#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  // #define USE_GETLINE
  /* #define SUBMIT_ONLY */
  #define DEBUG_OUTPUT
  #define SAMPLE_CHECK N
#endif
#ifdef INCLUDE_MAIN

VO Solve()
{
  CIN( int , N , Q );
  // 一旦子と親を全て同じ配列に格納
  vector<list<int>> children( N + 1 );
  FOR( i , 1 , N ){
    CIN( int , Ai , Bi );
    children[Ai].push_back( Bi );
    children[Bi].push_back( Ai );
  }
  // 子と親の決定
  vector<int> parent( N + 1 );
  // 解説では各vに対するf(1,v)の表示の第1項の計算畤に各xに対するx-(xの親)を参照しており
  // x=1の時に未定義参照となるが、等式自体は(xの親)を好きな定数に置き換えても成り立つので
  // ここでは(1の親)を定数で置き換える。
  // f(1,v)の表示の第3項を遅延平方分割から復元する際にv-(vの親)で割る必要があるので
  // (1の親)を置き換える定数は1以外にする必要がある。
  parent[1] = 0;
  vector<vector<int>> layer( N );
  vector<int> children_query( N );
  children_query[0] = 1;
  int start_children_query = 0;
  int size_children_query = 1;
  int size_children_query_copy = size_children_query;
  int depth = 0;
  while( start_children_query < size_children_query ){
    auto& layer_curr = layer[depth];
    FOR( i , start_children_query , size_children_query ){
      layer_curr.push_back( children_query[i] );
      auto& children_i = children[children_query[i]];
      RUN( children_i , j ){
	parent[j] = children_query[i];
	auto& children_ij = children[j];
	FOR_ITR( children_ij ){
	  if( children_query[i] == *itr_children_ij ){
	    children_ij.erase( itr_children_ij );
	    break;
	  }
	}
	children_query[size_children_query_copy++] = j;
      }
    }
    depth++;
    start_children_query = size_children_query;
    size_children_query = size_children_query_copy;
  }
  // HL分解のための重さ計算
  vector<int> weight( N + 1 );
  FOREQINV( d , depth - 1 , 0 ){
    auto& layer_curr = layer[d];
    RUN( layer_curr , i ){
      auto& children_curr = children[i];
      weight[i] = 1;
      RUN( children_curr , j ){
	weight[i] += weight[j];
      }
    }
  }
  // HL分解のためのheight計算
  FOREQ( i , 1 , N ){
    auto& children_i = children[i];
    int weight_max = -1;
    int children_H_i = -1;
    RUN( children_i , j_curr ){
      if( weight_max < weight[j_curr] ){
	weight_max = weight[j_curr];
	children_H_i = j_curr;
      }
    }
    FOR_ITR( children_i ){
      if( children_H_i == *itr_children_i ){
	children_i.erase( itr_children_i );
	children_i.push_front( children_H_i );
	break;
      }
    }
  }
  // HL分解
  // 添え字は0始まり、根に小さい番号づけ。
  // この向きでないとH側のpathが区間になるだけで部分木が１つの区間にならず区間族になることに注意。
  vector<int> HL( N );
  vector<int> HL_inv( N + 1 );
  list<int> root{ { 1 } };
  list<list<int>::iterator> children_itr_stack{} , children_end_stack{};
  children_itr_stack.push_back( root.begin() );
  children_end_stack.push_back( root.end() );
  int length = 0;
  while( ! children_itr_stack.empty() ){
    auto& itr_curr = children_itr_stack.back();
    auto& end_curr = children_end_stack.back();
    if( itr_curr == end_curr ){
      children_itr_stack.pop_back();
      children_end_stack.pop_back();
    } else {
      int& i_curr = *itr_curr;
      HL_inv[i_curr] = length;
      HL[length++] = i_curr;
      auto& children_curr = children[i_curr];
      children_itr_stack.push_back( children_curr.begin() );
      children_end_stack.push_back( children_curr.end() );
      // listでないと上のpush_backでitr_currが無効なイテレータになりえる。
      itr_curr++;
    }
  }
  // 1からのパスのなすHL上の半開区間[first,second)の計算（ここが各ノードでlogオーダーになることがHL分解特有の事情）
  using Interval = pair<int,int>;
  vector<vector<Interval>> path( N );
  path[0] = vector<Interval>( { Interval( 0 , 1 ) } );
  FOR( d , 1 , depth ){
    auto& layer_curr = layer[d];
    RUN( layer_curr , i ){
      auto& path_i = path[HL_inv[i]];
      path_i = path[HL_inv[parent[i]]];
      if( children[parent[i]].front() == i ){
	path_i.back().second = HL_inv[i] + 1;
      } else {
	path_i.push_back( Interval( HL_inv[i] , HL_inv[i] + 1 ) );
      }
    }
  }
  // ノードの子孫のなすHL上の半開区間[first,second)の計算
  vector<Interval> descendant( N );
