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

VO Solve()
{

  CIN( int , N , Q );
  CIN_A( int , 0 , N , A );
  using U = vector<ll>;

  int D = 6;
  vector<U> X_pre( N );
  auto val = [&]( const int& x ){
    U a( D );
    a[0] = x;
    FOR( d , 1 , D ){
      a[d] = RoundDownSqrt( a[d-1] );
    }
    return a;
  };
  FOR( i , 0 , N ){
    X_pre[i] = val( A[i] );
  }

  /* SetTheory/DirectProduct/AffineSpace/SqrtDecomposition/LazyEvaluation/ */
  LazySqrtDecomposition X{ AdditiveMonoid<int>{} , AbstractBiModule{ int{} , int{} , [&](const int& r,U u){ if( r < D ){ FOR( d , 0 , D ){ u[d] = d + r < D ? move( u[d+r] ) : u[d-1]; } } else { u = U( D , u[D-1] ); }  return move( u ); } , [&](U u,const int& r){ FOR( d , 0 , D ){ u[d] *= r; }  return move( u ); } , AbstractMonoid{ [&](U u0 ,const U& u1){ return move( u0 += u1 ); } , U( D ) } } , move( X_pre ) };

  FOR( q , 0 , Q ){
    CIN( int , type );
    if( type == 0 ){
      CIN( int , l , r ); --r;
      auto a = X.IntervalProduct( l , r );
      WHAT( a );
      COUT( a[0] );
      #ifdef DEBUG
        ll b = 0;
        FOREQ( i , l , r ){
          b += A[i];
        }
        assert( a[0] == b );
      #endif
    } else if( type == 1 ){
      CIN( int , l , r ); --r;
      CIN( int , x );
      X.IntervalSet( l , r , val( x ) );
      #ifdef DEBUG
        FOREQ( i , l , r ){
          A[i] = x;
        }
      #endif
    } else if( type == 2 ){
      CIN( int , l , r ); --r;
      X.IntervalAct( l , r , 1 );
      #ifdef DEBUG
        FOREQ( i , l , r ){
          A[i] = RoundDownSqrt( A[i] );
        }
      #endif
    }
    #ifdef DEBUG
      FOR( i , 0 , N ){
        CERRNS( X[i][0] , " \n"[i==N-1] );
        assert( X[i][0] == A[i] );
      }
    #endif
  }

}
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( "全てのケースを確認しました。" );
}

/* 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( "全てのケースを確認しました。" );
}

#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。*/

#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/AffineSpace/SqrtDecomposition/LazyEvaluation/a_Body.hpp"
#else
TE <TY L,TY R,TY U>CL VirtualBiModule:VI PU UnderlyingSet<U>{PU:VI U LAction(CO L& l,U u)= 0;VI U RAction(U u,CO R& r)= 0;IN U ScalarProduct(CO L& l,U u);IN U Power(U u,CO R& r);};TE <TY L,TY R,TY O_U_L,TY O_U_R,TY GROUP,TY U = inner_t<GROUP>>CL AbstractBiModule:PU VirtualBiModule<L,R,U>,PU GROUP{PU:O_U_L m_o_U_L;O_U_R m_o_U_R;IN AbstractBiModule(CO L& dummy_l,CO R& dummy_r,O_U_L o_U_L,O_U_R o_U_R,GROUP M);IN AbstractBiModule<L,R,U,O_U_L,O_U_R,GROUP>& OP=(CO AbstractBiModule<L,R,U,O_U_L,O_U_R,GROUP>&)NE;IN U LAction(CO L& l,U u);IN U RAction(U u,CO R& r);};TE <TY L,TY R,TY U>CL BiModule:VI PU VirtualBiModule<L,R,U>,PU AdditiveGroup<U>{PU:IN U LAction(CO L& r,U u);IN U RAction(U u,CO R& r);};
TE <TY L,TY R,TY O_U_L,TY O_U_R,TY GROUP,TY U> IN AbstractBiModule<L,R,O_U_L,O_U_R,GROUP,U>::AbstractBiModule(CO L& dummy_l,CO R& dummy_r,O_U_L o_U_L,O_U_R o_U_R,GROUP M):GROUP(MO(M)),m_o_U_L(MO(o_U_L)),m_o_U_R(MO(o_U_R)){ST_AS(is_same_v<U,inner_t<GROUP>> && is_invocable_r_v<U,O_U_L,CO L&,U> && is_invocable_r_v<U,O_U_R,U,CO R&>);}TE <TY L,TY R,TY O_U_L,TY O_U_R,TY GROUP,TY U> IN U AbstractBiModule<L,R,O_U_L,O_U_R,GROUP,U>::LAction(CO L& l,U u){RE m_o_U_L(l,MO(u));}TE <TY L,TY R,TY U> IN U BiModule<L,R,U>::LAction(CO L& l,U u){RE MO(u *= l);}TE <TY L,TY R,TY O_U_L,TY O_U_R,TY GROUP,TY U> IN U AbstractBiModule<L,R,O_U_L,O_U_R,GROUP,U>::RAction(U u,CO R& r){RE m_o_U_R(MO(u),r);}TE <TY L,TY R,TY U> IN U BiModule<L,R,U>::RAction(U u,CO R& r){RE MO(u *= r);}TE <TY L,TY R,TY U> IN U VirtualBiModule<L,R,U>::ScalarProduct(CO L& l,U u){RE LAction(l,MO(u));}TE <TY L,TY R,TY U> IN U VirtualBiModule<L,R,U>::Power(U u,CO R& r){RE RAction(MO(u),r);}

