != u){SolveSuspENedSubstitution(d,m_lazy_substitution_d);m_ai = u;m_bd = m_M.Product(m_M.PW(m_lazy_substitution_d,m_N_sqrt - 1),u);}}else{SolveSuspENedAction(d);if(m_ai != u){m_ai = u;SetProduct(d);}}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalSet(CRI i_start,CRI i_final,CO U& u){CO int i_min = max(i_start,0);CO int i_ulim = min(i_final + 1,m_N);CO int d_0 =(i_min + m_N_sqrt - 1)/ m_N_sqrt;CO int d_1 = max(d_0,i_ulim / m_N_sqrt);CO int d_0_N_sqrt = d_0 * m_N_sqrt;CO int d_1_N_sqrt = d_1 * m_N_sqrt;CO int i_0 = min(d_0_N_sqrt,i_ulim);CO int i_1 = max(i_0,d_1_N_sqrt);if(i_min < i_0){CO int d_0_minus = d_0 - 1;CO int d_0_N_sqrt_minus = d_0_N_sqrt - m_N_sqrt;U& m_bd = m_b[d_0_minus];VE<bool>::reference m_suspENed_d = m_suspENed[d_0_minus];if(m_suspENed_d){U& m_lazy_substitution_d = m_lazy_substitution[d_0_minus];IntervalSet_Body(d_0_N_sqrt_minus,i_min,m_lazy_substitution_d);IntervalSet_Body(i_min,i_0,u);IntervalSet_Body(i_0,d_0_N_sqrt,m_lazy_substitution_d);m_suspENed_d = false;m_bd = m_M.Product(m_M.PW(m_lazy_substitution_d,m_N_sqrt -(i_0 - i_min)),m_M.PW(u,i_0 - i_min));}else{SolveSuspENedAction(d_0_minus);IntervalSet_Body(i_min,i_0,u);m_bd = m_M.Product(m_M.Product(IntervalProduct_Body(d_0_N_sqrt_minus,i_min),m_M.PW(u,i_0 - i_min)),IntervalProduct_Body(i_0,d_0_N_sqrt));}}CO U PW = m_M.PW(u,m_N_sqrt);CO U& one = m_M.One();CO R& point = m_L.Point();for(int d = d_0;d < d_1;d++){m_b[d]= PW;m_lazy_substitution[d]= u;m_suspENed[d]= true;m_lazy_MU[d]= one;m_lazy_action[d]= point;}if(i_1 < i_ulim){CO int d_1_N_sqrt_plus = d_1_N_sqrt + m_N_sqrt;U& m_bd = m_b[d_1];VE<bool>::reference m_suspENed_d = m_suspENed[d_1];if(m_suspENed_d){U& m_lazy_substitution_d = m_lazy_substitution[d_1];IntervalSet_Body(d_1_N_sqrt,i_1,m_lazy_substitution_d);IntervalSet_Body(i_1,i_ulim,u);IntervalSet_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_substitution_d);m_suspENed_d = false;m_bd = m_M.Product(m_M.Product(m_M.PW(m_lazy_substitution_d,i_1 - d_1_N_sqrt),m_M.PW(u,i_ulim - i_1)),m_M.PW(m_lazy_substitution_d,d_1_N_sqrt_plus - i_ulim));}else{SolveSuspENedAction(d_1);IntervalSet_Body(i_1,i_ulim,u);m_bd = m_M.Product(m_M.Product(IntervalProduct_Body(d_1_N_sqrt,i_1),m_M.PW(u,i_ulim - i_1)),IntervalProduct_Body(i_ulim,d_1_N_sqrt_plus));}}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalAct(CRI i_start,CRI i_final,CO R& r){CO R& point = m_L.Point();if(r != point){CO U& one = m_M.One();CO int i_min = max(i_start,0);CO int i_ulim = min(i_final + 1,m_N);CO int d_0 =(i_min + m_N_sqrt - 1)/ m_N_sqrt;CO int d_1 = max(d_0,i_ulim / m_N_sqrt);CO int d_0_N_sqrt = d_0 * m_N_sqrt;CO int d_1_N_sqrt = d_1 * m_N_sqrt;CO int i_0 = min(d_0_N_sqrt,i_ulim);CO int i_1 = max(i_0,d_1_N_sqrt);if(i_min < i_0){CO int d_0_minus = d_0 - 1;CO int d_0_N_sqrt_minus = d_0_N_sqrt - m_N_sqrt;VE<bool>::reference m_suspENed_d = m_suspENed[d_0_minus];if(m_suspENed_d){U& m_lazy_substitution_d = m_lazy_substitution[d_0_minus];U& m_bd = m_b[d_0_minus];CO U u = m_M.Action(r,m_lazy_substitution_d);IntervalSet_Body(d_0_N_sqrt_minus,i_min,m_lazy_substitution_d);IntervalSet_Body(i_min,i_0,u);IntervalSet_Body(i_0,d_0_N_sqrt,m_lazy_substitution_d);m_suspENed_d = false;m_bd = m_M.Product(m_M.PW(m_lazy_substitution_d,m_N_sqrt -(i_0 - i_min)),m_M.PW(u,i_0 - i_min));}else{R& m_lazy_action_d = m_lazy_action[d_0_minus];if(m_lazy_action_d == point){IntervalAct_Body(i_min,i_0,r);}else{IntervalAct_Body(d_0_N_sqrt_minus,i_min,m_lazy_action_d);IntervalAct_Body(i_min,i_0,m_M.Action(r,m_lazy_action_d));IntervalAct_Body(i_0,d_0_N_sqrt,m_lazy_action_d);m_lazy_action_d = point;}U& m_lazy_MU_d = m_lazy_MU[d_0_minus];if(m_lazy_MU_d != one){IntervalMultiply_Body(d_0_N_sqrt_minus,i_min,m_lazy_MU_d);IntervalMultiply_Body(i_min,i_0,m_M.Action(r,m_lazy_MU_d));IntervalMultiply_Body(i_0,d_0_N_sqrt,m_lazy_MU_d);m_lazy_MU_d = one;}SetProduct(d_0_minus);}}for(int d = d_0;d < d_1;d++){U& m_bd = m_b[d];m_bd = m_M.Action(r,m_bd);if(m_suspENed[d]){U& m_lazy_substitution_d = m_lazy_substitution[d];m_lazy_substitution_d = m_M.Action(r,m_lazy_substitution_d);}else{R& m_lazy_action_d = m_lazy_action[d];m_lazy_action_d = m_M.Action(r,m_lazy_action_d);U& m_lazy_MU_d = m_lazy_MU[d];m_lazy_MU_d = m_M.Action(r,m_lazy_MU_d);}}if(i_1 < i_ulim){CO int d_1_N_sqrt_plus = d_1_N_sqrt + m_N_sqrt;VE<bool>::reference m_suspENed_d = m_suspENed[d_1];if(m_suspENed_d){U& m_lazy_substitution_d = m_lazy_substitution[d_1];U& m_bd = m_b[d_1];CO U u = m_M.Action(r,m_lazy_substitution_d);IntervalSet_Body(d_1_N_sqrt,i_1,m_lazy_substitution_d);IntervalSet_Body(i_1,i_ulim,u);IntervalSet_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_substitution_d);m_suspENed_d = false;m_bd = m_M.Product(m_M.PW(m_lazy_substitution_d,m_N_sqrt -(i_ulim - i_1)),m_M.PW(u,i_ulim - i_1));}else{R& m_lazy_action_d = m_lazy_action[d_1];if(m_lazy_action_d == point){IntervalAct_Body(i_1,i_ulim,r);}else{IntervalAct_Body(d_1_N_sqrt,i_1,m_lazy_action_d);IntervalAct_Body(i_1,i_ulim,m_M.Action(r,m_lazy_action_d));IntervalAct_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_action_d);m_lazy_action_d = point;}U& m_lazy_MU_d = m_lazy_MU[d_1];if(m_lazy_MU_d != one){IntervalMultiply_Body(d_1_N_sqrt,i_1,m_lazy_MU_d);IntervalMultiply_Body(i_1,i_ulim,m_M.Action(r,m_lazy_MU_d));IntervalMultiply_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_MU_d);m_lazy_MU_d = one;}SetProduct(d_1);}}}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalMultiply(CRI i_start,CRI i_final,CO U& u){CO U& one = m_M.One();if(u != one){CO R& point = m_L.Point();CO int i_min = max(i_start,0);CO int i_ulim = min(i_final + 1,m_N);CO int d_0 =(i_min + m_N_sqrt - 1)/ m_N_sqrt;CO int d_1 = max(d_0,i_ulim / m_N_sqrt);CO int d_0_N_sqrt = d_0 * m_N_sqrt;CO int d_1_N_sqrt = d_1 * m_N_sqrt;CO int i_0 = min(d_0_N_sqrt,i_ulim);CO int i_1 = max(i_0,d_1_N_sqrt);if(i_min < i_0){CO int d_0_minus = d_0 - 1;CO int d_0_N_sqrt_minus = d_0_N_sqrt - m_N_sqrt;U& m_bd = m_b[d_0_minus];m_bd = m_M.Product(m_bd,m_M.PW(u,i_0 - i_min));VE<bool>::reference m_suspENed_d = m_suspENed[d_0_minus];if(m_suspENed_d){U& m_lazy_substitution_d = m_lazy_substitution[d_0_minus];IntervalSet_Body(d_0_N_sqrt_minus,i_min,m_lazy_substitution_d);IntervalSet_Body(i_min,i_0,m_M.Product(m_lazy_substitution_d,u));IntervalSet_Body(i_0,d_0_N_sqrt,m_lazy_substitution_d);m_suspENed_d = false;}else{R& m_lazy_action_d = m_lazy_action[d_0_minus];if(m_lazy_action_d != point){IntervalAct_Body(d_0_N_sqrt_minus,d_0_N_sqrt,m_lazy_action_d);m_lazy_action_d = point;}U& m_lazy_MU_d = m_lazy_MU[d_0_minus];if(m_lazy_MU_d == one){IntervalMultiply_Body(i_min,i_0,u);}else{IntervalMultiply_Body(d_0_N_sqrt_minus,i_min,m_lazy_MU_d);IntervalMultiply_Body(i_min,i_0,m_M.Product(m_lazy_MU_d,u));IntervalMultiply_Body(i_0,d_0_N_sqrt,m_lazy_MU_d);m_lazy_MU_d = one;}}}CO U PW = m_M.PW(u,m_N_sqrt);for(int d = d_0;d < d_1;d++){U& m_bd = m_b[d];m_bd = m_M.Product(m_bd,PW);if(m_suspENed[d]){U& m_lazy_substitution_d = m_lazy_substitution[d];m_lazy_substitution_d = m_M.Product(m_lazy_substitution_d,u);}else{U& m_lazy_MU_d = m_lazy_MU[d];m_lazy_MU_d = m_M.Product(m_lazy_MU_d,u);}}if(i_1 < i_ulim){CO int d_1_N_sqrt_plus = d_1_N_sqrt + m_N_sqrt;U& m_bd = m_b[d_1];m_bd = m_M.Product(m_bd,m_M.PW(u,i_ulim - i_1));VE<bool>::reference m_suspENed_d = m_suspENed[d_1];if(m_suspENed_d){U& m_lazy_substitution_d = m_lazy_substitution[d_1];IntervalSet_Body(d_1_N_sqrt,i_1,m_lazy_substitution_d);IntervalSet_Body(i_1,i_ulim,m_M.Product(m_lazy_substitution_d,u));IntervalSet_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_substitution_d);m_suspENed_d = false;}else{R& m_lazy_action_d = m_lazy_action[d_1];if(m_lazy_action_d != point){IntervalAct_Body(d_1_N_sqrt,d_1_N_sqrt_plus,m_lazy_action_d);m_lazy_action_d = point;}U& m_lazy_MU_d = m_lazy_MU[d_1];if(m_lazy_MU_d == one){IntervalMultiply_Body(i_1,i_ulim,u);}else{IntervalMultiply_Body(d_1_N_sqrt,i_1,m_lazy_MU_d);IntervalMultiply_Body(i_1,i_ulim,m_M.Product(m_lazy_MU_d,u));IntervalMultiply_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_MU_d);m_lazy_MU_d = one;}}}}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN U IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalProduct_Body(CRI i_min,CRI i_ulim){U AN = m_M.One();for(int i = i_min;i < i_ulim;i++){AN = m_M.Product(AN,m_a[i]);}RE AN;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::SetProduct(CRI d){U& m_bd = m_b[d]= m_M.One();CO int i_min = d * m_N_sqrt;CO int i_ulim = i_min + m_N_sqrt;for(int i = i_min;i < i_ulim;i++){m_bd = m_M.Product(m_bd,m_a[i]);}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::SolveSuspENedSubstitution(CRI d,CO U& u){CO int i_min = d * m_N_sqrt;IntervalSet_Body(i_min,i_min + m_N_sqrt,u);m_suspENed[d]= false;RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalSet_Body(CRI i_min,CRI i_ulim,CO U& u){for(int i = i_min;i < i_ulim;i++){m_a[i]= u;}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::SolveSuspENedAction(CRI d){CO int i_min = d * m_N_sqrt;CO int i_ulim = i_min + m_N_sqrt;U& m_bd = m_b[d];R& m_lazy_action_d = m_lazy_action[d];if(m_lazy_action_d != m_L.Point()){IntervalAct_Body(i_min,i_ulim,m_lazy_action_d);m_bd = m_M.Action(m_lazy_action_d,m_bd);m_lazy_action_d = m_L.Point();}CO U& one = m_M.One();U& m_lazy_MU_d = m_lazy_MU[d];if(m_lazy_MU_d != one){IntervalMultiply_Body(i_min,i_ulim,m_lazy_MU_d);m_bd = m_M.Product(m_bd,m_M.PW(m_lazy_MU_d,m_N_sqrt));m_lazy_MU_d = one;}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalAct_Body(CRI i_min,CRI i_ulim,CO R& r){for(int i = i_min;i < i_ulim;i++){U& m_ai = m_a[i];m_ai = m_M.Action(r,m_ai);}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN VO IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalMultiply_Body(CRI i_min,CRI i_ulim,CO U& u){for(int i = i_min;i < i_ulim;i++){U& m_ai = m_a[i];m_ai = m_M.Product(m_ai,u);}RE;}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN U IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::Get(CRI i){CO int d = i / m_N_sqrt;RE m_suspENed[d]?m_lazy_substitution[d]:m_M.Product(m_M.Action(m_lazy_action[d],m_a[i]),m_lazy_MU[d]);}TE <TY R,TY PT_MAGMA,TY U,TY R_MODULE> IN U IntervalMultiplyLazySqrtDecomposition<R,PT_MAGMA,U,R_MODULE>::IntervalProduct(CRI i_start,CRI i_final){CO int i_min = max(i_start,0);CO int i_ulim = min(i_final + 1,m_N);CO int d_0 =(i_min + m_N_sqrt - 1)/ m_N_sqrt;CO int d_1 = max(d_0,i_ulim / m_N_sqrt);CO int i_0 = min(d_0 * m_N_sqrt,i_ulim);CO int i_1 = max(i_0,d_1 * m_N_sqrt);U AN = m_M.One();if(i_min < i_0){CO int d_0_minus = d_0 - 1;AN = m_suspENed[d_0_minus]?m_M.PW(m_lazy_substitution[d_0_minus],i_0 - i_min):m_M.Product(m_M.Action(m_lazy_action[d_0_minus],IntervalProduct_Body(i_min,i_0)),m_M.PW(m_lazy_MU[d_0_minus],i_0 - i_min));}for(int d = d_0;d < d_1;d++){AN = m_M.Product(AN,m_b[d]);}if(i_1 < i_ulim){AN = m_M.Product(AN,m_suspENed[d_1]?m_M.PW(m_lazy_substitution[d_1],i_ulim - i_1):m_M.Product(m_M.Action(m_lazy_action[d_1],IntervalProduct_Body(i_1,i_ulim)),m_M.PW(m_lazy_MU[d_1],i_ulim - i_1)));}RE AN;}