  FOREQINV( d , depth - 1 , 0 ){
    auto& layer_curr = layer[d];
    RUN( layer_curr , i ){
      auto& children_curr = children[i];
      descendant[HL_inv[i]] = Interval( HL_inv[i] , children_curr.empty() ? HL_inv[i] + 1 : descendant[HL_inv[children_curr.back()]].second );
    }
  }
  // 解説における(x-(xの親),(x-(xの親))c(1,x),(x-(xの親))c(1,x)^2)の管理
  using S1 = T3<ll>;
  using F1 = int;
  // vが(A,Ax,Ax^2)の形のベクトルの時にfvが(A,A(x+f),A(x+f)^2)=(A,Ax+fA,Ax^2+2fAx+f^2A)となるようなモノイド作用
  auto act1 = [&]( const F1& f , S1 v ){
    auto& [first,second,third] = v;
    third += ( f << 1 ) * second + ( f * f ) * first;
    second += f * first;
    return move( v );
  };
  AdditiveMonoid<F1> mag1{};
  AbstractModule mod1{ F1{} , act1 , AdditiveMonoid<S1>{} };
  vector<S1> v1_prep( N );
  FOR( i , 0 , N ){
    v1_prep[i] = { HL[i] - parent[HL[i]] , 0 , 0 };
  }
  LazySqrtDecomposition v1{ mag1 , mod1 , move( v1_prep ) };
  // 解説における(x-(xの親),(x-(xの親))c(1,x),(x-(xの親))(|S|-c(1,x)),(x-(xの親))c(1,x)(|S|-c(1,x)))の管理
  using S2 = T4<ll>;
  using F2 = pair<int,int>;
  // vが(A,Ax,Ay,Axy)の形のベクトルの時に(f_1,f_2)vが(A,A(x+f_1),A(y+f_2),A(x+f_1)(y+f_2))=(A,Ax+f_1A,Ay+f_2A,Axy+f_1Ay+f_2Ax+f_1f_2A)となるようなモノイド作用
  auto act2 = [&]( const F2& f , S2 v ){
    auto& [first,second,third,fourth] = v;
    fourth += f.first * third + f.second * second + ( f.first * f.second ) * first;
    third += f.second * first;
    second += f.first * first;
    return move( v );
  };
  auto scalar2 = [&]( S2 v , const int& n ){ return move( v *= n ); };
  AdditiveMonoid<F2> mag2{};
  AbstractBiModule mod2{ F2{} , int{} , act2 , scalar2 , AdditiveMonoid<S2>() };
  vector<S2> v2_prep( N );
  FOR( i , 0 , N ){
    v2_prep[i] = { HL[i] - parent[HL[i]] , 0 , 0 , 0 };
  }
  LazySqrtDecomposition v2{ mag2 , mod2 , move( v2_prep ) };
  // Xの総和を計算するための(x in S ? x : 0)の管理
  SqrtDecomposition v3{ vector<ll>( N ) };
  // 解説におけるf(1,v)の計算
  auto f1v = [&]( const int& v ){
    auto& descendant_v = descendant[HL_inv[v]];
    ll answer = get<2>( v1.IntervalProduct( descendant_v.first , descendant_v.second - 1 ) );
    // 解説におけるXの総和は(Ri,Vi)を(1,v)に置き換えてXを定義し直した時の値であることに注意。
    // 特に(Ri,Vi)=(v,v)!=(1,1)の時のf(v,v)（v!=1）の計算で呼び出されるf(1,1)の計算で参照されるXは問題文中で定義された集合とは別物であり(Ri,Vi)=(1,1)に置き換えて定義し直したものとなる。
    answer -= v3.IntervalSum( descendant_v.first , descendant_v.second - 1 );
    answer += get<2>( v1[HL_inv[v]] ) / ( v - parent[v] ) * parent[v];
    answer >>= 1;
    return answer;
  };
  // 解説におけるf(v,v)の計算
  auto fvv = [&]( const int& v ){
    ll answer = f1v( 1 );
    RUN( path[HL_inv[v]] , [first,second] ){
      // |S|-c(root,root) = 0であるので本来不要なrootの寄与は自動的に消える
      answer += get<3>( v2.IntervalProduct( first , second - 1 ) );
    }
    return answer;
  };
  vector<bool> S( N + 1 );
  // クエリ処理
  REPEAT( Q ){
    CIN( int , Ui , Ri , Vi );
    int sign;
    if( S[Ui] ){
      sign = -1;
      S[Ui] = false;
    } else {
      sign = 1;
      S[Ui] = true;
    }
    v3.Add( HL_inv[Ui] , sign * Ui );
    v2.IntervalAct( 0 , N - 1 , F2{ 0 , sign } );
    auto& path_Ui = path[HL_inv[Ui]];
    RUN( path_Ui , [first,second] ){
      v1.IntervalAct( first , second - 1 , sign );
      v2.IntervalAct( first , second - 1 , F2( sign , -sign ) );
    }
    if( Ri == Vi ){
      COUT( fvv( Vi ) );
    } else {
      bool belong = false;
      auto& path_Ri = path[HL_inv[Ri]];
      FOR_ITR( path_Ri ){
	auto& [first,second] = *itr_path_Ri;
	if( first <= HL_inv[Vi] && HL_inv[Vi] < second ){
	  belong = true;
          int HL_inv_x;
	  if( HL_inv[Vi] + 1 == second ){
	    itr_path_Ri++;
	    HL_inv_x = itr_path_Ri->first;
	  } else {
	    HL_inv_x = HL_inv[Vi] + 1;
	  }
	  COUT( fvv( Vi ) - f1v( HL[HL_inv_x] ) - get<3>( v2[HL_inv_x] ) / ( HL[HL_inv_x] - Vi ) * Vi );
	  break;
	}
      }
      if( !belong ){
	COUT( f1v( Vi ) );
      }
    }
  }
}
REPEAT_MAIN(1);