CL SqrtDecompositionCoordinate{PU:int m_N;int m_N_sqrt;int m_N_d;int m_N_m;IN SqrtDecompositionCoordinate(CRI N = 0);IN SqrtDecompositionCoordinate(CRI N,CRI N_sqrt);IN CRI size()CO NE;IN CRI BucketSize()CO NE;IN CRI BucketCount()CO NE;};
IN SqrtDecompositionCoordinate::SqrtDecompositionCoordinate(CRI N):SqrtDecompositionCoordinate(N,RoundUpSqrt(N)){};IN SqrtDecompositionCoordinate::SqrtDecompositionCoordinate(CRI N,CRI N_sqrt):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){}IN CRI SqrtDecompositionCoordinate::size()CO NE{RE m_N;}IN CRI SqrtDecompositionCoordinate::BucketSize()CO NE{RE m_N_sqrt;}IN CRI SqrtDecompositionCoordinate::BucketCount()CO NE{RE m_N_d;}

#define SFINAE_FOR_SD_S enable_if_t<is_invocable_r_v<bool,F,U,int>>*

TE <TY PT_MAGMA,TY RN_BIMODULE,TY R = inner_t<PT_MAGMA>,TY U = inner_t<RN_BIMODULE>>CL LazySqrtDecomposition:PU SqrtDecompositionCoordinate{PU:PT_MAGMA m_L;RN_BIMODULE m_M;VE<U> m_a;VE<U> m_b;VE<U> m_lazy_substitution;VE<bool> m_suspended;VE<R> m_lazy_action;TE <TY...Args> IN LazySqrtDecomposition(PT_MAGMA L,RN_BIMODULE M,CRI N = 0,CO Args&... args);TE <TY...Args> IN LazySqrtDecomposition(PT_MAGMA L,RN_BIMODULE M,VE<U> a,CO Args&... args);TE <TY...Args> IN VO Initialise(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 U OP[](CRI i);IN U Get(CRI i);IN U IntervalProduct(CRI i_start,CRI i_final);TE <TY F,SFINAE_FOR_SD_S = nullptr> IN int Search(CRI i_start,CO F& f,CO bool& reversed = false);IN int Search(CRI i_start,CO U& u,CO bool& reversed = false);IN VO COruct();IN VO SetProduct(CRI i);IN VO SolveSuspendedSubstitution(CRI d,CO U& u);IN VO IntervalSet_Body(CRI i_min,CRI i_ulim,CO U& u);IN VO SolveSuspendedAction(CRI d);IN VO IntervalAct_Body(CRI i_min,CRI i_ulim,CO R& r);IN U IntervalProduct_Body(CRI i_min,CRI i_ulim);TE <TY F> int Search_Body(CRI i_start,CO F& f,U product_temp);TE <TY F> int SearchReverse_Body(CRI i_final,CO F& f,U sum_temp);};
TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> TE <TY...Args> IN LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::LazySqrtDecomposition(PT_MAGMA L,RN_BIMODULE M,CRI N,CO Args&... args):SqrtDecompositionCoordinate(N,args...),m_L(MO(L)),m_M(MO(M)),m_a(N,m_M.One()),m_b(m_N_d,m_M.One()),m_lazy_substitution(m_b),m_suspended(m_N_d),m_lazy_action(m_N_d,m_L.Point()){COruct();}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> TE <TY...Args> IN LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::LazySqrtDecomposition(PT_MAGMA L,RN_BIMODULE M,VE<U> a,CO Args&... args):SqrtDecompositionCoordinate(a.SZ(),args...),m_L(MO(L)),m_M(MO(M)),m_a(MO(a)),m_b(m_N_d,m_M.One()),m_lazy_substitution(m_b),m_suspended(m_N_d),m_lazy_action(m_N_d,m_L.Point()){COruct();}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::COruct(){ST_AS(is_same_v<R,inner_t<PT_MAGMA>> && is_same_v<U,inner_t<RN_BIMODULE>>);m_a.resize(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(MO(m_bd),m_a[i]);}i_min = i_ulim;i_ulim += m_N_sqrt;}}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> TE <TY...Args> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::Initialise(Args&&...args){LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U> temp{m_L,m_M,forward<Args>(args)...};SqrtDecompositionCoordinate::OP=(temp);m_a = MO(temp.m_a);m_b = MO(temp.m_b);m_lazy_substitution = MO(temp.m_lazy_substitution);m_suspended = MO(temp.m_suspended);m_lazy_action = MO(temp.m_lazy_action);}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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];if(m_suspended[d]){U& m_lazy_substitution_d = m_lazy_substitution[d];if(m_lazy_substitution_d != u){SolveSuspendedSubstitution(d,m_lazy_substitution_d);m_ai = u;m_b[d]= m_M.Product(m_M.Product(m_M.Power(m_lazy_substitution_d,i - i_min),u),m_M.Power(m_lazy_substitution_d,i_ulim -(i + 1)));}}else{SolveSuspendedAction(d);if(m_ai != u){m_ai = u;SetProduct(d);}}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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_suspended_d = m_suspended[d_0_minus];if(m_suspended_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_suspended_d = false;m_bd = m_M.Product(m_M.Product(m_M.Power(m_lazy_substitution_d,i_min - d_0_N_sqrt_minus),m_M.Power(u,i_0 - i_min)),m_M.Power(m_lazy_substitution_d,d_0_N_sqrt - i_0));}else{SolveSuspendedAction(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.Power(u,i_0 - i_min)),IntervalProduct_Body(i_0,d_0_N_sqrt));}}CO U pw = m_M.Power(u,m_N_sqrt);for(int d = d_0;d < d_1;d++){m_b[d]= pw;m_lazy_substitution[d]= u;m_suspended[d]= true;m_lazy_action[d]= m_L.