// 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
  #define REPEAT_MAIN( BOUND ) START_MAIN; signal( SIGABRT , &AlertAbort ); AutoCheck( exec_mode , use_getline ); if( exec_mode == sample_debug_mode || exec_mode == submission_debug_mode || exec_mode == library_search_mode ){ RE 0; } else if( exec_mode == experiment_mode ){ Experiment(); RE 0; } else if( exec_mode == small_test_mode ){ SmallTest(); RE 0; }; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if( exec_mode == solve_mode ){ if CE( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } } else if( exec_mode == random_test_mode ){ CERR( "ランダムテストを行う回数を指定してください。" ); SET_LL( test_case_num ); } FINISH_MAIN
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) if( exec_mode == solve_mode ){ SET_LL( A ); ASSERT( A , MIN , MAX ); } else if( exec_mode == random_test_mode ){ CERR( #A , " = " , ( A = GetRand( MIN , MAX ) ) ); } else { AS( false ); }
  #define SOLVE_ONLY ST_AS( __FUNCTION__[0] == 'S' )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define COUT( ... ) VariadicCout( cout << "出力： " , __VA_ARGS__ ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << endl
  #define COUT_A( A , N ) cout << "出力： "; OUTPUT_ARRAY( cout , A , N ) << endl
  #define CERR_ITR( A ) OUTPUT_ITR( cerr , A ) << endl
  #define COUT_ITR( A ) cout << "出力： "; OUTPUT_ITR( cout , A ) << endl
#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 ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) SET_LL( A ); ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #define CERR( ... ) 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define CERR_A( A , N ) 
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout , A , N ) << ENDL
  #define CERR_ITR( A ) 
  #define COUT_ITR( A ) OUTPUT_ITR( cout , A ) << ENDL
#endif
#ifdef REACTIVE
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#ifdef USE_GETLINE
  #define SET_LL( A ) { GETLINE( A ## _str ); A = stoll( A ## _str ); }
  #define GETLINE_SEPARATE( SEPARATOR , ... ) SOLVE_ONLY; string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) SOLVE_ONLY; GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
#else
  #define SET_LL( A ) cin >> A
  #define CIN( LL , ... ) SOLVE_ONLY; LL __VA_ARGS__; VariadicCin( cin , __VA_ARGS__ )
  #define SET_A( A , N ) SOLVE_ONLY; FOR( VARIABLE_FOR_SET_A , 0 , N ){ cin >> A[VARIABLE_FOR_SET_A]; }
  #define CIN_A( LL , A , N ) VE<LL> A( N ); SET_A( A , 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 FINISH_MAIN REPEAT( test_case_num ){ if CE( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } Solve(); CERR( "" ); } }
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now()
#define CURRENT_TIME static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME < TL_MS - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( A , N , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A] , MIN , MAX ); }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( A , N , MIN , MAX ) vector<decldecay_t( MAX )> A( N ); SET_A_ASSERT( A , N , 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 AUTO_ITR( ARRAY ) auto itr_ ## ARRAY = ARRAY .BE() , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( AUTO_ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT_ ## HOW_MANY_TIMES , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS )
#define OUTPUT_ARRAY( OS , A , N ) FOR( VARIABLE_FOR_OUTPUT_ARRAY , 0 , N ){ OS << A[VARIABLE_FOR_OUTPUT_ARRAY] << (VARIABLE_FOR_OUTPUT_ARRAY==N-1?"":" "); } OS
#define OUTPUT_ITR( OS , A ) { auto ITERATOR_FOR_OUTPUT_ITR = A.BE() , EN_FOR_OUTPUT_ITR = A.EN(); bool VARIABLE_FOR_OUTPUT_ITR = ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR; WH( VARIABLE_FOR_OUTPUT_ITR ){ OS << *ITERATOR_FOR_COUT_ITR; ( VARIABLE_FOR_OUTPUT_ITR = ++ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR ) ? OS : OS << " "; } } OS
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ ); auto answer = Answer( __VA_ARGS__ ); bool match = naive == answer; COUT( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive == " , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ 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 begind
#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 ST_AS static_assert
#define reMO_CO remove_const
#define is_COructible_v is_constructible_v
#define rBE rbegin
#define reSZ resize