#else /* INCLUDE_MAIN */
#ifdef INCLUDE_SUB

/* 圧縮時は中身だけ削除する。*/
IN VO Experiment()
{ 
}

/* 圧縮時は中身だけ削除する。*/
IN VO SmallTest()
{
}

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

#define INCLUDE_MAIN
#include __FILE__
#else /* INCLUDE_SUB */
#ifdef INCLUDE_LIBRARY

/* VVV 常設でないライブラリは以下に挿入する。*/

#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/AffineSpace/SqrtDecomposition/a_Body.hpp"
#else
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);ST IN int Sqrt(CRI N)NE;};
IN SqrtDecompositionCoordinate::SqrtDecompositionCoordinate(CRI N):SqrtDecompositionCoordinate(N,Sqrt(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 int SqrtDecompositionCoordinate::Sqrt(CRI N)NE{if(N <= 1){RE 1;}int l = 0,r = N;WH(l + 1 < r){int m =(l + r)>> 1;(m <=(N - 1)/ m?l:r)= m;}RE r;}

#define SFINAE_FOR_SD_S enable_if_t<is_invocable_r_v<bool,F,U,int>>*
TE <TY U,TY ABELIAN_GROUP>CL AbstractSqrtDecomposition:PU SqrtDecompositionCoordinate{PU:ABELIAN_GROUP m_M;VE<U> m_a;VE<U> m_b;TE <TY...Args> IN AbstractSqrtDecomposition(ABELIAN_GROUP M,CRI N = 0,CO Args&... args);TE <TY...Args> IN AbstractSqrtDecomposition(ABELIAN_GROUP 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 Add(CRI i,CO U& u);IN CO U& OP[](CRI i)CO;IN CO U& Get(CRI i)CO;IN U IntervalSum(CRI i_start,CRI i_final);TE <TY F,SFINAE_FOR_SD_S = nullptr> IN int Search(CRI i_start,CO F& f);IN int Search(CRI i_start,CO U& u);VO COruct();TE <TY F> int Search_Body(CRI i_start,CO F& f,U sum_temp);};TE <TY ABELIAN_GROUP,TY...Args> AbstractSqrtDecomposition(ABELIAN_GROUP M,Args&&...args)-> AbstractSqrtDecomposition<inner_t<ABELIAN_GROUP>,ABELIAN_GROUP>;TE <TY U = ll>CL SqrtDecomposition:PU AbstractSqrtDecomposition<U,AdditiveGroup<U>>{PU:TE <TY...Args> IN SqrtDecomposition(Args&&... args);};TE <TY U,TY...Args> SqrtDecomposition(VE<U> a,Args&&...args)-> SqrtDecomposition<U>;
TE <TY U,TY ABELIAN_GROUP> TE <TY...Args> IN AbstractSqrtDecomposition<U,ABELIAN_GROUP>::AbstractSqrtDecomposition(ABELIAN_GROUP M,CRI N,CO Args&... args):SqrtDecompositionCoordinate(N,args...),m_M(MO(M)),m_a(m_N_m,m_M.Zero()),m_b(m_N_d,m_M.Zero()){ST_AS(! is_same_v<U,int> && is_same_v<U,inner_t<ABELIAN_GROUP>>);}TE <TY U,TY ABELIAN_GROUP> TE <TY...Args> IN AbstractSqrtDecomposition<U,ABELIAN_GROUP>::AbstractSqrtDecomposition(ABELIAN_GROUP M,VE<U> a,CO Args&... args):SqrtDecompositionCoordinate(a.SZ(),args...),m_M(MO(M)),m_a(MO(a)),m_b(m_N_d,m_M.Zero()){COruct();}TE <TY U> TE <TY...Args> IN SqrtDecomposition<U>::SqrtDecomposition(Args&&... args):AbstractSqrtDecomposition<U,AdditiveGroup<U>>(AdditiveGroup<U>(),forward<Args>(args)...){}TE <TY U,TY ABELIAN_GROUP> IN VO AbstractSqrtDecomposition<U,ABELIAN_GROUP>::COruct(){ST_AS(! is_same_v<U,int> && is_same_v<U,inner_t<ABELIAN_GROUP>>);m_a.resize(m_N_m);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.Sum(MO(m_bd),m_a[i]);}i_min = i_ulim;i_ulim += m_N_sqrt;}}TE <TY U,TY ABELIAN_GROUP> TE <TY...Args> IN VO AbstractSqrtDecomposition<U,ABELIAN_GROUP>::Initialise(Args&&... args){AbstractSqrtDecomposition<U,ABELIAN_GROUP> temp{m_M,forward<Args>(args)...};SqrtDecompositionCoordinate::OP=(temp);m_a = MO(temp.m_a);m_b = MO(temp.m_b);}TE <TY U,TY ABELIAN_GROUP> IN VO AbstractSqrtDecomposition<U,ABELIAN_GROUP>::Set(CRI i,CO U& u){U& m_ai = m_a[i];U& m_bd = m_b[i / m_N_sqrt];m_bd = m_M.Sum(MO(m_bd),m_M.Sum(m_M.Inverse(m_ai),u));m_ai = u;}TE <TY U,TY ABELIAN_GROUP> IN VO AbstractSqrtDecomposition<U,ABELIAN_GROUP>::Add(CRI i,CO U& u){U& m_ai = m_a[i];U& m_bd = m_b[i / m_N_sqrt];m_bd = m_M.Sum(MO(m_bd),u);m_ai = m_M.Sum(MO(m_ai),u);}TE <TY U,TY ABELIAN_GROUP> IN CO U& AbstractSqrtDecomposition<U,ABELIAN_GROUP>::OP[](CRI i)CO{AS(0 <= i && i < m_N);RE m_a[i];}TE <TY U,TY ABELIAN_GROUP> IN CO U& AbstractSqrtDecomposition<U,ABELIAN_GROUP>::Get(CRI i)CO{RE OP[](i);}TE <TY U,TY ABELIAN_GROUP> IN U AbstractSqrtDecomposition<U,ABELIAN_GROUP>::IntervalSum(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.Zero();for(int i = i_min;i < i_0;i++){AN = m_M.Sum(MO(AN),m_a[i]);}for(int d = d_0;d < d_1;d++){AN = m_M.Sum(MO(AN),m_b[d]);}for(int i = i_1;i < i_ulim;i++){AN = m_M.Sum(MO(AN),m_a[i]);}RE AN;}TE <TY U,TY ABELIAN_GROUP> TE <TY F,SFINAE_FOR_SD_S> IN int AbstractSqrtDecomposition<U,ABELIAN_GROUP>::Search(CRI i_start,CO F& f){RE Search_Body(i_start,f,m_M.Zero());}TE <TY U,TY ABELIAN_GROUP> IN int AbstractSqrtDecomposition<U,ABELIAN_GROUP>::Search(CRI i_start,CO U& u){RE Search(i_start,[&](CO U& sum,CRI){RE !(sum < u);});}TE <TY U,TY ABELIAN_GROUP> TE <TY F> int AbstractSqrtDecomposition<U,ABELIAN_GROUP>::Search_Body(CRI i_start,CO F& f,U sum_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);for(int i = i_min;i < i_0;i++){sum_temp = m_M.Sum(MO(sum_temp),m_a[i]);if(f(sum_temp,i)){RE i;}}for(int d = d_0;d < m_N_d;d++){U sum_next = m_M.Sum(sum_temp,m_b[d]);if(f(sum_next,min((d + 1)* m_N_sqrt,m_N)- 1)){RE Search_Body(d * m_N_sqrt,f,sum_temp);}sum_temp = MO(sum_next);}RE -1;}
#endif

#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 PW(U u,CO R& r);};TE <TY L,TY R,TY U,TY O_U_L,TY O_U_R,TY 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 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 U,TY O_U_L,TY O_U_R,TY GROUP> IN AbstractBiModule<L,R,U,O_U_L,O_U_R,GROUP>::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 U,TY O_U_L,TY O_U_R,TY GROUP> IN U AbstractBiModule<L,R,U,O_U_L,O_U_R,GROUP>::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 U,TY O_U_L,TY O_U_R,TY GROUP> IN U AbstractBiModule<L,R,U,O_U_L,O_U_R,GROUP>::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);}

