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

VO Solve()
{
  CIN( int , N , Q );
  CIN( string , S );
  CIN_A( int , 0 , N + 1 , W );
  vector<int> X_pre( N + 1 );
  FOREQ( i , 1 , N ){
    X_pre[i] = S[i-1] == 'B';
  }
  IntervalMultiplyLazySqrtDecomposition X{ MultiplicativeMonoid<int>{ 1 } , Module<int,int>{} , move( X_pre ) } , Y{ MultiplicativeMonoid<int>{ 1 } , Module<int,int>{} , move( W ) };
  FOR( q , 0 , Q ){
    CIN( int , type );
    if( type == 1 ){
      CIN( int , l , r );
      X.IntervalAct( l , r , -1 );
      X.IntervalMultiply( l , r , 1 );
    } else if( type == 2 ){
      CIN( int , l , r , a );
      Y.IntervalMultiply( l , r , a );
    } else if( type == 3 ){
      CIN( int , v , K );
      CIN_A( int , 0 , K , u );
      int v_next = X.Search( v , 1 );
      if( v_next == -1 ){
        v_next = N;
      }
      int num = Y.IntervalProduct( v , v_next ) , den = Y.IntervalProduct( 0 , v_next );
      int k0 = 0;
      while( k0 < K && u[k0] < v ){
        k0++;
      }
      int k1 = k0;
      while( k1 < K && u[k1] <= v_next ){
        k1++;
      }
      vector<MP> a( K + 1 );
      if( num == den ){
        assert( v == 0 && k0 == 0 );
        a[K] = 1;
      } else {
        vector<int> multiplicity( K + 1 );
        int k_prev = k1;
        FOREQ( k , k1 + 1 , K ){
          if( k == K || X.IntervalProduct( u[k-1] , u[k] - 1 ) > 0 ){
            multiplicity[k-k_prev]++;
            k_prev = k;
          }
        }
        MP r = MP{ num } / MP{ den - num };
        FPS<MP> log{ K + 1 };
        FOREQ( k , 1 , K ){
          if( multiplicity[k] > 0 ){
            MP power = multiplicity[k];
            FOREQ( d , k , K , k ){
              power *= r;
              int i = d / k;
              log[d] += ( i & 1 ? 1 : -1 ) * power / i;
            }
          }
        }
        FPS<MP> f = Exp( log ) * Power( MP{ den - num } / MP{ den } , Sum<int>( multiplicity ) );
        FOREQ( k , k1 - k0 , K ){
          a[k] = f[k-(k1-k0)];
        }
      }
      COUT( a );
    }
  }
}
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/IntervalMultiply/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 PW(U u,CO R& r);};TE <TY L,TY R,TY O_U_L,TY O_U_R,TY GROUP,TY U>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,O_U_L,O_U_R,GROUP,U>& OP=(CO AbstractBiModule<L,R,O_U_L,O_U_R,GROUP,U>&)NE;IN U LAction(CO L& l,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> AbstractBiModule(CO L& dummy_l,CO R& dummy_r,O_U_L o_U_L,O_U_R o_U_R,GROUP M)-> AbstractBiModule<L,R,inner_t<GROUP>,O_U_L,O_U_R,GROUP>;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>::PW(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 IntervalMultiplyLazySqrtDecomposition: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;VE<U> m_lazy_MU;TE <TY...Args> IN IntervalMultiplyLazySqrtDecomposition(PT_MAGMA L,RN_BIMODULE M,CRI N = 0,CO Args&... args);TE <TY...Args> IN IntervalMultiplyLazySqrtDecomposition(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 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);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 VO IntervalMultiply_Body(CRI i_min,CRI i_ulim,CO U& u);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 IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::IntervalMultiplyLazySqrtDecomposition(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()),m_lazy_MU(m_b){COruct();}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> TE <TY...Args> IN IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::IntervalMultiplyLazySqrtDecomposition(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()),m_lazy_MU(m_b){COruct();}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::Initialise(Args&&...args){IntervalMultiplyLazySqrtDecomposition<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);m_lazy_MU = MO(temp.m_lazy_MU);}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::Set(CRI i,CO U& u){CO int d = i / m_N_sqrt;U& m_ai = m_a[i];U& m_bd = m_b[d];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_bd = m_M.Product(m_M.Power(m_lazy_substitution_d,m_N_sqrt - 1),u);}}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 IntervalMultiplyLazySqrtDecomposition<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){CO 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.Power(m_lazy_substitution_d,m_N_sqrt -(i_0 - i_min)),m_M.Power(u,i_0 - i_min));}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);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_suspended[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_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 IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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_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.Power(m_lazy_substitution_d,m_N_sqrt -(i_0 - i_min)),m_M.Power(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_L.Product(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.ScalarProduct(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.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);U& m_lazy_MU_d = m_lazy_MU[d];m_lazy_MU_d = m_M.ScalarProduct(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_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.Power(m_lazy_substitution_d,m_N_sqrt -(i_ulim - i_1)),m_M.Power(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_L.Product(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.ScalarProduct(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 PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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(MO(m_bd),m_M.Power(u,i_0 - i_min));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];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_suspended_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.Power(u,m_N_sqrt);for(int d = d_0;d < d_1;d++){U& m_bd = m_b[d];m_bd = m_M.Product(MO(m_bd),pw);if(m_suspended[d]){U& m_lazy_substitution_d = m_lazy_substitution[d];m_lazy_substitution_d = m_M.Product(MO(m_lazy_substitution_d),u);}else{U& m_lazy_MU_d = m_lazy_MU[d];m_lazy_MU_d = m_M.Product(MO(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(MO(m_bd),m_M.Power(u,i_ulim - i_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,m_M.Product(m_lazy_substitution_d,u));IntervalSet_Body(i_ulim,d_1_N_sqrt_plus,m_lazy_substitution_d);m_suspended_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 PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN U IntervalMultiplyLazySqrtDecomposition<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.Product(m_M.ScalarProduct(m_lazy_action[d],m_a[i]),m_lazy_MU[d]);}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN U IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<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.Product(m_M.ScalarProduct(m_lazy_action[d_0_minus],IntervalProduct_Body(i_min,i_0)),m_M.Power(m_lazy_MU[d_0_minus],i_0 - i_min));}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.Product(m_M.ScalarProduct(m_lazy_action[d_1],IntervalProduct_Body(i_1,i_ulim)),m_M.Power(m_lazy_MU[d_1],i_ulim - i_1)));}RE AN;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::SolveSuspendedAction(CRI d){CO int i_min = d * m_N_sqrt;CO int i_ulim = i_min + m_N_sqrt;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_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_lazy_MU_d = one;}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN VO IntervalMultiplyLazySqrtDecomposition<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> IN VO IntervalMultiplyLazySqrtDecomposition<PT_MAGMA,RN_BIMODULE,R,U>::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(MO(m_ai),u);}RE;}TE <TY PT_MAGMA,TY RN_BIMODULE,TY R,TY U> IN U IntervalMultiplyLazySqrtDecomposition<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> TE <TY F,SFINAE_FOR_SD_S> IN int IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<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 IntervalMultiplyLazySqrtDecomposition<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