// 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;
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>;
US path = pair<int,ll>;

// 入出力用
TE <CL Traits> IN basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is ) { RE is; }
TE <CL Traits , TY Arg , TY... ARGS> IN basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is , Arg& arg , ARGS&... args ) { RE VariadicCin( is >> arg , args... ); }
TE <CL Traits> IN basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , CO char& separator ) { RE is; }
TE <CL Traits , TY Arg , TY... ARGS> IN basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , CO char& separator , Arg& arg , ARGS&... args ) { RE VariadicGetline( getline( is , arg , separator ) , separator , args... ); }
TE <CL Traits , TY Arg> IN basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , CO VE<Arg>& arg ) { auto BE = arg.BE() , EN = arg.EN(); auto itr = BE; WH( itr != EN ){ ( itr == BE ? os : os << " " ) << *itr; itr++; } RE os; }
TE <CL Traits , TY Arg1 , TY Arg2> IN basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , CO pair<Arg1,Arg2>& arg ) { RE os << arg.first << " " << arg.second; }
TE <CL Traits , TY Arg> IN basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , CO Arg& arg ) { RE os << arg; }
TE <CL Traits , TY Arg1 , TY Arg2 , TY... ARGS> IN basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , CO Arg1& arg1 , CO Arg2& arg2 , CO ARGS&... args ) { RE VariadicCout( os << arg1 << " " , arg2 , args... ); }

// 算術用
TE <TY T> CE T PositiveBaseResidue( CO T& a , CO T& p ){ RE a >= 0 ? a % p : p - 1 - ( ( - ( a + 1 ) ) % p ); }
TE <TY T> CE T Residue( CO T& a , CO T& p ){ RE PositiveBaseResidue( a , p < 0 ? -p : p ); }
TE <TY T> CE T PositiveBaseQuotient( CO T& a , CO T& p ){ RE ( a - PositiveBaseResidue( a , p ) ) / p; }
TE <TY T> CE T Quotient( CO T& a , CO T& p ){ RE p < 0 ? PositiveBaseQuotient( -a , -p ) : PositiveBaseQuotient( a , p ); }

#define POWER( ANSWER , ARGUMENT , EXPONENT )				\
  ST_AS( ! is_same<decldecay_t( ARGUMENT ),int>::value && ! is_same<decldecay_t( ARGUMENT ),uint>::value ); \
  decldecay_t( ARGUMENT ) ANSWER{ 1 };					\
  {									\
    decldecay_t( ARGUMENT ) ARGUMENT_FOR_SQUARE_FOR_POWER = ( ARGUMENT ); \
    decldecay_t( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT ); \
    WH( EXPONENT_FOR_SQUARE_FOR_POWER != 0 ){				\
      if( EXPONENT_FOR_SQUARE_FOR_POWER % 2 == 1 ){			\
	ANSWER *= ARGUMENT_FOR_SQUARE_FOR_POWER;			\
      }									\
      ARGUMENT_FOR_SQUARE_FOR_POWER *= ARGUMENT_FOR_SQUARE_FOR_POWER;	\
      EXPONENT_FOR_SQUARE_FOR_POWER /= 2;				\
    }									\
  }									\