TE <TY R,TY PT_MAGMA,TY U,TY 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);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 PT_MAGMA,TY RN_BIMODULE,TY...Args> LazySqrtDecomposition(PT_MAGMA L,RN_BIMODULE M,CO Args&... args)-> LazySqrtDecomposition<inner_t<PT_MAGMA>,PT_MAGMA,inner_t<RN_BIMODULE>,RN_BIMODULE>;
TE <TY R,TY PT_MAGMA,TY U,TY RN_BIMODULE> TE <TY...Args> IN LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> TE <TY...Args> IN LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> TE <TY...Args> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::Initialise(Args&&...args){LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE> 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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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.PW(m_lazy_substitution_d,i - i_min),u),m_M.PW(m_lazy_substitution_d,i_ulim -(i + 1)));}}else{SolveSuspendedAction(d);if(m_ai != u){m_ai = u;SetProduct(d);}}RE;}TE <TY R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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.PW(m_lazy_substitution_d,i_min - d_0_N_sqrt_minus),m_M.PW(u,i_0 - i_min)),m_M.PW(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.PW(u,i_0 - i_min)),IntervalProduct_Body(i_0,d_0_N_sqrt));}}CO U PW = m_M.PW(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.PW(m_lazy_substitution_d,i_1 - d_1_N_sqrt),m_M.PW(u,i_ulim - i_1)),m_M.PW(m_lazy_substitution_d,d_1_N_sqrt_plus - i_ulim));}else{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.PW(u,i_ulim - i_1)),IntervalProduct_Body(i_ulim,d_1_N_sqrt_plus));}}RE;}TE <TY R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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.PW(m_lazy_substitution_d,i_min - d_0_N_sqrt_minus),m_M.PW(u,i_0 - i_min)),m_M.PW(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.PW(m_lazy_substitution_d,i_1 - d_1_N_sqrt),m_M.PW(u,i_ulim - i_1)),m_M.PW(m_lazy_substitution_d,d_1_N_sqrt_plus - i_ulim));}else{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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN U LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::IntervalSet_Body(CRI i_min,CRI i_ulim,CO U& u){for(int i = i_min;i < i_ulim;i++){m_a[i]= u;}RE;}TE <TY R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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);U& m_bd = m_b[d];m_bd = m_M.ScalarProduct(m_lazy_action_d,m_bd);m_lazy_action_d = m_L.Point();}RE;}TE <TY R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN U LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN U LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::Get(CRI i){RE OP[](i);}TE <TY R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN U LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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.PW(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.PW(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 R,TY PT_MAGMA,TY U,TY RN_BIMODULE> IN VO LazySqrtDecomposition<R,PT_MAGMA,U,RN_BIMODULE>::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;}
#endif

/* AAA 常設でないライブラリは以上に挿入する。*/

#define INCLUDE_SUB
#include __FILE__
#else /* INCLUDE_LIBRARY */
#ifdef DEBUG
  #define _GLIBCXX_DEBUG
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if CE( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } FINISH_MAIN
  #define FINISH_MAIN REPEAT( test_case_num ){ if CE( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } Solve(); CERR( "" ); } }
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #ifdef USE_GETLINE
    #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
    #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
    #define SET_LL( A ) { GETLINE( A ## _str ); A = stoll( A ## _str ); }
  #else
    #define CIN( LL , ... ) LL __VA_ARGS__; VariadicCin( cin , __VA_ARGS__ )
    #define SET_A( I , N , ... ) VariadicResize( N + I , __VA_ARGS__ ); FOR( VARIABLE_FOR_SET_A , 0 , N ){ VariadicSet( cin , VARIABLE_FOR_SET_A + I , __VA_ARGS__ ); }
    #define CIN_A( LL , I , N , ... ) VE<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
    #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) VE<VE<LL>> VAR( N0 + I0 ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , VAR[VARIABLE_FOR_CIN_AA + I0] ); }
    #define SET_LL( A ) cin >> A
  #endif
  #define SET_ASSERT( A , MIN , MAX ) SET_LL( 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 ) 
#endif
#ifdef REACTIVE
  #ifdef DEBUG
    #define RSET( A , ... ) A = __VA_ARGS__
  #else
    #define RSET( A , ... ) cin >> A
  #endif
  #define RCIN( LL , A , ... ) LL A; RSET( A , __VA_ARGS__ )
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now(); double loop_average_time = 0.0 , loop_start_time = 0.0 , current_time = 0.0; int loop_count = 0
#define CURRENT_TIME ( current_time = static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 ) )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME , loop_count == 0 ? loop_start_time = current_time : loop_average_time = ( current_time - loop_start_time ) / loop_count , ++loop_count , current_time < TL_MS - loop_average_time * 2 - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( I , N , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A + I] , MIN , MAX ); }
#define SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_AA0 , 0 , N0 ){ FOR( VARIABLE_FOR_SET_AA1 , 0 , N1 ){ SET_ASSERT( A[VARIABLE_FOR_SET_AA0 + I0][VARIABLE_FOR_SET_AA1 + I1] , MIN , MAX ); } }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); SET_A_ASSERT( I , N , A , MIN , MAX )
#define CIN_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) vector A( N0 + I0 , vector<decldecay_t( MAX )>( N1 + I1 ) ); SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX )
#define OUTPUT_ARRAY( C , I , N , A ) FOR( VARIABLE_FOR_OUTPUT_ARRAY , I , N ){ C << A[VARIABLE_FOR_OUTPUT_ARRAY] << " \n"[VARIABLE_FOR_OUTPUT_ARRAY==(N)-1]; }
#define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( decldecay_t( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( decldecay_t( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( decldecay_t( INITIAL ) VAR = INITIAL ; VAR + 1 > FINAL ; VAR -- )
#define ITR( ARRAY ) auto begin_ ## ARRAY = ARRAY .BE() , itr_ ## ARRAY = begin_ ## ARRAY , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define RUN( ARRAY , ... ) for( auto&& __VA_ARGS__ : ARRAY )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT_ ## HOW_MANY_TIMES , 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__ , true ); auto answer = Answer( __VA_ARGS__ ); bool match = naive == answer; CERR( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive == " , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ RE; }