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

TE <TY T,int EX_lim>CL Power3Power_CE{PU:T m_val[EX_lim];CE Power3Power_CE(CO T& t);CE CO T& OP[](CRI i)CO;CE CO T(&Get()CO)[EX_lim];};
TE <TY T,int EX_lim> CE Power3Power_CE<T,EX_lim>::Power3Power_CE(CO T& t):m_val(){T pw{t};for(uint EX = EX_lim - 1;EX + 1 > 0;EX--){m_val[EX]= -pw;m_val[EX]*= pw *= pw;}}TE <TY T,int EX_lim> CE CO T& Power3Power_CE<T,EX_lim>::OP[](CRI i)CO{AS(i < EX_lim);RE m_val[i];}TE <TY T,int EX_lim> CE CO T(&Power3Power_CE<T,EX_lim>::Get()CO)[EX_lim]{RE m_val;}

#define DC_OF_AR_FOR_PO(FUNC)IN PO<T> OP FUNC(PO<T> f)CO;IN PO<T> OP FUNC(T t)CO
#define DF_OF_AR_FOR_PO(FUNC,DEF)TE <TY T> IN PO<T> PO<T>::OP FUNC(PO<T> f)CO{RE MO(DEF);};TE <TY T> IN PO<T> PO<T>::OP FUNC(T t)CO{RE *TH FUNC PO<T>(MO(t));}
TE <TY T>CL PO{PU:VE<T> m_f;int m_SZ;IN PO();IN PO(CO PO<T>& f);IN PO(PO<T>&& f);IN PO(VE<T> f);IN PO(T t);IN PO(CRI i,T t);IN PO<T>& OP=(T n);IN PO<T>& OP=(PO<T> f);IN PO<T>& OP=(VE<T> f);IN CO T& OP[](CRI i)CO;IN T& OP[](CRI i);T OP()(CO T& t)CO;PO<T>& OP+=(CO PO<T>& f);PO<T>& OP-=(CO PO<T>& f);PO<T>& OP*=(PO<T> f);IN PO<T>& OP/=(CO PO<T>& f);PO<T>& OP/=(CO T& t);PO<T>& OP%=(CO PO<T>& f);PO<T>& OP%=(CO T& t);bool OP==(CO PO<T>& f)CO;bool OP==(CO T& t)CO;TE <TY P> IN bool OP!=(CO P& f)CO;DC_OF_AR_FOR_PO(+);IN PO<T> OP-()CO;DC_OF_AR_FOR_PO(-);DC_OF_AR_FOR_PO(*);IN PO<T> OP/(CO PO<T>& f)CO;IN PO<T> OP/(CO T& t)CO;IN PO<T> OP%(CO PO<T>& f)CO;IN PO<T> OP%(CO T& t)CO;IN CO VE<T>& GetCoefficient()CO NE;IN CRI SZ()CO NE;IN VO resize(CRI deg_plus)NE;int Valuation()CO NE;IN VO swap(PO<T>& f);IN VO swap(VE<T>& f);VO Reduce();VO TP(CRI N_trunc);ST PO<T> NaiveCN(PO<T> f0,CRI valuation0,CO PO<T>& f1,CRI valuation1,CRI N_trunc);ST PO<T> NaiveQuotient(PO<T> f0,CO PO<T>& f1);ST PO<T> NaiveResidue(PO<T> f0,CO PO<T>& f1);ST IN CO PO<T>& zero();ST IN CO PO<T>& one();ST IN CO PO<T>& x();ST IN CO T& c_zero();ST IN CO T& c_one();ST IN CO T& c_minus_one();IN PO<T>& SignInvert();};

