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

#ifdef INCLUDE_MAIN

IN VO Solve()
{
  CIN( ll , N , M );
  vector<ll> A( N + M ) , B( N + M ); FOR( i , 0 , N + M ){ cin >> A[i] >> B[i]; }
  CIN( ll , K );
  CIN_A( ll , C , K );
  CoordinateCompress<ll> cc{};
  FOR( i , 0 , N ){
    cc.insert( A[i] );
    cc.insert( B[i] );
  }
  auto range = [&]( const int& k , const int& j ){
    return pair<ll,ll>{ C[k] - ( B[j+N] - C[k] ) , C[k] - ( A[j+N] - C[k] ) };
  };
  FOR( k , 0 , K ){
    FOR( j , 0 , M ){
      auto [L,R] = range( k , j );
      cc.insert( L );
      cc.insert( R );
    }
  }
  const int& L = cc.size();
  DifferenceSequence<ll> ds{ L };
  FOR( i , 0 , N ){
    ds.IntervalAdd( cc.GetOrder( A[i] ) , cc.GetOrder( B[i] ) , 1 );
  }
  vector<ll> count( L );
  bool multiple = false;
  FOR( i , 0 , L ){
    multiple |= ( count[i] = ds.Get( i ) ) > 1;
  }
  CumulativeSum<ll> cs{ count };
  FOR( k , 0 , K ){
    bool valid = multiple;
    FOR( j , 0 , M ){
      auto [L,R] = range( k , j );
      valid |= cs.IntervalSum( cc.GetOrder( L ) , cc.GetOrder( R ) ) > 0;
    }
    cout << ( valid ? 1 : 0 ) << " \n"[k==K-1];
  }
}
REPEAT_MAIN(1);

#else // INCLUDE_MAIN

#ifdef INCLUDE_SUB

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

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

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

  // }
  return answer;
}

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

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

// 圧縮時は中身だけ削除する。
IN VO RandomTest()
{
  // CEXPR( int , bound_N , 1e5 ); CIN_ASSERT( N , 1 , bound_N );
  // CEXPR( ll , bound_M , 1e18 ); CIN_ASSERT( M , 1 , bound_M );
  // CEXPR( ll , bound_K , 1e9 ); CIN_ASSERT( K , 1 , bound_K );
  // COMPARE( N , M , N );
}

#define INCLUDE_MAIN
#include __FILE__

#else // INCLUDE_SUB

#ifdef INCLUDE_LIBRARY

/*

C-x 3 C-x o C-x C-fによるファイル操作用

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

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

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

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

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

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

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

*/

// VVV 常設でないライブラリは以下に挿入する。
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/CoordinateCompress/a_Body.hpp"
#else
TE <TY T,TE <TY...> TY MAP = Map>CL CoordinateCompress{PU:VE<T> m_a;MAP<T,int> m_enum;bool m_compressed;int m_SZ;IN CoordinateCompress();TE <TY U> IN CoordinateCompress(CO VE<U>& a);IN VO insert(CO T& t);TE <TY U> IN VO insert(CO VE<U>& a);IN CO T& OP[](CRI i);IN CO T& GetSmallest(CRI i = 0);IN CO T& GetLargest(CRI i = 0);IN int GetOrder(CO T& t);IN CRI SZ();IN TY MAP<T,int>::iterator BE();IN TY MAP<T,int>::iterator EN();IN VO Compress();};
TE <TY T,TE <TY...> TY MAP> IN CoordinateCompress<T,MAP>::CoordinateCompress():m_a(),m_enum(),m_compressed(),m_SZ(){}TE <TY T,TE <TY...> TY MAP> TE <TY U> IN CoordinateCompress<T,MAP>::CoordinateCompress(CO VE<U>& a):CoordinateCompress(){insert(a);}TE <TY T,TE <TY...> TY MAP> IN VO CoordinateCompress<T,MAP>::insert(CO T& t){m_enum[t];m_SZ = m_enum.SZ();m_compressed = false;}TE <TY T,TE <TY...> TY MAP> TE <TY U> IN VO CoordinateCompress<T,MAP>::insert(CO VE<U>& a){CO int LE = a.SZ();if(LE > 0){for(int i = 0;i < LE;i++){m_enum[a[i]];}m_SZ = m_enum.SZ();m_compressed = false;}}TE <TY T,TE <TY...> TY MAP> IN CO T& CoordinateCompress<T,MAP>::OP[](CRI i){RE GetSmallest(i);}TE <TY T,TE <TY...> TY MAP> IN CO T& CoordinateCompress<T,MAP>::GetSmallest(CRI i){if(! m_compressed){Compress();}AS(i < m_SZ);RE m_a[i];}TE <TY T,TE <TY...> TY MAP> IN CO T& CoordinateCompress<T,MAP>::GetLargest(CRI i){if(! m_compressed){Compress();}AS(i < m_SZ);RE m_a[m_SZ - i - 1];}TE <TY T,TE <TY...> TY MAP> IN int CoordinateCompress<T,MAP>::GetOrder(CO T& t){if(! m_compressed){Compress();}RE m_enum.count(t)== 1?m_enum[t]:-1;}TE <TY T,TE <TY...> TY MAP> IN CRI CoordinateCompress<T,MAP>::SZ(){RE m_SZ;}TE <TY T,TE <TY...> TY MAP> IN TY MAP<T,int>::iterator CoordinateCompress<T,MAP>::BE(){RE m_enum.BE();}TE <TY T,TE <TY...> TY MAP> IN TY MAP<T,int>::iterator CoordinateCompress<T,MAP>::EN(){RE m_enum.EN();}TE <TY T,TE <TY...> TY MAP> IN VO CoordinateCompress<T,MAP>::Compress(){m_a.reSZ(m_SZ);m_SZ = 0;for(auto IT = m_enum.BE(),EN = m_enum.EN();IT != EN;IT++){m_a[IT->second = m_SZ++]= IT->first;}m_compressed = true;}
#endif