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_suspended_d = m_suspended[d_1];if(m_suspended_d){CO 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_suspended_d = false;m_bd = m_M.Product(m_M.Product(m_M.Power(m_lazy_substitution_d,i_1 - d_1_N_sqrt),m_M.Power(u,i_ulim - i_1)),m_M.Power(m_lazy_substitution_d,d_1_N_sqrt_plus - i_ulim));}else{SolveSuspendedAction(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.Power(u,i_ulim - i_1)),IntervalProduct_Body(i_ulim,d_1_N_sqrt_plus));}}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::IntervalAct(CRI i_start,CRI i_final,CO R& r){if(r != 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;VE<bool>::reference m_suspended_d = m_suspended[d_0_minus];if(m_suspended_d){CO U& m_lazy_substitution_d = m_lazy_substitution[d_0_minus];U& m_bd = m_b[d_0_minus];CO U u = m_M.ScalarProduct(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_suspended_d = false;m_bd = m_M.Product(m_M.Product(m_M.Power(m_lazy_substitution_d,i_min - d_0_N_sqrt_minus),m_M.Power(u,i_0 - i_min)),m_M.Power(m_lazy_substitution_d,d_0_N_sqrt - i_0));}else{R& m_lazy_action_d = m_lazy_action[d_0_minus];if(m_lazy_action_d == m_L.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_L.Product(r,m_lazy_action_d));IntervalAct_Body(i_0,d_0_N_sqrt,m_lazy_action_d);m_lazy_action_d = m_L.Point();}SetProduct(d_0_minus);}}for(int d = d_0;d < d_1;d++){U& m_bd = m_b[d];m_bd = m_M.ScalarProduct(r,m_bd);if(m_suspended[d]){U& m_lazy_substitution_d = m_lazy_substitution[d];m_lazy_substitution_d = m_M.ScalarProduct(r,m_lazy_substitution_d);}else{R& m_lazy_action_d = m_lazy_action[d];m_lazy_action_d = m_L.Product(r,m_lazy_action_d);}}if(i_1 < i_ulim){CO int d_1_N_sqrt_plus = d_1_N_sqrt + m_N_sqrt;VE<bool>::reference m_suspended_d = m_suspended[d_1];if(m_suspended_d){CO U& m_lazy_substitution_d = m_lazy_substitution[d_1];U& m_bd = m_b[d_1];CO U u = m_M.ScalarProduct(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_suspended_d = false;m_bd = m_M.Product(m_M.Product(m_M.Power(m_lazy_substitution_d,i_1 - d_1_N_sqrt),m_M.Power(u,i_ulim - i_1)),m_M.Power(m_lazy_substitution_d,d_1_N_sqrt_plus - i_ulim));}else{R& m_lazy_action_d = m_lazy_action[d_1];if(m_lazy_action_d == m_L.Point()){IntervalAct_Body(i_1,i_ulim,r);SetProduct(d_1);}else{IntervalAct_Body(d_1_N_sqrt,i_1,m_lazy_action_d);IntervalAct_Body(i_1,i_ulim,m_L.Product(r,m_lazy_action_d));IntervalAct_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_action_d);m_lazy_action_d = m_L.Point();SetProduct(d_1);}}}}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN U LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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(MO(AN),m_a[i]);}RE AN;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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(MO(m_bd),m_a[i]);}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::SolveSuspendedSubstitution(CRI d,CO U& u){CO int i_min = d * m_N_sqrt;IntervalSet_Body(i_min,i_min + m_N_sqrt,u);m_suspended[d]= false;RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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 PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::SolveSuspendedAction(CRI d){R& m_lazy_action_d = m_lazy_action[d];if(m_lazy_action_d != m_L.Point()){CO int i_min = d * m_N_sqrt;CO int i_ulim = i_min + m_N_sqrt;IntervalAct_Body(i_min,i_ulim,m_lazy_action_d);m_lazy_action_d = m_L.Point();}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN U LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::OP[](CRI i){AS(0 <= i && i < m_N);CO int d = i / m_N_sqrt;RE m_suspended[d]?m_lazy_substitution[d]:m_M.ScalarProduct(m_lazy_action[d],m_a[i]);}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN U LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::Get(CRI i){RE OP[](i);}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN U LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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_suspended[d_0_minus]?m_M.Power(m_lazy_substitution[d_0_minus],i_0 - i_min):m_M.ScalarProduct(m_lazy_action[d_0_minus],IntervalProduct_Body(i_min,i_0));}for(int d = d_0;d < d_1;d++){AN = m_M.Product(MO(AN),m_b[d]);}if(i_1 < i_ulim){AN = m_M.Product(MO(AN),m_suspended[d_1]?m_M.Power(m_lazy_substitution[d_1],i_ulim - i_1):m_M.ScalarProduct(m_lazy_action[d_1],IntervalProduct_Body(i_1,i_ulim)));}RE AN;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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.ScalarProduct(r,m_ai);}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> TE <TY F,SFINAE_FOR_SD_S> IN int LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::Search(CRI i_start,CO F& f,CO bool& reversed){RE reversed?SearchReverse_Body(i_start,f,m_M.One()):Search_Body(i_start,f,m_M.One());}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN int LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::Search(CRI i_start,CO U& u,CO bool& reversed){RE Search(i_start,[&](CO U& product,CRI){RE !(product < u);},reversed);}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> TE <TY F> int LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::Search_Body(CRI i_start,CO F& f,U product_temp){CO int i_min = max(i_start,0);CO int d_0 = i_min / m_N_sqrt + 1;CO int i_0 = min(d_0 * m_N_sqrt,m_N);if(i_min < i_0){CO int d_0_minus = d_0 - 1;if(m_suspended[d_0_minus]){SolveSuspendedSubstitution(d_0_minus,m_lazy_substitution[d_0_minus]);}else{SolveSuspendedAction(d_0_minus);}}for(int i = i_min;i < i_0;i++){product_temp = m_M.Product(MO(product_temp),m_a[i]);if(f(product_temp,i)){RE i;}}for(int d = d_0;d < m_N_d;d++){U product_next = m_M.Product(product_temp,m_b[d]);if(f(product_next,min((d + 1)* m_N_sqrt,m_N)- 1)){RE Search_Body(d * m_N_sqrt,f,MO(product_temp));}product_temp = MO(product_next);}RE -1;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> TE <TY F> int LazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::SearchReverse_Body(CRI i_final,CO F& f,U product_temp){CO int i_max = min(i_final,m_N - 1);CO int d_1 = i_max / m_N_sqrt;CO int i_1 = max(d_1 * m_N_sqrt,0);if(m_suspended[d_1]){SolveSuspendedSubstitution(d_1,m_lazy_substitution[d_1]);}else{SolveSuspendedAction(d_1);}for(int i = i_max;i >= i_1;i--){product_temp = m_M.Product(m_a[i],product_temp);if(f(product_temp,i)){RE i;}}for(int d = d_1 - 1;d >= 0;d--){U product_next = m_M.Product(m_b[d],product_temp);if(f(product_next,d * m_N_sqrt)){RE Search_Body((d + 1)* m_N_sqrt - 1,f,MO(product_temp));}product_temp = MO(product_next);}RE -1;}
#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 VAR( N0 + I0 , VE<LL>( N1 + I1 ) ); 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 WHAT( ... ) 
  #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 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"
  US MP = DMod;
#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){auto IT = S.find(t);if(IT != S.EN()){S.erase(IT);}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)> 0;}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);}TE <TY RET,TE <TY...> TY PAIR,TY INT> T2<RET> cast(CO PAIR<INT,INT>& t){RE{get<0>(t),get<1>(t)};}TE <TY RET,TY INT> T3<RET> cast(CO tuple<INT,INT,INT>& t){RE{get<0>(t),get<1>(t),get<2>(t)};}TE <TY RET,TY INT> T4<RET> cast(CO tuple<INT,INT,INT,INT>& t){RE{get<0>(t),get<1>(t),get<2>(t),get<3>(t)};}