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

#define PS_FOR_FFT_BODY(MOD,LE,PR,IPR,TYPE)ST_AS((TYPE::DeRP(PR)*= TYPE::DeRP(IPR))== TYPE::DeRP(1));TE <> CE CO uint LimitOfPowerForFFT<TYPE> = LE - 1;TE <> IN CO TYPE(&PrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<TYPE>]{ST CE Power3Power_CE<TYPE,LimitOfPowerForFFT<TYPE>> PRT{PR};ST_AS(PRT.m_val[0]== TYPE::DeRP(1));RE PRT.Get();}TE <> IN CO TYPE(&InversePrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<TYPE>]{ST CE Power3Power_CE<TYPE,LimitOfPowerForFFT<TYPE>> IPRT{IPR};ST_AS(IPRT.m_val[0]== TYPE::DeRP(1)&&(TYPE::DeRP(PR)*= TYPE::DeRP(IPR))== TYPE::DeRP(1));RE IPRT.Get();}TE <> IN PO<TYPE>& PO<TYPE>::OP*=(PO<TYPE> f){CO int SZ = m_SZ + f.m_SZ - 1;RE *TH = FFTCN(MO(*TH),MO(f),SZ);}TE <> IN PO<TYPE>& PO<TYPE>::OP/=(CO PO<TYPE>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();if(m_SZ < f.m_SZ){RE *TH = zero();}RE *TH = FFTQuotient(MO(*TH),f);}TE <> IN PO<TYPE>& PO<TYPE>::OP%=(CO PO<TYPE>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();RE *TH = FFTResidue(MO(*TH),f);}
#define PS_FOR_FFT(MOD,LE,PR,IPR,MINT)PS_FOR_FFT_BODY(MOD,LE,PR,IPR,MINT<MOD>)
TE <TY T> CE CO int LimitOfPowerForFFT;TE <TY T> IN CO T(&PrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<T>];TE <TY T> IN CO T(&InversePrimitiveRootOfTwoForFFT()NE)[LimitOfPowerForFFT<T>];
TE <TY T>VO CooleyTukey(VE<T>& f,CRI N_shift,CRI N_input_start,CRI N_input_lim,CRI N_trunc,CRI two_pw,CRI EX,CO T(&PRT)[LimitOfPowerForFFT<T>]){AS(N_input_lim - N_input_start <= two_pw);CO int N_zero = N_shift + N_input_start,le = N_zero + two_pw;AS(N_zero <= N_trunc);CO int N_input_final = min(N_input_start + two_pw,int(f.SZ()));for(int i = N_input_lim;i < N_input_final;i++){f[i]= T{};}WH(int(f.SZ())< le){f.push_back(T{});}ST VE<int> bit_reverse[32]={VE<int>(1)};ST int e_next = 1;ST int two_pw_next = 1;ST int two_pw_next2 = 2;ST VE<int>* p_bit_reverse_prev = bit_reverse;ST VE<int>* p_bit_reverse_curr = p_bit_reverse_prev + 1;WH(e_next <= EX){*p_bit_reverse_curr = VE<int>(two_pw_next2);int* p_bit_reverse_curr_i = &((*p_bit_reverse_curr)[0]);int* p_bit_reverse_curr_i_plus = p_bit_reverse_curr_i + two_pw_next;int* p_bit_reverse_prev_i = &((*p_bit_reverse_prev)[0]);for(int i = 0;i < two_pw_next;i++){(*(p_bit_reverse_curr_i_plus++)= *(p_bit_reverse_curr_i++)= *(p_bit_reverse_prev_i++)* 2)+= 1;}e_next++;swap(two_pw_next,two_pw_next2);two_pw_next2 *= 4;p_bit_reverse_prev++;p_bit_reverse_curr++;}CO VE<int>& bit_reverse_EX = bit_reverse[EX];int bit_num = 0;CO int* p_bit_num_reverse = &(bit_reverse_EX[bit_num]);WH(bit_num < two_pw){if(*p_bit_num_reverse < bit_num){swap(f[*p_bit_num_reverse + N_input_start],f[bit_num + N_input_start]);}bit_num++;p_bit_num_reverse++;}CO T& one = PRT[0];T zeta,diff;int i,j,j_lim,two_pw_curr = 1,two_pw_curr_2 = 2;WH(two_pw_curr < two_pw){CO int N_input_final_curr = N_input_start + two_pw_curr;bit_num = 0;i = 0;zeta = one;WH(i < two_pw){j = i;j_lim = i + two_pw_curr;WH(j < j_lim){diff = f[j + N_input_start]- f[j + N_input_final_curr];f[j + N_input_start]+= f[j + N_input_final_curr];f[j + N_input_final_curr]= zeta * diff;j++;}bit_num++;i += two_pw_curr_2;j = 0;WH(true){if(((bit_num >> j)& 1)== 1){zeta *= PRT[j+1];break;}j++;}}two_pw_curr <<= 1;two_pw_curr_2 <<= 1;}if(N_trunc < le){f.resize(N_trunc);}if(N_shift > 0){for(int i = N_trunc - 1;i >= N_zero;i--){f[i]= MO(f[i - N_shift]);}for(int i = N_zero - 1;i >= N_input_start;i--){f[i]= T{};}}RE;}TE <TY T> IN VO FFT(VE<T>& f,CRI N_input_start,CRI N_input_lim,CRI two_pw,CRI EX){CooleyTukey<T>(f,0,N_input_start,N_input_lim,N_input_start + two_pw,two_pw,EX,PrimitiveRootOfTwoForFFT<T>());}TE <TY T> IN VO IFFT(VE<T>& f,CRI N_shift,CRI N_input_start,CRI N_input_lim,CRI N_trunc,CRI two_pw,CO T& two_pw_inv,CRI EX){CooleyTukey<T>(f,N_shift,N_input_start,N_input_lim,N_trunc,two_pw,EX,InversePrimitiveRootOfTwoForFFT<T>());CO int SZ = f.SZ();for(int i = N_shift + N_input_start;i < SZ;i++){f[i]*= two_pw_inv;}}TE <TY T>PO<T> FFTCN(PO<T> f0,PO<T> f1,CRI N_trunc){f0.Reduce();if(f0.m_SZ == 0){RE MO(f0);}f1.Reduce();if(f1.m_SZ == 0){RE MO(f1);}AS(f0.m_SZ <= N_trunc);CO int valuation0 = f0.Valuation();CO int valuation1 = f1.Valuation();if(N_trunc <= valuation0 + valuation1){RE f0.zero();}CO int le0 = f0.m_SZ - valuation0;CO int le1 = min(f1.m_SZ,N_trunc)- valuation1;CO int le = le0 + le1 - 1;CO int EX = Log(le);if(min(le0,le1)<= EX){RE f0.NaiveCN(MO(f0),valuation0,MO(f1),valuation1,min(f0.m_SZ + f1.m_SZ - 1,N_trunc));}CO int two_pw = 1 << EX;FFT(f0.m_f,valuation0,f0.m_SZ,two_pw,EX);FFT(f1.m_f,valuation1,valuation1 + le1,two_pw,EX);for(int i = 0;i < two_pw;i++){f0.m_f[i + valuation0]*= f1.m_f[i + valuation1];}IFFT(f0.m_f,valuation1,valuation0,valuation0 + two_pw,N_trunc,two_pw,f0.c_one()/ two_pw,EX);f0.m_SZ = f0.m_f.SZ();RE MO(f0);}TE <TY T>PO<T> FFTQuotient(PO<T> f0,PO<T> f1){AS(f1.m_SZ > 0 && f1[f1.m_SZ-1]!= f0.c_zero());if(f0.m_SZ < f1.m_SZ){RE PO<T>::zero();}CO int f0_TP_SZ = f0.m_SZ - f1.m_SZ + 1;CO int f1_TP_SZ = min(f0_TP_SZ,f1.m_SZ);f1.TP(f1_TP_SZ);CO FPS<T> f1_TP_inverse = FPS<T>(f0_TP_SZ,MO(f1)).Inverse();f0.TP(f0_TP_SZ);FPS<T> f0_TP{f0_TP_SZ,MO(f0)};f0_TP *= f1_TP_inverse;f0_TP.TP(f0_TP_SZ);RE f0_TP;}TE <TY T>PO<T> FFTResidue(PO<T> f0,CO PO<T>& f1){if(f0.m_SZ >= f1.m_SZ){f0 -=(f0 / f1)* f1;f0.Reduce();}RE MO(f0);}
PS_FOR_FFT(P,24,31,128805723,Mod);
PS_FOR_FFT(167772161,26,17,29606852,Mod);
PS_FOR_FFT(469762049,27,30,15658735,Mod);
PS_FOR_FFT(754974721,25,362,415027540,Mod);