#define POWER_MOD( ANSWER , ARGUMENT , EXPONENT , MODULO )		\
  ll ANSWER{ 1 };							\
  {									\
    ll ARGUMENT_FOR_SQUARE_FOR_POWER = ( ( ARGUMENT ) % ( MODULO ) ) % ( MODULO ); \
    ARGUMENT_FOR_SQUARE_FOR_POWER < 0 ? ARGUMENT_FOR_SQUARE_FOR_POWER += ( MODULO ) : ARGUMENT_FOR_SQUARE_FOR_POWER; \
    decldecay_t( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT ); \
    WH( EXPONENT_FOR_SQUARE_FOR_POWER != 0 ){				\
      if( EXPONENT_FOR_SQUARE_FOR_POWER % 2 == 1 ){			\
	ANSWER = ( ANSWER * ARGUMENT_FOR_SQUARE_FOR_POWER ) % ( MODULO ); \
      }									\
      ARGUMENT_FOR_SQUARE_FOR_POWER = ( ARGUMENT_FOR_SQUARE_FOR_POWER * ARGUMENT_FOR_SQUARE_FOR_POWER ) % ( MODULO ); \
      EXPONENT_FOR_SQUARE_FOR_POWER /= 2;				\
    }									\
  }									\

#define FACTORIAL_MOD( ANSWER , ANSWER_INV , INVERSE , MAX_INDEX , CE_LENGTH , MODULO ) \
  ll ANSWER[CE_LENGTH];							\
  ll ANSWER_INV[CE_LENGTH];						\
  ll INVERSE[CE_LENGTH];						\
  {									\
    ll VARIABLE_FOR_PRODUCT_FOR_FACTORIAL = 1;				\
    ANSWER[0] = VARIABLE_FOR_PRODUCT_FOR_FACTORIAL;			\
    FOREQ( i , 1 , MAX_INDEX ){						\
      ANSWER[i] = ( VARIABLE_FOR_PRODUCT_FOR_FACTORIAL *= i ) %= ( MODULO ); \
    }									\
    ANSWER_INV[0] = ANSWER_INV[1] = INVERSE[1] = VARIABLE_FOR_PRODUCT_FOR_FACTORIAL = 1; \
    FOREQ( i , 2 , MAX_INDEX ){						\
      ANSWER_INV[i] = ( VARIABLE_FOR_PRODUCT_FOR_FACTORIAL *= INVERSE[i] = ( MODULO ) - ( ( ( ( MODULO ) / i ) * INVERSE[ ( MODULO ) % i ] ) % ( MODULO ) ) ) %= ( MODULO ); \
    }									\
  }									\

// 二分探索用
// EXPRESSIONがANSWERの広義単調関数の時、EXPRESSION >= CO_TARGETの整数解を格納。
#define BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , DESIRED_INEQUALITY , CO_TARGET , INEQUALITY_FOR_CHECK , UPDATE_U , UPDATE_L , UPDATE_ANSWER ) \
  ST_AS( ! is_same<decldecay_t( CO_TARGET ),uint>::value && ! is_same<decldecay_t( CO_TARGET ),ull>::value ); \
  ll ANSWER = MINIMUM;							\
  {									\
    ll L_BS = MINIMUM;							\
    ll U_BS = MAXIMUM;							\
    ANSWER = UPDATE_ANSWER;						\
    ll EXPRESSION_BS;							\
    CO ll CO_TARGET_BS = ( CO_TARGET );			\
    ll DIFFERENCE_BS;							\
    WH( L_BS < U_BS ){						\
      DIFFERENCE_BS = ( EXPRESSION_BS = ( EXPRESSION ) ) - CO_TARGET_BS; \
      CERR( "二分探索中：" , "L_BS =" , L_BS , "<=" , #ANSWER , "=" , ANSWER , "<=" , U_BS , "= U_BS :" , #EXPRESSION , "=" , EXPRESSION_BS , DIFFERENCE_BS > 0 ? ">" : DIFFERENCE_BS < 0 ? "<" : "=" , CO_TARGET_BS , "=" , #CO_TARGET ); \
      if( DIFFERENCE_BS INEQUALITY_FOR_CHECK 0 ){			\
	U_BS = UPDATE_U;						\
      } else {								\
	L_BS = UPDATE_L;						\
      }									\
      ANSWER = UPDATE_ANSWER;						\
    }									\
    if( L_BS > U_BS ){							\
      CERR( "二分探索失敗：" , "L_BS =" , L_BS , ">" , U_BS , "= U_BS :" , #ANSWER , ":=" , #MAXIMUM , "+ 1 =" , MAXIMUM + 1  ); \
      CERR( "二分探索マクロにミスがある可能性があります。変更前の版に戻してください。" ); \
      ANSWER = MAXIMUM + 1;						\
    } else {								\
      CERR( "二分探索終了：" , "L_BS =" , L_BS , "<=" , #ANSWER , "=" , ANSWER , "<=" , U_BS , "= U_BS" ); \
      CERR( "二分探索が成功したかを確認するために" , #EXPRESSION , "を計算します。" ); \
      CERR( "成功判定が不要な場合はこの計算を削除しても構いません。" );	\
      EXPRESSION_BS = ( EXPRESSION );					\
      CERR( "二分探索結果：" , #EXPRESSION , "=" , EXPRESSION_BS , ( EXPRESSION_BS > CO_TARGET_BS ? ">" : EXPRESSION_BS < CO_TARGET_BS ? "<" : "=" ) , CO_TARGET_BS ); \
      if( EXPRESSION_BS DESIRED_INEQUALITY CO_TARGET_BS ){		\
	CERR( "二分探索成功：" , #ANSWER , ":=" , ANSWER );		\
      } else {								\
	CERR( "二分探索失敗：" , #ANSWER , ":=" , #MAXIMUM , "+ 1 =" , MAXIMUM + 1 ); \
	CERR( "単調でないか、単調増加性と単調減少性を逆にしてしまったか、探索範囲内に解が存在しません。" ); \
	ANSWER = MAXIMUM + 1;						\
      }									\
    }									\
  }									\