/* 圧縮用 */
#define TE template
#define TY typename
#define US using
#define ST static
#define AS assert
#define IN inline
#define CL class
#define PU public
#define OP operator
#define CE constexpr
#define CO const
#define NE noexcept
#define RE return 
#define WH while
#define VO void
#define VE vector
#define LI list
#define BE 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
#define reSZ resize

/* 型のエイリアス */
#define decldecay_t(VAR)decay_t<decltype(VAR)>
TE <TY F,TY...Args> US ret_t = decltype(declval<F>()(declval<Args>()...));
TE <TY T> US inner_t = TY T::type;
US uint = unsigned int;
US ll = long long;
US ull = unsigned long long;
US ld = long double;
US lld = __float128;
TE <TY INT> US T2 = pair<INT,INT>;
TE <TY INT> US T3 = tuple<INT,INT,INT>;
TE <TY INT> US T4 = tuple<INT,INT,INT,INT>;
US path = pair<int,ll>;

/* VVV 常設ライブラリは以下に挿入する。*/
#ifdef DEBUG
  #include "C:/Users/user/Documents/Programming/Contest/Template/Local/a_Body.hpp"
#else
/* BinarySearch (2KB)*/
/* EXPRESSIONがANSWERの広義単調関数の時、EXPRESSION >= CONST_TARGETの整数解を格納。*/
#define BS(AN,MINIMUM,MAXIMUM,EXPRESSION,DESIRED_INEQUALITY,CO_TARGET,INEQUALITY_FOR_CHECK,UPDATE_U,UPDATE_L,UPDATE_AN)ST_AS(! is_same<decldecay_t(CO_TARGET),uint>::value && ! is_same<decldecay_t(CO_TARGET),ull>::value);ll AN = MINIMUM;{ll AN ## _L = MINIMUM;ll AN ## _R = MAXIMUM;AN = UPDATE_AN;ll EXPRESSION_BS;CO ll CO_TARGET_BS =(CO_TARGET);ll DIFFERENCE_BS;WH(AN ## _L < AN ## _R){DIFFERENCE_BS =(EXPRESSION_BS =(EXPRESSION))- CO_TARGET_BS;if(DIFFERENCE_BS INEQUALITY_FOR_CHECK 0){AN ## _R = UPDATE_U;}else{AN ## _L = UPDATE_L;}AN = UPDATE_AN;}if(AN ## _L > AN ## _R || !((EXPRESSION)DESIRED_INEQUALITY CO_TARGET_BS)){AN = MAXIMUM + 1;}}
/* 単調増加の時にEXPRESSION >= CONST_TARGETの最小解を格納。*/
#define BS1(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,>=,CO_TARGET,>=,AN,AN + 1,(AN ## _L + AN ## _R)/ 2)
/* 単調増加の時にEXPRESSION <= CONST_TARGETの最大解を格納。*/
#define BS2(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,<=,CO_TARGET,>,AN - 1,AN,(AN ## _L + 1 + AN ## _R)/ 2)
/* 単調減少の時にEXPRESSION >= CONST_TARGETの最大解を格納。*/
#define BS3(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,>=,CO_TARGET,<,AN - 1,AN,(AN ## _L + 1 + AN ## _R)/ 2)
/* 単調減少の時にEXPRESSION <= CONST_TARGETの最小解を格納。*/
#define BS4(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,<=,CO_TARGET,<=,AN,AN + 1,(AN ## _L + AN ## _R)/ 2)

/* TwoPoitnterApproach (2KB)*/
/* VAR_TPAは尺取り法用の変数名の接頭辞で、実際の変数名ではなく、_Lと_Rと_infoがつく。
ANSWER ## _temp = {VAR_TPA ## _L,VAR_TPA ## _R,VPA_TPA ## _info}を
{INIT,INIT,INFO_init}で初期化する。VPA_TPA ## _infoは区間和など。
ANSWER ## _tempがCONTINUE_CONDITIONを満たす限り、ANSWER ## _tempが
条件ON_CONDITIONを満たすか否かを判定し、それがtrueになるか
VAR_TAR ## _LがVAR_TAR ## _Rに追い付くまでVAR_TPA ## _LとVPA_TPA ## _infoの
更新操作UPDATE_Lを繰り返し、その後VAR_TPA ## _RとVPA_TPA ## _infoの
更新操作UPDATE_Rを行う。（マクロとコンマの制約上、関数オブジェクトを用いる）
ON_CONDITIONがtrueとなる極大閉区間とその時点でのinfoをANSWERに格納する。
例えば長さNの非負整数値配列Aで極大な正値区間とそこでの総和を取得したい場合
auto update_L = [&]( int& i_L , ll& i_info ){ i_info -= A[i_L++]; };
auto update_R = [&]( int& i_R , ll& i_info ){ ++i_R < N ? i_info += A[i_R] : i_info; };
TPA( interval , i , 0 , i_R < N , update_L( i_L , i_info ) , update_R( i_R , i_info ) , A[i_L] > 0 && A[i_R] > 0 , ll( A[0] ) );
とすればtuple<int,int,ll>値配列intervalに{左端,右端,総和}の列が格納される。*/
#define TPA(AN,VAR_TPA,INIT,CONTINUE_CONDITION,UPDATE_L,UPDATE_R,ON_CONDITION,INFO_init)VE<tuple<decldecay_t(INIT),decldecay_t(INIT),decldecay_t(INFO_init)>> AN{};{auto init_TPA = INIT;decldecay_t(AN.front())AN ## _temp ={init_TPA,init_TPA,INFO_init};auto AN ## _prev = AN ## _temp;auto& VAR_TPA ## _L = get<0>(AN ## _temp);auto& VAR_TPA ## _R = get<1>(AN ## _temp);auto& VAR_TPA ## _info = get<2>(AN ## _temp);bool on_TPA_prev = false;WH(true){bool continuing = CONTINUE_CONDITION;bool on_TPA = continuing &&(ON_CONDITION);if(on_TPA_prev && ! on_TPA){AN.push_back(AN ## _prev);}if(continuing){if(on_TPA || VAR_TPA ## _L == VAR_TPA ## _R){AN ## _prev = AN ## _temp;UPDATE_R;}else{UPDATE_L;}}else{break;}on_TPA_prev = on_TPA;}}