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

TE <TY T> IN PO<T>::PO():m_f(),m_SZ(0){}TE <TY T> IN PO<T>::PO(CO PO<T>& f):m_f(f.m_f),m_SZ(f.m_SZ){}TE <TY T> IN PO<T>::PO(PO<T>&& f):m_f(MO(f.m_f)),m_SZ(f.m_SZ){}TE <TY T> IN PO<T>::PO(VE<T> f):m_f(MO(f)),m_SZ(m_f.SZ()){}TE <TY T> IN PO<T>::PO(T t):PO(){if(t != c_zero()){OP[](0)= MO(t);}}TE <TY T> IN PO<T>::PO(CRI i,T t):PO(){if(t != c_zero()){OP[](i)= MO(t);}}TE <TY T> IN PO<T>& PO<T>::OP=(T n){m_f.clear();m_SZ = 0;OP[](0)= MO(n);RE *TH;}TE <TY T> IN PO<T>& PO<T>::OP=(PO<T> f){m_f = MO(f.m_f);m_SZ = f.m_SZ;RE *TH;}TE <TY T> IN PO<T>& PO<T>::OP=(VE<T> f){m_f = MO(f);m_SZ = m_f.SZ();RE *TH;}TE <TY T> IN CO T& PO<T>::OP[](CRI i)CO{RE m_SZ <= i?c_zero():m_f[i];}TE <TY T> IN T& PO<T>::OP[](CRI i){if(m_SZ <= i){CO T& z = c_zero();WH(m_SZ <= i){m_f.push_back(z);m_SZ++;}}RE m_f[i];}TE <TY T> T PO<T>::OP()(CO T& t)CO{T AN =(*TH)[0];T t_pw = c_one();for(int d = 1;d < m_SZ;d++){AN += m_f[d]*(t_pw *= t);}RE AN;}TE <TY T>PO<T>& PO<T>::OP+=(CO PO<T>& f){if(m_SZ < f.m_SZ){for(int i = 0;i < m_SZ;i++){m_f[i]+= f.m_f[i];}for(int i = m_SZ;i < f.m_SZ;i++){m_f.push_back(f.m_f[i]);}m_SZ = f.m_SZ;}else{for(int i = 0;i < f.m_SZ;i++){m_f[i]+= f.m_f[i];}}RE *TH;}TE <TY T>PO<T>& PO<T>::OP-=(CO PO<T>& f){if(m_SZ < f.m_SZ){for(int i = 0;i < m_SZ;i++){m_f[i]-= f.m_f[i];}for(int i = m_SZ;i < f.m_SZ;i++){m_f.push_back(- f.m_f[i]);}m_SZ = f.m_SZ;}else{for(int i = 0;i < f.m_SZ;i++){m_f[i]-= f.m_f[i];}}RE *TH;}TE <TY T>PO<T>& PO<T>::OP/=(CO T& t){if(t == c_one()){RE *TH;}CO T t_inv{c_one()/ t};for(int i = 0;i < m_SZ;i++){OP[](i)*= t_inv;}RE *TH;}TE <TY T>PO<T>& PO<T>::OP%=(CO T& t){if(t == c_one()){RE *TH = zero();}for(int i = 0;i < m_SZ;i++){m_f[i]%= t;}RE *TH;}TE <TY T>bool PO<T>::OP==(CO PO<T>& f)CO{CRI SZ0 = SZ();CRI SZ1 = f.SZ();CRI SZ_max = SZ0 < SZ1?SZ1:SZ0;for(int i = 0;i < SZ_max;i++){if(OP[](i)!= f[i]){RE false;}}RE true;}TE <TY T>bool PO<T>::OP==(CO T& t)CO{CRI SZ_max = SZ();CO T& zero = PO<T>::c_zero();for(int i = 1;i < SZ_max;i++){if(m_f[i]!= zero){RE false;}}RE OP[](0)== t;}TE <TY T> TE<TY P> IN bool PO<T>::OP!=(CO P& f)CO{RE !(*TH == f);}DF_OF_AR_FOR_PO(+,f += *TH);TE <TY T> IN PO<T>& PO<T>::SignInvert(){Reduce();for(auto& fi:m_f){fi = -fi;}RE *TH;}TE <TY T> IN PO<T> PO<T>::OP-()CO{RE MO(PO<T>(*TH).SignInvert());}DF_OF_AR_FOR_PO(-,f.SignInvert()+= *TH);DF_OF_AR_FOR_PO(*,f *= *TH);TE <TY T> IN PO<T> PO<T>::OP/(CO T& t)CO{RE MO(PO<T>(*TH)/= t);}TE <TY T> IN PO<T> PO<T>::OP/(CO PO<T>& f)CO{RE MO(PO<T>(*TH)/= f);}TE <TY T> IN PO<T> PO<T>::OP%(CO T& t)CO{RE MO(PO<T>(*TH)%= t);}TE <TY T> IN PO<T> PO<T>::OP%(CO PO<T>& f)CO{RE MO(PO<T>(*TH)%= f);}TE <TY T> IN CO VE<T>& PO<T>::GetCoefficient()CO NE{RE m_f;}TE <TY T> IN CRI PO<T>::SZ()CO NE{RE m_SZ;}TE <TY T> IN VO PO<T>::resize(CRI deg_plus)NE{m_f.resize(m_SZ = deg_plus);}TE <TY T>int PO<T>::Valuation()CO NE{for(int i = 0;i < m_SZ;i++){if(m_f[i]!