// 単調増加の時にEXPRESSION >= CO_TARGETの最小解を格納。
#define BS1( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CO_TARGET , >= , ANSWER , ANSWER + 1 , ( L_BS + U_BS ) / 2 )
// 単調増加の時にEXPRESSION <= CO_TARGETの最大解を格納。
#define BS2( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CO_TARGET , > , ANSWER - 1 , ANSWER , ( L_BS + 1 + U_BS ) / 2 )
// 単調減少の時にEXPRESSION >= CO_TARGETの最大解を格納。
#define BS3( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CO_TARGET , < , ANSWER - 1 , ANSWER , ( L_BS + 1 + U_BS ) / 2 )
// 単調減少の時にEXPRESSION <= CO_TARGETの最小解を格納。
#define BS4( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CO_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CO_TARGET , <= , ANSWER , ANSWER + 1 , ( L_BS + U_BS ) / 2 )
// t以下の値が存在すればその最大値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MaximumLeq( set<T>& S , CO T& t ) { CO auto EN = S.EN(); if( S.empty() ){ RE EN; } auto itr = S.upper_bound( t ); RE itr == EN ? S.find( *( S.rBE() ) ) : itr == S.BE() ? EN : --itr; }
// t未満の値が存在すればその最大値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MaximumLt( set<T>& S , CO T& t ) { CO auto EN = S.EN(); if( S.empty() ){ RE EN; } auto itr = S.lower_bound( t ); RE itr == EN ? S.find( *( S.rBE() ) ) : itr == S.BE() ? EN : --itr; }
// t以上の値が存在すればその最小値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MinimumGeq( set<T>& S , CO T& t ) { RE S.lower_bound( t ); }
// tより大きい値が存在すればその最小値のiterator、存在しなければend()を返す。
TE <TY T> IN TY set<T>::iterator MinimumGt( set<T>& S , CO T& t ) { RE S.upper_bound( t ); }

// 尺取り法用
// VAR_TPAがINITからUPDATEを繰り返しCONTINUE_CONDITIONを満たす限り、ON_CONDITIONを判定して
// trueならON、falseならOFFとなる。直近のONの区間を[VAR_TPA_L,VAR_TPA_R)で管理する。
#define TPA( VAR_TPA , INIT , UPDATE , CONTINUE_CONDITION , ON_CONDITION , ONON , ONOFF , OFFON , OFFOFF , FINISH ) \
  {									\
    auto VAR_TPA = INIT;						\
    auto VAR_TPA ## _L = VAR_TPA;					\
    auto VAR_TPA ## _R = VAR_TPA;					\
    bool on_TPA = false;						\
    int state_TPA = 3;							\
    WH( CONTINUE_CONDITION ){						\
      bool on_TPA_next = ON_CONDITION;					\
      state_TPA = ( ( on_TPA ? 1 : 0 ) << 1 ) | ( on_TPA_next ? 1 : 0 ); \
      CERR( "尺取り中： [L,R) = [" , VAR_TPA ## _L , "," , VAR_TPA ## _R , ") ," , #VAR_TPA , "=" , VAR_TPA , "," , ( ( state_TPA >> 1 ) & 1 ) == 1 ? "on" : "off" , " ->" , ( state_TPA & 1 ) == 1 ? "on" : "off" ); \
      if( state_TPA == 0 ){						\
	OFFOFF; VAR_TPA ## _L = VAR_TPA ## _R = VAR_TPA; UPDATE;	\
      } else if( state_TPA == 1 ){					\
	OFFON; VAR_TPA ## _L = VAR_TPA; UPDATE; VAR_TPA ## _R = VAR_TPA; \
      } else if( state_TPA == 2 ){					\
	ONOFF; VAR_TPA ## _L = VAR_TPA ## _R = VAR_TPA; UPDATE;		\
      } else {								\
	ONON; UPDATE; VAR_TPA ## _R = VAR_TPA;				\
      }									\
      on_TPA = on_TPA_next;						\
    }									\
    CERR( "尺取り終了： [L,R) = [" , VAR_TPA ## _L , "," , VAR_TPA ## _R , ") ," , #VAR_TPA , "=" , VAR_TPA ); \
    FINISH;								\
  }									\