#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 = decldecay_t(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<decldecay_t(a[0]),decldecay_t(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);}TE <TY RET,TE <TY...> TY V,TY T> IN V<RET> cast(CO V<T>& a){V<RET> AN{};for(auto& x:a){AN <<= 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;}

/* Module (6KB)*/
#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(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 AbstractNSet<U,F_U>& OP=(CO AbstractNSet&)NE;IN U Transfer(CO U& u);};TE <TY U>CL VirtualMagma:VI PU UnderlyingSet<U>{PU:VI U Product(U u0,CO U& u1)= 0;IN U Sum(U u0,CO U& u1);};TE <TY U = ll>CL AdditiveMagma:VI PU VirtualMagma<U>{PU:IN U Product(U u0,CO U& u1);};TE <TY U = ll>CL MultiplicativeMagma:VI PU VirtualMagma<U>{PU:IN U Product(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 AbstractMagma<U,M_U>& OP=(CO AbstractMagma<U,M_U>&)NE;IN U Product(U u0,CO U& u1);};
TE <TY U> IN PointedSet<U>::PointedSet(U b_U):m_b_U(MO(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 AbstractNSet<U,F_U>& AbstractNSet<U,F_U>::operator=(CO AbstractNSet<U,F_U>&)NE{RE *TH;}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,TY M_U> IN AbstractMagma<U,M_U>& AbstractMagma<U,M_U>::OP=(CO AbstractMagma<U,M_U>&)NE{RE *TH;}TE <TY U> IN U AdditiveMagma<U>::Product(U u0,CO U& u1){RE MO(u0 += u1);}TE <TY U> IN U MultiplicativeMagma<U>::Product(U u0,CO U& u1){RE MO(u0 *= u1);}TE <TY U,TY M_U> IN U AbstractMagma<U,M_U>::Product(U u0,CO U& u1){RE m_m_U(MO(u0),u1);}TE <TY U> IN U VirtualMagma<U>::Sum(U u0,CO U& u1){RE Product(MO(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(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,U e_U);};
TE <TY U> IN MultiplicativeMonoid<U>::MultiplicativeMonoid(U e_U):PointedSet<U>(MO(e_U)){}TE <TY U,TY M_U> IN AbstractMonoid<U,M_U>::AbstractMonoid(M_U m_U,U e_U):AbstractMagma<U,M_U>(MO(m_U)),PointedSet<U>(MO(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,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,U e_U,I_U i_U):AbstractMonoid<U,M_U>(MO(m_U),MO(e_U)),AbstractNSet<U,I_U>(MO(i_U)){}TE <TY U> IN U AdditiveGroup<U>::Transfer(CO U& u){RE -u;}