= c_zero()){RE i;}}RE -1;}TE <TY T> IN VO PO<T>::swap(PO<T>& f){m_f.swap(f.m_f);swap(m_SZ,f.m_SZ);}TE <TY T> IN VO PO<T>::swap(VE<T>& f){m_f.swap(f);m_SZ = m_f.SZ();}TE <TY T>VO PO<T>::Reduce(){CO T& z = c_zero();WH(m_SZ > 0 && m_f[m_SZ - 1]== z){m_f.pop_back();m_SZ--;}RE;}TE <TY T>VO PO<T>::TP(CRI N_trunc){WH(N_trunc > m_SZ){m_f.push_back(c_zero());m_SZ++;}CO int N_half = min(N_trunc,(m_SZ + 1)/ 2);for(int d = 0;d < N_half;d++){::swap(m_f[d],m_f[m_SZ - 1 - d]);}m_f.resize(N_trunc);m_SZ = N_trunc;RE;}TE <TY T> IN CO PO<T>& PO<T>::zero(){ST CO PO<T> z{};RE z;}TE <TY T> IN CO PO<T>& PO<T>::one(){ST CO PO<T> o{c_one()};RE o;}TE <TY T> IN CO PO<T>& PO<T>::x(){ST CO PO<T> f{1,c_one()};RE f;}TE <TY T> IN CO T& PO<T>::c_zero(){ST CO T z{0};RE z;}TE <TY T> IN CO T& PO<T>::c_one(){ST CO T o{1};RE o;}TE <TY T> IN CO T& PO<T>::c_minus_one(){ST CO T m{-1};RE m;}TE <TY T>PO<T> Differential(CRI n,CO PO<T>& f){CRI SZ = f.SZ();if(SZ < n){RE PO<T>::zero();}VE<T> df(SZ - n);T coef = T::Factorial(n);int i = n;WH(i < SZ){df[i - n]= f[i]* coef;i++;(coef *= i)/=(i - n);}RE PO<T>(MO(df));}TE <TY T> IN PO<T> Differential(CO PO<T>& f){RE Differential(1,f);}
TE <TY T>PO<T> PO<T>::NaiveCN(PO<T> f0,CRI valuation0,CO PO<T>& f1,CRI valuation1,CRI N_trunc){CO int SZ0 = f0.SZ();AS(0 <= valuation0 && valuation0 < SZ0);CO int SZ1 = f1.SZ();AS(0 <= valuation1 && valuation1 < SZ1);for(int i = SZ0 - 1;i >= valuation0;i--){CO T f0i = f0[i];f0[i]*= f1[0];CO int j_ulim = min(SZ1,N_trunc - i),j_min = max(valuation1,1);for(int j = j_ulim - 1;j >= j_min;j--){f0[i+j]+= f0i * f1[j];}}RE MO(f0);}TE <TY T,CL Traits> IN OS& OP<<(OS& os,CO PO<T>& f){auto& SZ = f.SZ();for(int i = 0;i < SZ;i++){(i == 0?os:os << " ")<< f[i];}RE os;}
TE <TY T>PO<T> PO<T>::NaiveQuotient(PO<T> f0,CO PO<T>& f1){CO int diff = f0.m_SZ - f1.m_SZ;if(diff < 0){RE MO(f0);}CO T r = f0[f0.m_SZ-1]/ f1[f1.m_SZ-1];f0.m_f.pop_back();f0.m_SZ--;f0.Reduce();for(int i = diff;i < f0.m_SZ;i++){f0[i]-= r * f1[i - diff];}f0.Reduce();f0 = NaiveQuotient(MO(f0),f1);f0[diff]= r;RE MO(f0);}TE <TY T>PO<T> PO<T>::NaiveResidue(PO<T> f0,CO PO<T>& f1){CO int diff = f0.m_SZ - f1.m_SZ;if(diff < 0){RE MO(f0);}CO T r = f0[f0.m_SZ-1]/ f1[f1.m_SZ-1];f0.m_f.pop_back();f0.m_SZ--;f0.Reduce();for(int i = diff;i < f0.m_SZ;i++){f0[i]-= r * f1[i - diff];}f0.Reduce();RE NaiveResidue(MO(f0),f1);}TE <TY T>PO<T>& PO<T>::OP*=(PO<T> f){Reduce();if(m_SZ == 0){RE *TH;}f.Reduce();if(f.m_SZ == 0){RE *TH = MO(f);}CO int valuation0 = TH->Valuation();CO int valuation1 = f.Valuation();CO int le0 = m_SZ - valuation0;CO int le1 = f.m_SZ - valuation1;CO int SZ = m_SZ + f.m_SZ - 1;m_f = NaiveCN(MO(*TH),valuation0,f,valuation1,SZ);m_SZ = m_f.SZ();RE *TH;}TE <TY T>PO<T>& PO<T>::OP/=(CO PO<T>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();if(m_SZ < f.m_SZ){RE *TH = zero();}RE *TH = NaiveQuotient(MO(*TH),f);}TE <TY T>PO<T>& PO<T>::OP%=(CO PO<T>& f){AS(f.m_SZ > 0 && f[f.m_SZ-1]!= c_zero());Reduce();RE *TH = NaiveResidue(MO(*TH),f);}
#endif