// データ構造用
TE <TY T , TE <TY...> TY V> IN V<T> OP+( CO V<T>& a0 , CO V<T>& a1 ) { if( a0.empty() ){ RE a1; } if( a1.empty() ){ RE a0; } AS( a0.SZ() == a1.SZ() ); V<T> answer{}; for( auto itr0 = a0.BE() , itr1 = a1.BE() , EN0 = a0.EN(); itr0 != EN0 ; itr0++ , itr1++ ){ answer.push_back( *itr0 + *itr1 ); } RE answer; }
TE <TY T , TY U> IN pair<T,U> OP+( CO pair<T,U>& t0 , CO pair<T,U>& t1 ) { RE { t0.first + t1.first , t0.second + t1.second }; }
TE <TY T , TY U , TY V> IN tuple<T,U,V> OP+( CO tuple<T,U,V>& t0 , CO tuple<T,U,V>& t1 ) { RE { get<0>( t0 ) + get<0>( t1 ) , get<1>( t0 ) + get<1>( t1 ) , get<2>( t0 ) + get<2>( t1 ) }; }
TE <TY T , TY U , TY V , TY W> IN tuple<T,U,V,W> OP+( CO tuple<T,U,V,W>& t0 , CO tuple<T,U,V,W>& t1 ) { RE { get<0>( t0 ) + get<0>( t1 ) , get<1>( t0 ) + get<1>( t1 ) , get<2>( t0 ) + get<2>( t1 ) , get<3>( t0 ) + get<3>( t1 ) }; }
TE <TY T> IN T Add( CO T& t0 , CO T& t1 ) { RE t0 + t1; }
TE <TY T> IN T XorAdd( CO T& t0 , CO T& t1 ){ RE t0 ^ t1; }
TE <TY T> IN T Multiply( CO T& t0 , CO T& t1 ) { RE t0 * t1; }
TE <TY T> IN CO T& Zero() { ST CO T z{}; RE z; }
TE <TY T> IN CO T& One() { ST CO T o = 1; RE o; }\
TE <TY T> IN T AddInv( CO T& t ) { RE -t; }
TE <TY T> IN T Id( CO T& v ) { RE v; }
TE <TY T> IN T Min( CO T& a , CO T& b ){ RE a < b ? a : b; }
TE <TY T> IN T Max( CO T& a , CO T& b ){ RE a < b ? b : a; }
TE <TY T , TE <TY...> TY V> IN auto Get( CO V<T>& a ) { return [&]( CRI i = 0 ){ RE a[i]; }; }

// グリッド問題用
int H , W , H_minus , W_minus , HW;
VE<VE<bool>> non_wall;
IN T2<int> EnumHW( CRI v ) { RE { v / W , v % W }; }
IN int EnumHW_inv( CO T2<int>& ij ) { auto& [i,j] = ij; RE i * W + j; }
CO string direction[4] = {"U","R","D","L"};
// (i,j)->(k,h)の方向番号を取得
IN int DirectionNumberOnGrid( CRI i , CRI j , CRI k , CRI h ){RE i<k?2:i>k?0:j<h?1:j>h?3:(AS(false),-1);}
// v->wの方向番号を取得
IN int DirectionNumberOnGrid( CRI v , CRI w ){auto [i,j]=EnumHW(v);auto [k,h]=EnumHW(w);RE DirectionNumberOnGrid(i,j,k,h);}
// 方向番号の反転U<->D、R<->L
IN int ReverseDirectionNumberOnGrid( CRI n ){AS(0<=n&&n<4);RE(n+2)%4;}
IN VO SetEdgeOnGrid( CO string& Si , CRI i , VE<LI<int>>& e , CO char& walkable = '.' ){FOR(j,0,W){if(Si[j]==walkable){int v = EnumHW_inv({i,j});if(i>0){e[EnumHW_inv({i-1,j})].push_back(v);}if(i+1<H){e[EnumHW_inv({i+1,j})].push_back(v);}if(j>0){e[EnumHW_inv({i,j-1})].push_back(v);}if(j+1<W){e[EnumHW_inv({i,j+1})].push_back(v);}}}}
IN VO SetEdgeOnGrid( CO string& Si , CRI i , VE<LI<path>>& e , CO char& walkable = '.' ){FOR(j,0,W){if(Si[j]==walkable){CO int v=EnumHW_inv({i,j});if(i>0){e[EnumHW_inv({i-1,j})].push_back({v,1});}if(i+1<H){e[EnumHW_inv({i+1,j})].push_back({v,1});}if(j>0){e[EnumHW_inv({i,j-1})].push_back({v,1});}if(j+1<W){e[EnumHW_inv({i,j+1})].push_back({v,1});}}}}
IN VO SetWallOnGrid( CO string& Si , CRI i , VE<VE<bool>>& non_wall , CO char& walkable = '.'  , CO char& unwalkable = '#' ){non_wall.push_back(VE<bool>(W));auto& non_wall_i=non_wall[i];FOR(j,0,W){non_wall_i[j]=Si[j]==walkable?true:(assert(Si[j]==unwalkable),false);}}

// デバッグ用
#ifdef DEBUG
  IN VO AlertAbort( int n ) { CERR( "abort関数が呼ばれました。assertマクロのメッセージが出力されていない場合はオーバーフローの有無を確認をしてください。" ); }
  VO AutoCheck( int& exec_mode , CO bool& use_getline );
  IN VO Solve();
  IN VO Experiment();
  IN VO SmallTest();
  IN VO RandomTest();
  ll GetRand( CRL Rand_min , CRL Rand_max );
  IN VO BreakPoint( CRI LINE ) {}
  int exec_mode;
  CEXPR( int , solve_mode , 0 );
  CEXPR( int , sample_debug_mode , 1 );
  CEXPR( int , submission_debug_mode , 2 );
  CEXPR( int , library_search_mode , 3 );
  CEXPR( int , experiment_mode , 4 );
  CEXPR( int , small_test_mode , 5 );
  CEXPR( int , random_test_mode , 6 );
  #ifdef USE_GETLINE
    CEXPR( bool , use_getline , true );
  #else
    CEXPR( bool , use_getline , false );
  #endif
#else
  ll GetRand( CRL Rand_min , CRL Rand_max ) { ll answer = time( NULL ); RE answer * rand() % ( Rand_max + 1 - Rand_min ) + Rand_min; }
#endif

// VVV 常設ライブラリは以下に挿入する。
// Map (1KB)
// c:/Users/user/Documents/Programming/Mathematics/Function/Map/compress.txt
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 , 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>>;