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

/* Set (1KB)*/
#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>>;

/* Tuple (5KB)*/
#define DF_OF_AR_FOR_TUPLE(OPR)TE <TY T,TY U,TE <TY...> TY V> IN auto OP OPR ## =(V<T,U>& t0,CO V<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> IN tuple<T,U,V>& OP OPR ## =(tuple<T,U,V>& t0,CO tuple<T,U,V>& t1){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> IN tuple<T,U,V,W>& OP OPR ## =(tuple<T,U,V,W>& t0,CO tuple<T,U,V,W>& t1){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 V> IN auto OP OPR ## =(V<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> IN tuple<T,U,V>& OP OPR ## =(tuple<T,U,V>& t0,CO ARG& t1){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> IN tuple<T,U,V,W>& OP OPR ## =(tuple<T,U,V,W>& t0,CO ARG& t1){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 V,TY...ARGS,TY ARG> IN auto OP OPR(CO V<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 V> IN auto OP INCR(V<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> IN tuple<T,U,V>& OP INCR(tuple<T,U,V>& t){INCR get<0>(t);INCR get<1>(t);INCR get<2>(t);RE t;}TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W>& OP INCR(tuple<T,U,V,W>& t){INCR get<0>(t);INCR get<1>(t);INCR get<2>(t);INCR get<3>(t);RE t;}
TE <CL Traits,TY T> IN IS& OP>>(IS& is,tuple<T>& arg){RE is >> get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY...> TY V> IN auto OP>>(IS& is,V<T,U>& arg)-> decltype((get<0>(arg),is))&{RE is >> get<0>(arg)>> get<1>(arg);}TE <CL Traits,TY T,TY U,TY V> IN IS& OP>>(IS& is,tuple<T,U,V>& arg){RE is >> get<0>(arg)>> get<1>(arg)>> get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W> IN IS& OP>>(IS& is,tuple<T,U,V,W>& arg){RE is >> get<0>(arg)>> get<1>(arg)>> get<2>(arg)>> get<3>(arg);}TE <CL Traits,TY T> IN OS& OP<<(OS& os,CO tuple<T>& arg){RE os << get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY...> TY V> IN auto OP<<(OS& os,CO V<T,U>& arg)-> decltype((get<0>(arg),os))&{RE os << get<0>(arg)<< " " << get<1>(arg);}TE <CL Traits,TY T,TY U,TY V> IN OS& OP<<(OS& os,CO tuple<T,U,V>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W> IN OS& OP<<(OS& os,CO tuple<T,U,V,W>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg)<< " " << get<3>(arg);}DF_OF_AR_FOR_TUPLE(+);TE <TY T,TY U,TE <TY...> TY V> IN auto OP-(CO V<T,U>& t)-> decltype(get<0>(t),t){RE{-get<0>(t),-get<1>(t)};}TE <TY T,TY U,TY V> IN tuple<T,U,V> OP-(CO tuple<T,U,V>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t)};}TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W> OP-(CO tuple<T,U,V,W>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t),-get<3>(t)};}DF_OF_AR_FOR_TUPLE(-);DF_OF_AR_FOR_TUPLE(*);DF_OF_AR_FOR_TUPLE(/);DF_OF_AR_FOR_TUPLE(%);DF_OF_INCREMENT_FOR_TUPLE(++);DF_OF_INCREMENT_FOR_TUPLE(--);

#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 (2KB)*/
#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;}
#define DF_OF_AR_FOR_VE(V,OPR)TE <TY T> IN V<T>& OP OPR ## =(V<T>& a,CO T& t){for(auto& s:a){s OPR ## = t;}RE a;}TE <TY T> IN V<T>& OP OPR ## =(V<T>& a0,CO V<T>& a1){AS(a0.SZ()<= a1.SZ());auto IT0 = a0.BE(),EN0 = a0.EN();auto IT1 = a1.BE();WH(IT0 != EN0){*(IT0++)OPR ## = *(IT1++);}RE a0;}TE <TY T,TY U> IN V<T> OP OPR(V<T> a,CO U& u){RE MO(a OPR ## = u);}
#define DF_OF_INCREMENT_FOR_VE(V,INCR)TE <TY T> IN V<T>& OP INCR(V<T>& a){for(auto& i:a){INCR i;}RE a;}
#define DF_OF_ARS_FOR_VE(V)DF_OF_AR_FOR_VE(V,+);DF_OF_AR_FOR_VE(V,-);DF_OF_AR_FOR_VE(V,*);DF_OF_AR_FOR_VE(V,/);DF_OF_AR_FOR_VE(V,%);DF_OF_INCREMENT_FOR_VE(V,++);DF_OF_INCREMENT_FOR_VE(V,--);TE <TY T> IN V<T> OP*(CO T& scalar,V<T> v){for(auto& t:v){t *= scalar;}RE MO(v);}
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_ARS_FOR_VE(VE);DF_OF_ARS_FOR_VE(LI);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...);}TE <TY V> IN auto Get(V& a){RE[&](CRI i = 0)-> CO decldecay_t(a[0])&{RE a[i];};}TE <TY T = int> IN VE<T> id(CRI SZ){VE<T> AN(SZ);FOR(i,0,SZ){AN[i]= i;}RE AN;}TE <TY T> VO Sort(VE<T>& a,CO bool& reversed = false){if(reversed){ST auto comp =[](CO T& t0,CO T& t1){RE t1 < t0;};sort(a.BE(),a.EN(),comp);}else{sort(a.BE(),a.EN());}}TE <TY T> IN VE<int> IndexSort(CO VE<T>& a,CO bool& reversed = false){auto index = id<int>(a.SZ());if(reversed){sort(index.BE(),index.EN(),[&](CRI i,CRI j){RE a[j]< a[i];});}else{sort(index.BE(),index.EN(),[&](CRI i,CRI j){RE a[i]< a[j];});}RE index;}

/* 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 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));}