/* AAA 常設でないライブラリは以上に挿入する。*/

#define INCLUDE_SUB
#include __FILE__
#else /* INCLUDE_LIBRARY */
#ifdef DEBUG
  #define _GLIBCXX_DEBUG
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , test_case_num_bound , BOUND ); int test_case_num = 1; if CE( test_case_num_bound > 1 ){ FINISH_MAIN
  #ifdef USE_GETLINE
  #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
#else
  #define ENDL "\n"
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now(); double loop_average_time = 0.0 , loop_start_time = loop_average_time , current_time = loop_start_time; int loop_count = current_time; assert( loop_count == 0 )
#define CURRENT_TIME ( current_time = static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 ) )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME , loop_count == 0 ? loop_start_time = current_time : loop_average_time = ( current_time - loop_start_time ) / loop_count , ++loop_count , current_time < TL_MS - loop_average_time * 2 - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( I , N , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A + I] , MIN , MAX ); }
#define SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_AA0 , 0 , N0 ){ FOR( VARIABLE_FOR_SET_AA1 , 0 , N1 ){ SET_ASSERT( A[VARIABLE_FOR_SET_AA0 + I0][VARIABLE_FOR_SET_AA1 + I1] , MIN , MAX ); } }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); SET_A_ASSERT( I , N , A , MIN , MAX )
#define CIN_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) vector A( N0 + I0 , vector<decldecay_t( MAX )>( N1 + I1 ) ); SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX )
#define PR1( A1 , ... ) A1
#define PR2( A1 , A2 , ... ) A2
#define PR3( A1 , A2 , A3 , ... ) A3
#define FOR_( VAR , INITIAL , FINAL , UPPER , COMP , INCR ) for( decldecay_t( UPPER ) VAR = INITIAL ; VAR COMP ( FINAL ) ; VAR INCR )
#define FOR( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , < , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQ( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , <= , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQINV( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , INITIAL , + 1 > , PR3( __VA_ARGS__ , -= PR2( __VA_ARGS__ , ? ) , -- ) )
#define ITR( ARRAY ) auto begin_ ## ARRAY = ARRAY .BE() , itr_ ## ARRAY = begin_ ## ARRAY , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define RUN( ARRAY , ... ) for( auto&& __VA_ARGS__ : ARRAY )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS ); cerr << fixed << setprecision( DECIMAL_DIGITS )
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ , false ); auto answer = Answer( __VA_ARGS__ , false ); bool match = naive == answer; CERR( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive ==" , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ CERR( "出力の不一致が検出されました。" ); RE; }
#define CHECK( ... ) auto answer = Answer( __VA_ARGS__ , false ); CERR( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Answer == " , answer )

/* 圧縮用 */
#define TE template
#define TY typename
#define US using
#define ST static
#define AS assert
#define IN inline
#define CL class
#define PU public
#define OP operator
#define CE constexpr
#define CO const
#define NE noexcept
#define RE return 
#define WH while
#define VO void
#define VE vector
#define LI list
#define BE begin
#define EN end
#define SZ size
#define LE length
#define PW Power
#define MO move
#define TH this
#define CRI CO int&
#define CRUI CO uint&
#define CRL CO ll&
#define VI virtual 
#define IS basic_istream<char,Traits>
#define OS basic_ostream<char,Traits>
#define ST_AS static_assert
#define reMO_CO remove_const
#define is_COructible_v is_constructible_v
#define rBE rbegin

/* 型のエイリアス */
#define decldecay_t(VAR)decay_t<decltype(VAR)>
TE <TY F,TY...Args> US ret_t = decltype(declval<F>()(declval<Args>()...));
TE <TY T> US inner_t = TY T::type;
US uint = unsigned int;
US ll = long long;
US ull = unsigned long long;
US ld = long double;
US lld = __float128;

/* VVV 常設ライブラリは以下に挿入する。*/
#ifdef DEBUG
  #include "C:/Users/user/Documents/Programming/Contest/Template/Local/a_Body.hpp"
  US MP = Mod<P>;
#else
/* Set (2KB)*/
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>>;

/* 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...);}

/* Map (1KB)*/
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>>;

/* 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,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)...);}

/* 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 PW(U u,CO R& r);IN U ScalarProduct(CO R& r,U u);};TE <TY U,TY 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 MAGMA> RegularRSet(MAGMA magma)-> RegularRSet<inner_t<MAGMA>,MAGMA>;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 U,TY O_U,TY 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 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 VirtualRSet<R,U>,PU AdditiveGroup<U>{PU:IN U Action(CO R& r,U u);};
TE <TY R,TY MAGMA> IN RegularRSet<R,MAGMA>::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 U,TY O_U,TY GROUP> IN AbstractModule<R,U,O_U,GROUP>::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 U,TY MAGMA> IN U RegularRSet<U,MAGMA>::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>::PW(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));}