// Algebra (4KB)
// c:/Users/user/Documents/Programming/Mathematics/Algebra/compress.txt
#define DC_OF_CPOINT(POINT)IN CO U& POINT()CO NE
#define DC_OF_POINT(POINT)IN U& POINT()NE
#define DF_OF_CPOINT(POINT)TE <TY U> IN CO U& VirtualPointedSet<U>::POINT()CO NE{RE Point();}
#define DF_OF_POINT(POINT)TE <TY U> IN U& VirtualPointedSet<U>::POINT()NE{RE Point();}
TE <TY U>CL UnderlyingSet{PU:US type = U;};TE <TY U>CL VirtualPointedSet:VI PU UnderlyingSet<U>{PU:VI CO U& Point()CO NE = 0;VI U& Point()NE = 0;DC_OF_CPOINT(Unit);DC_OF_CPOINT(Zero);DC_OF_CPOINT(One);DC_OF_CPOINT(Infty);DC_OF_POINT(init);DC_OF_POINT(root);};TE <TY U>CL PointedSet:VI PU VirtualPointedSet<U>{PU:U m_b_U;IN PointedSet(CO U& b_u = U());IN CO U& Point()CO NE;IN U& Point()NE;};TE <TY U>CL VirtualNSet:VI PU UnderlyingSet<U>{PU:VI U Transfer(CO U& u)= 0;IN U Inverse(CO U& u);};TE <TY U,TY F_U>CL AbstractNSet:VI PU VirtualNSet<U>{PU:F_U m_f_U;IN AbstractNSet(F_U f_U);IN U Transfer(CO U& u);};TE <TY U>CL VirtualMagma:VI PU UnderlyingSet<U>{PU:VI U Product(CO U& u0,CO U& u1)= 0;IN U Sum(CO U& u0,CO U& u1);};TE <TY U = ll>CL AdditiveMagma:VI PU VirtualMagma<U>{PU:IN U Product(CO U& u0,CO U& u1);};TE <TY U = ll>CL MultiplicativeMagma:VI PU VirtualMagma<U>{PU:IN U Product(CO U& u0,CO U& u1);};TE <TY U,TY M_U>CL AbstractMagma:VI PU VirtualMagma<U>{PU:M_U m_m_U;IN AbstractMagma(M_U m_U);IN U Product(CO U& u0,CO U& u1);};
TE <TY U> IN PointedSet<U>::PointedSet(CO U& b_U):m_b_U(b_U){}TE <TY U> IN CO U& PointedSet<U>::Point()CO NE{RE m_b_U;}TE <TY U> IN U& PointedSet<U>::Point()NE{RE m_b_U;}DF_OF_CPOINT(Unit);DF_OF_CPOINT(Zero);DF_OF_CPOINT(One);DF_OF_CPOINT(Infty);DF_OF_POINT(init);DF_OF_POINT(root);TE <TY U,TY F_U> IN AbstractNSet<U,F_U>::AbstractNSet(F_U f_U):m_f_U(MO(f_U)){ST_AS(is_invocable_r_v<U,F_U,U>);}TE <TY U,TY F_U> IN U AbstractNSet<U,F_U>::Transfer(CO U& u){RE m_f_U(u);}TE <TY U> IN U VirtualNSet<U>::Inverse(CO U& u){RE Transfer(u);}TE <TY U,TY M_U> IN AbstractMagma<U,M_U>::AbstractMagma(M_U m_U):m_m_U(MO(m_U)){ST_AS(is_invocable_r_v<U,M_U,U,U>);}TE <TY U> IN U AdditiveMagma<U>::Product(CO U& u0,CO U& u1){RE u0 + u1;}TE <TY U> IN U MultiplicativeMagma<U>::Product(CO U& u0,CO U& u1){RE u0 * u1;}TE <TY U,TY M_U> IN U AbstractMagma<U,M_U>::Product(CO U& u0,CO U& u1){RE m_m_U(u0,u1);}TE <TY U> IN U VirtualMagma<U>::Sum(CO U& u0,CO U& u1){RE Product(u0,u1);}

TE <TY U>CL VirtualMonoid:VI PU VirtualMagma<U>,VI PU VirtualPointedSet<U>{};TE <TY U = ll>CL AdditiveMonoid:VI PU VirtualMonoid<U>,PU AdditiveMagma<U>,PU PointedSet<U>{};TE <TY U = ll>CL MultiplicativeMonoid:VI PU VirtualMonoid<U>,PU MultiplicativeMagma<U>,PU PointedSet<U>{PU:IN MultiplicativeMonoid(CO U& e_U);};TE <TY U,TY M_U>CL AbstractMonoid:VI PU VirtualMonoid<U>,PU AbstractMagma<U,M_U>,PU PointedSet<U>{PU:IN AbstractMonoid(M_U m_U,CO U& e_U);};
TE <TY U> IN MultiplicativeMonoid<U>::MultiplicativeMonoid(CO U& e_U):PointedSet<U>(e_U){}TE <TY U,TY M_U> IN AbstractMonoid<U,M_U>::AbstractMonoid(M_U m_U,CO U& e_U):AbstractMagma<U,M_U>(MO(m_U)),PointedSet<U>(e_U){}

TE <TY U>CL VirtualGroup:VI PU VirtualMonoid<U>,VI PU VirtualPointedSet<U>,VI PU VirtualNSet<U>{};TE <TY U = ll>CL AdditiveGroup:VI PU VirtualGroup<U>,PU AdditiveMonoid<U>{PU:IN U Transfer(CO U& u);};TE <TY U,TY M_U,TY I_U>CL AbstractGroup:VI PU VirtualGroup<U>,PU AbstractMonoid<U,M_U>,PU AbstractNSet<U,I_U>{PU:IN AbstractGroup(M_U m_U,CO U& e_U,I_U i_U);};
TE <TY U,TY M_U,TY I_U> IN AbstractGroup<U,M_U,I_U>::AbstractGroup(M_U m_U,CO U& e_U,I_U i_U):AbstractMonoid<U,M_U>(MO(m_U),e_U),AbstractNSet<U,I_U>(MO(i_U)){}TE <TY U> IN U AdditiveGroup<U>::Transfer(CO U& u){RE -u;}

// 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