/* Graph (5KB)*/
TE <TY T,TY R1,TY R2,TY E>CL VirtualGraph:VI PU UnderlyingSet<T>{PU:VI R1 Enumeration(CRI i)= 0;IN R2 Enumeration_inv(CO T& t);TE <TY PATH> IN R2 Enumeration_inv(CO PATH& p);IN VO Reset();VI CRI SZ()CO NE = 0;VI E& edge()NE = 0;VI ret_t<E,T> Edge(CO T& t)= 0;TE <TY PATH> IN ret_t<E,T> Edge(CO PATH& p);ST IN CO T& Vertex(CO T& t)NE;TE <TY PATH> ST IN CO T& Vertex(CO PATH& e)NE;VI R2 Enumeration_inv_Body(CO T& t)= 0;};TE <TY T,TY R1,TY R2,TY E>CL EdgeImplimentation:VI PU VirtualGraph<T,R1,R2,E>{PU:int m_SZ;E m_edge;IN EdgeImplimentation(CRI SZ,E edge);IN CRI SZ()CO NE;IN E& edge()NE;IN ret_t<E,T> Edge(CO T& t);};TE <TY E>CL Graph:PU EdgeImplimentation<int,CRI,CRI,E>{PU:IN Graph(CRI SZ,E edge);IN CRI Enumeration(CRI i);TE <TY F> IN Graph<F> GetGraph(F edge)CO;IN CRI Enumeration_inv_Body(CRI t);};TE <TY T,TY Enum_T,TY Enum_T_inv,TY E>CL EnumerationGraph:PU EdgeImplimentation<T,ret_t<Enum_T,int>,ret_t<Enum_T_inv,T>,E>{PU:Enum_T m_enum_T;Enum_T_inv m_enum_T_inv;IN EnumerationGraph(CRI SZ,Enum_T enum_T,Enum_T_inv enum_T_inv,E edge);IN ret_t<Enum_T,int> Enumeration(CRI i);TE <TY F> IN EnumerationGraph<T,Enum_T,Enum_T_inv,F> GetGraph(F edge)CO;IN ret_t<Enum_T_inv,T> Enumeration_inv_Body(CO T& t);};TE <TY Enum_T,TY Enum_T_inv,TY E> EnumerationGraph(CRI SZ,Enum_T enum_T,Enum_T_inv enum_T_inv,E edge)-> EnumerationGraph<decldecay_t(declval<Enum_T>()(0)),Enum_T,Enum_T_inv,E>;TE <TY T,TY E>CL MemorisationGraph:PU EdgeImplimentation<T,T,CRI,E>{PU:int m_LE;VE<T> m_memory;Map<T,int> m_memory_inv;IN MemorisationGraph(CRI SZ,CO T& dummy,E edge);IN T Enumeration(CRI i);IN VO Reset();TE <TY F> IN MemorisationGraph<T,F> GetGraph(F edge)CO;IN CRI Enumeration_inv_Body(CO T& t);};
TE <TY T,TY R1,TY R2,TY E> IN EdgeImplimentation<T,R1,R2,E>::EdgeImplimentation(CRI SZ,E edge):m_SZ(SZ),m_edge(MO(edge)){ST_AS(is_COructible_v<T,R1> && is_COructible_v<int,R2> && is_invocable_v<E,T>);}TE <TY E> IN Graph<E>::Graph(CRI SZ,E edge):EdgeImplimentation<int,CRI,CRI,E>(SZ,MO(edge)){}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> IN EnumerationGraph<T,Enum_T,Enum_T_inv,E>::EnumerationGraph(CRI SZ,Enum_T enum_T,Enum_T_inv enum_T_inv,E edge):EdgeImplimentation<T,ret_t<Enum_T,int>,ret_t<Enum_T_inv,T>,E>(SZ,MO(edge)),m_enum_T(MO(enum_T)),m_enum_T_inv(MO(enum_T_inv)){}TE <TY T,TY E> IN MemorisationGraph<T,E>::MemorisationGraph(CRI SZ,CO T& dummy,E edge):EdgeImplimentation<T,T,CRI,E>(SZ,MO(edge)),m_LE(),m_memory(),m_memory_inv(){ST_AS(is_invocable_v<E,T>);}TE <TY E> IN CRI Graph<E>::Enumeration(CRI i){RE i;}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> IN ret_t<Enum_T,int> EnumerationGraph<T,Enum_T,Enum_T_inv,E>::Enumeration(CRI i){RE m_enum_T(i);}TE <TY T,TY E> IN T MemorisationGraph<T,E>::Enumeration(CRI i){AS(0 <= i && i < m_LE);RE m_memory[i];}TE <TY T,TY R1,TY R2,TY E> IN R2 VirtualGraph<T,R1,R2,E>::Enumeration_inv(CO T& t){RE Enumeration_inv_Body(t);}TE <TY T,TY R1,TY R2,TY E> TE <TY PATH> IN R2 VirtualGraph<T,R1,R2,E>::Enumeration_inv(CO PATH& p){RE Enumeration_inv_Body(get<0>(p));}TE <TY E> IN CRI Graph<E>::Enumeration_inv_Body(CRI i){RE i;}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> IN ret_t<Enum_T_inv,T> EnumerationGraph<T,Enum_T,Enum_T_inv,E>::Enumeration_inv_Body(CO T& t){RE m_enum_T_inv(t);}TE <TY T,TY E> IN CRI MemorisationGraph<T,E>::Enumeration_inv_Body(CO T& t){if(m_memory_inv.count(t)== 0){AS(m_LE < TH->SZ());m_memory.push_back(t);RE m_memory_inv[t]= m_LE++;}RE m_memory_inv[t];}TE <TY T,TY R1,TY R2,TY E> VO VirtualGraph<T,R1,R2,E>::Reset(){}TE <TY T,TY E> IN VO MemorisationGraph<T,E>::Reset(){m_LE = 0;m_memory.clear();m_memory_inv.clear();}TE <TY T,TY R1,TY R2,TY E> IN CRI EdgeImplimentation<T,R1,R2,E>::SZ()CO NE{RE m_SZ;}TE <TY T,TY R1,TY R2,TY E> IN E& EdgeImplimentation<T,R1,R2,E>::edge()NE{RE m_edge;}TE <TY T,TY R1,TY R2,TY E> IN ret_t<E,T> EdgeImplimentation<T,R1,R2,E>::Edge(CO T& t){RE m_edge(t);}TE <TY T,TY R1,TY R2,TY E> TE <TY PATH> IN ret_t<E,T> VirtualGraph<T,R1,R2,E>::Edge(CO PATH& p){RE Edge(get<0>(p));}TE <TY E> TE <TY F> IN Graph<F> Graph<E>::GetGraph(F edge)CO{RE Graph<F>(TH->SZ(),MO(edge));}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> TE <TY F> IN EnumerationGraph<T,Enum_T,Enum_T_inv,F> EnumerationGraph<T,Enum_T,Enum_T_inv,E>::GetGraph(F edge)CO{RE EnumerationGraph<T,Enum_T,Enum_T_inv,F>(TH->SZ(),m_enum_T,m_enum_T_inv,MO(edge));}TE <TY T,TY E> TE <TY F> IN MemorisationGraph<T,F> MemorisationGraph<T,E>::GetGraph(F edge)CO{RE MemorisationGraph<T,F>(TH->SZ(),MO(edge));}TE <TY T,TY R1,TY R2,TY E> IN CO T& VirtualGraph<T,R1,R2,E>::Vertex(CO T& t)NE{RE t;}TE <TY T,TY R1,TY R2,TY E> TE <TY PATH> IN CO T& VirtualGraph<T,R1,R2,E>::Vertex(CO PATH& e)NE{RE Vertex(get<0>(e));}