TE <TY R,TY U>CL VirtualRSet:VI PU UnderlyingSet<U>{PU:VI U Action(CO R& r,U u)= 0;IN U Power(U u,CO R& r);IN U ScalarProduct(CO R& r,U u);};TE <TY MAGMA,TY U = inner_t<MAGMA>>CL RegularRSet:VI PU VirtualRSet<U,U>,PU MAGMA{PU:IN RegularRSet(MAGMA magma);IN U Action(CO U& r,U u);};TE <TY R,TY U,TY O_U>CL AbstractRSet:VI PU VirtualRSet<R,U>{PU:O_U m_o_U;IN AbstractRSet(CO R& dummy0,CO U& dummy1,O_U o_U);IN AbstractRSet<R,U,O_U>& OP=(CO AbstractRSet<R,U,O_U>&)NE;IN U Action(CO R& r,U u);};TE <TY R,TY O_U,TY GROUP,TY U = inner_t<GROUP>>CL AbstractModule:PU AbstractRSet<R,U,O_U>,PU GROUP{PU:IN AbstractModule(CO R& dummy,O_U o_U,GROUP M);};TE <TY R,TY U>CL Module:VI PU VirtualRSet<R,U>,PU AdditiveGroup<U>{PU:IN U Action(CO R& r,U u);};
TE <TY MAGMA,TY U> IN RegularRSet<MAGMA,U>::RegularRSet(MAGMA magma):MAGMA(MO(magma)){}TE <TY R,TY U,TY O_U> IN AbstractRSet<R,U,O_U>::AbstractRSet(CO R& dummy0,CO U& dummy1,O_U o_U):m_o_U(MO(o_U)){ST_AS(is_invocable_r_v<U,O_U,R,U>);}TE <TY R,TY O_U,TY GROUP,TY U> IN AbstractModule<R,O_U,GROUP,U>::AbstractModule(CO R& dummy,O_U o_U,GROUP M):AbstractRSet<R,U,O_U>(dummy,M.One(),MO(o_U)),GROUP(MO(M)){ST_AS(is_same_v<U,inner_t<GROUP>>);}TE <TY R,TY U,TY O_U> IN AbstractRSet<R,U,O_U>& AbstractRSet<R,U,O_U>::OP=(CO AbstractRSet<R,U,O_U>&)NE{RE *TH;}TE <TY MAGMA,TY U> IN U RegularRSet<MAGMA,U>::Action(CO U& r,U u){RE TH->Product(r,MO(u));}TE <TY R,TY U,TY O_U> IN U AbstractRSet<R,U,O_U>::Action(CO R& r,U u){RE m_o_U(r,MO(u));}TE <TY R,TY U> IN U Module<R,U>::Action(CO R& r,U u){RE MO(u *= r);}TE <TY R,TY U> IN U VirtualRSet<R,U>::Power(U u,CO R& r){RE Action(r,MO(u));}TE <TY R,TY U> IN U VirtualRSet<R,U>::ScalarProduct(CO R& r,U u){RE Action(r,MO(u));}

/* 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(CO V<U>& f){RE LeftConnectiveProd(T{0},f,[](T t0,CO U& u1){RE MO(t0 += u1);});}TE <TY T,TY U,TE <TY...> TY V> IN T Prod(CO V<U>& f){RE LeftConnectiveProd(T{1},f,[](T t0,CO U& u1){RE MO(t0 *= u1);});}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> IN CO T& Max(CO VE<T>& f){RE *max_element(f.BE(),f.EN());}TE <TY T,TE <TY...> TY SET> IN CO T& Max(CO SET<T>& f){RE *--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> IN CO T& Min(CO VE<T>& f){RE *min_element(f.BE(),f.EN());}TE <TY T,TE <TY...> TY SET> IN CO T& Min(CO SET<T>& f){RE *f.BE();}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)/ d + 1)/ 2;}TE <TY INT> IN INT SpecialisedArithmeticProgressionSum(CO INT& l,CO INT& r,CO INT& d){AS(l - 1 <= r);CO INT c =(r - l)/ d;RE l - 1 == r?0:(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 INT> IN INT SquareSum(CO INT& r){RE r *(r + 1)*(2 * r + 1)/ 6;}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 */