/* ConstexprModulo (8KB) */
CEXPR(uint,P,998244353);
#define RP Represent
#define DeRP Derepresent
TE <uint M,TY INT> CE INT Residue(INT n)NE{RE MO(n < 0?((((++n)*= -1)%= M)*= -1)+= M - 1:n < INT(M)?n:n %= M);}TE <TY INT> CE INT& ResidueP(INT& n)NE{CE CO uint trunc =(1 << 23)- 1;INT n_u = n >> 23;n &= trunc;INT n_uq =(n_u / 7)/ 17;n_u -= n_uq * 119;n += n_u << 23;RE n < n_uq?n += P - n_uq:n -= n_uq;}
TE <TY INT> IN INT ModularInverse(CO INT& base,ll c){AS(base > 0);ll a[2]={0,1 % base};INT b[2]={base,INT((c %= base)< 0?c += base:c)};WH(b[1]!= 0){CO INT q = b[0]/ b[1];(a[0]-= q * a[1]% base)< 0?a[0]+= base:a[0];b[0]-= q * b[1];swap(a[0],a[1]);swap(b[0],b[1]);}AS(b[0]== 1 &&(a[0]* c - 1)% base == 0);RE a[0];}
TE <uint M> CL Mod;TE <uint M>CL COantsForMod{PU:COantsForMod()= delete;ST CE CO uint g_memory_bound = 2e6;ST CE CO uint g_memory_le = M < g_memory_bound?M:g_memory_bound;ST CE uint g_M_minus = M - 1;ST CE int g_order = M - 1;ST CE int g_order_minus = g_order - 1;};
#define SFINAE_FOR_MOD enable_if_t<is_COructible_v<uint,decay_t<T>>>*
#define DC_OF_CM_FOR_MOD(OPR)CE bool OP OPR(CO Mod<M>& n)CO NE
#define DC_OF_AR_FOR_MOD(OPR,EX)CE Mod<M> OP OPR(Mod<M> n)CO EX;
#define DF_OF_CM_FOR_MOD(OPR)TE <uint M> CE bool Mod<M>::OP OPR(CO Mod<M>& n)CO NE{RE m_n OPR n.m_n;}
#define DF_OF_AR_FOR_MOD(OPR,EX,LEFT,OPR2)TE <uint M> CE Mod<M> Mod<M>::OP OPR(Mod<M> n)CO EX{RE MO(LEFT OPR2 ## = *TH);}TE <uint M,TY T,SFINAE_FOR_MOD = nullptr> CE Mod<M> OP OPR(T n0,CO Mod<M>& n1)EX{RE MO(Mod<M>(MO(n0))OPR ## = n1);}
TE <uint M>CL Mod{PU:uint m_n;CE Mod()NE;CE Mod(CO Mod<M>& n)NE;CE Mod(Mod<M>&& n)NE;TE <TY T,SFINAE_FOR_MOD = nullptr> CE Mod(T n)NE;CE Mod<M>& OP=(Mod<M> n)NE;CE Mod<M>& OP+=(CO Mod<M>& n)NE;CE Mod<M>& OP-=(CO Mod<M>& n)NE;CE Mod<M>& OP*=(CO Mod<M>& n)NE;IN Mod<M>& OP/=(Mod<M> n);CE Mod<M>& OP^=(ll EX);CE Mod<M>& OP<<=(ll n);CE Mod<M>& OP>>=(ll n);CE Mod<M>& OP++()NE;CE Mod<M> OP++(int)NE;CE Mod<M>& OP--()NE;CE Mod<M> OP--(int)NE;DC_OF_CM_FOR_MOD(==);DC_OF_CM_FOR_MOD(!=);DC_OF_CM_FOR_MOD(<);DC_OF_CM_FOR_MOD(<=);DC_OF_CM_FOR_MOD(>);DC_OF_CM_FOR_MOD(>=);DC_OF_AR_FOR_MOD(+,NE);DC_OF_AR_FOR_MOD(-,NE);DC_OF_AR_FOR_MOD(*,NE);DC_OF_AR_FOR_MOD(/,);CE Mod<M> OP^(ll EX)CO;CE Mod<M> OP<<(ll n)CO;CE Mod<M> OP>>(ll n)CO;CE Mod<M> OP-()CO NE;CE VO swap(Mod<M>& n)NE;CE CRUI RP()CO NE;ST CE Mod<M> DeRP(uint n)NE;ST IN CO Mod<M>& Factorial(CRL n);ST IN CO Mod<M>& FactorialInverse(CRL n);ST IN Mod<M> Combination(CRL n,CRL i);ST IN CO Mod<M>& zero()NE;ST IN CO Mod<M>& one()NE;ST CE uint GetModulo()NE;CE Mod<M>& SignInvert()NE;IN Mod<M>& Invert();CE Mod<M>& PPW(ll EX)NE;CE Mod<M>& NNPW(ll EX)NE;ST IN CO Mod<M>& Inverse(CRI n);ST IN CO Mod<M>& TwoPower(CRI n);ST IN CO Mod<M>& TwoPowerInverse(CRI n);US COants = COantsForMod<M>;};
US MP = Mod<P>;
TE <uint M> CE Mod<M>::Mod()NE:m_n(){}TE <uint M> CE Mod<M>::Mod(CO Mod<M>& n)NE:m_n(n.m_n){}TE <uint M> CE Mod<M>::Mod(Mod<M>&& n)NE:m_n(MO(n.m_n)){}TE <uint M> TE <TY T,SFINAE_FOR_MOD> CE Mod<M>::Mod(T n)NE:m_n(Residue<M>(MO(n))){}TE <uint M> CE Mod<M>& Mod<M>::OP=(Mod<M> n)NE{m_n = MO(n.m_n);RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP+=(CO Mod<M>& n)NE{(m_n += n.m_n)< M?m_n:m_n -= M;RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP-=(CO Mod<M>& n)NE{m_n < n.m_n?(m_n += M)-= n.m_n:m_n -= n.m_n;RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP*=(CO Mod<M>& n)NE{m_n = MO(ull(m_n)* n.m_n)% M;RE *TH;}TE <> CE MP& MP::OP*=(CO MP& n)NE{ull m_n_copy = m_n;m_n = MO((m_n_copy *= n.m_n)< P?m_n_copy:ResidueP(m_n_copy));RE *TH;}TE <uint M> IN Mod<M>& Mod<M>::OP/=(Mod<M> n){RE OP*=(n.Invert());}TE <uint M> CE Mod<M>& Mod<M>::PPW(ll EX)NE{Mod<M> pw{*TH};EX--;WH(EX != 0){(EX & 1)== 1?*TH *= pw:*TH;EX >>= 1;pw *= pw;}RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::NNPW(ll EX)NE{RE EX == 0?