/* Grid (2KB)*/
#define SET_GRID H_minus = H - 1;W_minus = W - 1;HW = ll(H)* W
#define SET_HW(h,w)H = h;W = w;SET_GRID
#define CIN_HW cin >> H >> W;SET_GRID
TE <TY E>CL GridGraph:PU EnumerationGraph<T2<int>,T2<int>(&)(CRI),int(&)(CO T2<int>&),E>{PU:IN GridGraph(E e);};int H,W,H_minus,W_minus;ll HW;VE<string> grid;char walkable = '.',unwalkable = '#';
IN T2<int> EnumHW(CRI v){RE{v / W,v % W};}IN int EnumHW_inv(CO T2<int>& ij){auto&[i,j]= ij;RE i * W + j;}TE <TY E> IN GridGraph<E>::GridGraph(E e):EnumerationGraph<T2<int>,T2<int>(&)(CRI),int(&)(CO T2<int>&),E>(HW,EnumHW,EnumHW_inv,MO(e)){AS(HW >> 31 == 0 && H * W == HW);}VE<T2<int>> EdgeOnGrid(CO T2<int>& v){VE<T2<int>> AN{};auto&[i,j]= v;if(grid[i][j]== walkable){if(i > 0 && grid[i-1][j]== walkable){AN.push_back({i-1,j});}if(i+1 < H && grid[i+1][j]== walkable){AN.push_back({i+1,j});}if(j > 0 && grid[i][j-1]== walkable){AN.push_back({i,j-1});}if(j+1 < W && grid[i][j+1]== walkable){AN.push_back({i,j+1});}}RE AN;}VE<pair<T2<int>,ll>> WEdgeOnGrid(CO T2<int>& v){VE<pair<T2<int>,ll>> AN{};auto&[i,j]= v;if(grid[i][j]== walkable){if(i>0 && grid[i-1][j]== walkable){AN.push_back({{i-1,j},1});}if(i+1 < H && grid[i+1][j]== walkable){AN.push_back({{i+1,j},1});}if(j>0 && grid[i][j-1]== walkable){AN.push_back({{i,j-1},1});}if(j+1 < W && grid[i][j+1]== walkable){AN.push_back({{i,j+1},1});}}RE AN;}IN VO SetWallStringOnGrid(CRI i,VE<string>& S){if(S.empty()){S.resize(H);}cin >> S[i];AS(int(S[i].SZ())== W);}CO string direction="URDL";IN int DirectionNumberOnGrid(CRI i,CRI j,CRI k,CRI h){RE i < k?2:i > k?0:j < h?1:(AS(j > h),3);}IN int DirectionNumberOnGrid(CO T2<int>& v,CO T2<int>& w){auto&[i,j]= v;auto&[k,h]= w;RE DirectionNumberOnGrid(i,j,k,h);}IN int DirectionNumberOnGrid(CRI v,CRI w){RE DirectionNumberOnGrid(EnumHW(v),EnumHW(w));}IN int ReverseDirectionNumberOnGrid(CRI n){AS(0 <= n && n<4);RE n ^ 2;}

/* Loop (1KB)*/
TE <TY INT> bool NextLoop(CRI SZ,CO VE<INT>& lower_bound,CO VE<INT>& upper_limit,VE<INT>& index){int depth = 0;WH(depth < SZ){if(++index[depth]< upper_limit[depth]){break;}index[depth]= lower_bound[depth];depth++;}RE depth < SZ;}TE <TY INT> bool NextLoop(CO VE<INT>& lower_bound,CO VE<INT>& upper_limit,VE<INT>& index){RE NextLoop(index.SZ(),lower_bound,upper_limit,index);}TE <TY INT> bool NextLoopEq(CRI SZ,CO VE<INT>& lower_bound,CO VE<INT>& upper_bound,VE<INT>& index){int depth = 0;WH(depth < SZ){if(++index[depth]<= upper_bound[depth]){break;}index[depth]= lower_bound[depth];depth++;}RE depth < SZ;}TE <TY INT> bool NextLoopEq(CO VE<INT>& lower_bound,CO VE<INT>& upper_bound,VE<INT>& index){RE NextLoopEq(index.SZ(),lower_bound,upper_bound,index);}

/* string (1KB)*/
TE <TY INT> IN char IntToChar(CO INT& i,CO char& c = 'a'){RE c + i;}TE <TY INT> IN INT CharToInt(CO char& i){RE i -(i < 'a'?'A':'a');}TE <TY INT>string ArrayToString(CO VE<INT>& A,CO char& c = 'a'){CO int N = A.SZ();string S(N,c);for(int i = 0;i < N;i++){S[i]= IntToChar<INT>(A[i],c);}RE S;}TE <TY INT>VE<INT> StringToArray(CO string& S){CO int N = S.SZ();VE<int> A(N);for(int i = 0;i < N;i++){A[i]= CharToInt<INT>(S[i]);}RE A;}
#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 */