(m_n = 1,*TH):PPW(MO(EX));}TE <uint M> CE Mod<M>& Mod<M>::OP^=(ll EX){if(EX < 0){m_n = ModularInverse(M,MO(m_n));EX *= -1;}RE NNPW(MO(EX));}TE <uint M> CE Mod<M>& Mod<M>::OP<<=(ll n){RE *TH *=(n < 0 && -n < int(COants::g_memory_le))?TwoPowerInverse(- int(n)):(n >= 0 && n < int(COants::g_memory_le))?TwoPower(int(n)):Mod<M>(2)^= MO(n);}TE <uint M> CE Mod<M>& Mod<M>::OP>>=(ll n){RE *TH <<= MO(n *= -1);}TE <uint M> CE Mod<M>& Mod<M>::OP++()NE{m_n < COants::g_M_minus?++m_n:m_n = 0;RE *TH;}TE <uint M> CE Mod<M> Mod<M>::OP++(int)NE{Mod<M> n{*TH};OP++();RE n;}TE <uint M> CE Mod<M>& Mod<M>::OP--()NE{m_n == 0?m_n = COants::g_M_minus:--m_n;RE *TH;}TE <uint M> CE Mod<M> Mod<M>::OP--(int)NE{Mod<M> n{*TH};OP--();RE n;}DF_OF_CM_FOR_MOD(==);DF_OF_CM_FOR_MOD(!=);DF_OF_CM_FOR_MOD(>);DF_OF_CM_FOR_MOD(>=);DF_OF_CM_FOR_MOD(<);DF_OF_CM_FOR_MOD(<=);DF_OF_AR_FOR_MOD(+,NE,n,+);DF_OF_AR_FOR_MOD(-,NE,n.SignInvert(),+);DF_OF_AR_FOR_MOD(*,NE,n,*);DF_OF_AR_FOR_MOD(/,,n.Invert(),*);TE <uint M> CE Mod<M> Mod<M>::OP^(ll EX)CO{RE MO(Mod<M>(*TH)^= MO(EX));}TE <uint M> CE Mod<M> Mod<M>::OP<<(ll n)CO{RE MO(Mod<M>(*TH)<<= MO(n));}TE <uint M> CE Mod<M> Mod<M>::OP>>(ll n)CO{RE MO(Mod<M>(*TH)>>= MO(n));}TE <uint M> CE Mod<M> Mod<M>::OP-()CO NE{RE MO(Mod<M>(*TH).SignInvert());}TE <uint M> CE Mod<M>& Mod<M>::SignInvert()NE{m_n > 0?m_n = M - m_n:m_n;RE *TH;}TE <uint M> IN Mod<M>& Mod<M>::Invert(){m_n = m_n < COants::g_memory_le?Inverse(int(m_n)).m_n:ModularInverse(M,MO(m_n));RE *TH;}TE <uint M> CE VO Mod<M>::swap(Mod<M>& n)NE{std::swap(m_n,n.m_n);}TE <uint M> IN CO Mod<M>& Mod<M>::Inverse(CRI n){AS(0 < n && n < int(COants::g_memory_le));ST VE<Mod<M>> memory ={zero(),one()};ST int le_curr = 2;WH(le_curr <= n){memory.push_back(DeRP(M - memory[M % le_curr].m_n * ull(M / le_curr)% M));le_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::TwoPower(CRI n){AS(0 <= n && n < int(COants::g_memory_le));ST VE<Mod<M>> memory ={one()};ST int le_curr = 1;WH(le_curr <= n){memory.push_back(memory.back()+ memory.back());le_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::TwoPowerInverse(CRI n){AS(0 <= n && n < int(COants::g_memory_le));ST VE<Mod<M>> memory ={one()};ST int le_curr = 1;WH(le_curr <= n){auto& m = memory.back().m_n;memory.push_back(DeRP(((m & 1)== 0?m:m + M)>> 1));le_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::Factorial(CRL n){AS(n >= 0);if(ll(M)<= n){RE zero();}ST VE<Mod<M>> memory ={one(),one()};ST int le_curr = 2;WH(le_curr <= n){memory.push_back(memory[le_curr - 1]* le_curr);le_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::FactorialInverse(CRL n){AS(0 <= n && n < ll(M));ST VE<Mod<M>> memory ={one(),one()};ST int le_curr = 2;WH(le_curr <= n){memory.push_back(memory[le_curr - 1]* Inverse(le_curr));le_curr++;}RE memory[n];}TE <uint M> IN Mod<M> Mod<M>::Combination(CRL n,CRL i){RE 0 <= i && i <= n?Factorial(n)* FactorialInverse(i)* FactorialInverse(n - i):zero();}TE <uint M> CE CRUI Mod<M>::RP()CO NE{RE m_n;}TE <uint M> CE Mod<M> Mod<M>::DeRP(uint n)NE{Mod<M> n_copy{};n_copy.m_n = MO(n);RE n_copy;}TE <uint M> IN CO Mod<M>& Mod<M>::zero()NE{ST CE CO Mod<M> z{};RE z;}TE <uint M> IN CO Mod<M>& Mod<M>::one()NE{ST CE CO Mod<M> o{1};RE o;}TE <uint M> CE uint Mod<M>::GetModulo()NE{RE M;}TE <uint M> IN Mod<M> Inverse(CO Mod<M>& n){RE MO(Mod<M>(n).Invert());}TE <uint M> CE Mod<M> Power(Mod<M> n,ll EX){RE MO(n ^= MO(EX));}TE <uint M> CE VO swap(Mod<M>& n0,Mod<M>& n1)NE{n0.swap(n1);}TE <uint M> IN string to_string(CO Mod<M>& n)NE{RE to_string(n.RP())+ " + " + to_string(M)+ "Z";}TE <uint M,CL Traits> IN IS& OP>>(IS& is,Mod<M>& n){ll m;is >> m;n = m;RE is;}TE <uint M,CL Traits> IN OS& OP<<(OS& os,CO Mod<M>& n){RE os << n.RP();}

/* Iteration (3KB) */
 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);});}

/* Sqrt (1KB) */
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;}
#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 */