#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/AffineSpace/DifferenceSequence/a_Body.hpp"
#else
CL LinearEdge{PU:int m_SZ;bool m_directed;IN LinearEdge(CRI SZ,CO bool& directed = true);IN VE<int> OP()(CRI t);};CL LinearGraph:PU Graph<LinearEdge>{PU:IN LinearGraph(CRI SZ,CO bool& directed = true);};
IN LinearEdge::LinearEdge(CRI SZ,CO bool& directed):m_SZ(SZ),m_directed(directed){}IN VE<int> LinearEdge::OP()(CRI t){VE<int> AN{};if(!m_directed && t > 0){AN.push_back(t - 1);}if(t + 1 < m_SZ){AN.push_back(t + 1);}RE AN;}IN LinearGraph::LinearGraph(CRI SZ,CO bool& directed):Graph<LinearEdge>(SZ,LinearEdge(SZ,directed)){}

TE <TY T,TY U>CL VirtualDifferenceSequence{PU:VI VO Set(CO T& t,CO U& u)= 0;VI CO U& Get(CO T& t)= 0;VI U& Ref(CO T& t)= 0;VI VO Add(CO T& t,CO U& u)= 0;VI VO SubTreeAdd(CO T& t_start,CO VE<T>& t_finals,CO U& u)= 0;};TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP>CL AbstractDifferenceSequence{PU:FOREST m_G;PREV m_Prev;GROUP m_M;VE<U> m_a;VE<U> m_lazy_addition;bool m_updated;IN AbstractDifferenceSequence(FOREST G,PREV Prev,GROUP M);IN AbstractDifferenceSequence(FOREST G,PREV Prev,GROUP M,VE<U> a);IN VO Initialise(VE<U> a);IN VO Set(CO T& t,CO U& u);IN CO U& Get(CO T& t);IN U& Ref(CO T& t);IN VO Add(CO T& t,CO U& u);VO SubTreeAdd(CO T& t_start,CO VE<T>& t_finals,CO U& u);IN AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>& OP+=(CO VE<U>& a);VO Update();};TE <TY FOREST,TY PREV,TY GROUP> AbstractDifferenceSequence(FOREST G,PREV Prev,GROUP M)-> AbstractDifferenceSequence<inner_t<FOREST>,FOREST,PREV,inner_t<GROUP>,GROUP>;TE <TY FOREST,TY PREV,TY U,TY GROUP> AbstractDifferenceSequence(FOREST G,PREV Prev,GROUP M,VE<U>)-> AbstractDifferenceSequence<inner_t<FOREST>,FOREST,PREV,U,GROUP>;CL PrevOfLinearGraph{PU:IN int OP()(CRI t_start)CO NE;};TE <TY U>CL DifferenceSequence:VI PU AbstractDifferenceSequence<int,LinearGraph,PrevOfLinearGraph,U,AdditiveGroup<U>>{PU:IN DifferenceSequence(CRI SZ = 0);IN DifferenceSequence(VE<U> a);IN VO InitialSegmentAdd(CRI t_start,CO U& u);IN VO FinalSegmentAdd(CRI t_final,CO U& u);IN VO IntervalAdd(CRI t_start,CRI t_final,CO U& u);IN DifferenceSequence(CRI SZ,VE<U>& a);};
TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP> IN AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::AbstractDifferenceSequence(FOREST G,PREV Prev,GROUP M):m_G(MO(G)),m_Prev(MO(Prev)),m_M(MO(M)),m_a(m_G.SZ(),m_M.Zero()),m_lazy_addition(m_a),m_updated(false){ST_AS(is_invocable_r_v<T,PREV,CO T&>);}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP> IN AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::AbstractDifferenceSequence(FOREST G,PREV Prev,GROUP M,VE<U> a):m_G(MO(G)),m_Prev(MO(m_Prev)),m_M(MO(M)),m_a(MO(a)),m_lazy_addition(m_G.SZ(),m_M.Zero()),m_updated(false){ST_AS(!is_same_v<U,int> && is_invocable_r_v<T,PREV,CO T&>);}TE <TY U> IN DifferenceSequence<U>::DifferenceSequence(CRI SZ):DifferenceSequence(VE<U>(SZ)){}TE <TY U> IN DifferenceSequence<U>::DifferenceSequence(VE<U> a):DifferenceSequence<U>(a.SZ(),a){}TE <TY U> IN DifferenceSequence<U>::DifferenceSequence(CRI SZ,VE<U>& a):AbstractDifferenceSequence<int,LinearGraph,PrevOfLinearGraph,U,AdditiveGroup<U>>(LinearGraph(SZ,true),PrevOfLinearGraph(),AdditiveGroup<U>(),MO(a)){}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP> IN VO AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::Initialise(VE<U> a){*TH = AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>(MO(m_G),MO(m_Prev),MO(m_M),MO(a));}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP> IN VO AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::Set(CO T& t,CO U& u){Update();m_a[m_G.Enumeration_inv(t)]= u;}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP> IN CO U& AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::Get(CO T& t){Update();RE m_a[m_G.Enumeration_inv(t)];}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP> IN U& AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::Ref(CO T& t){RE m_a[m_G.Enumeration_inv(t)];}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP> IN VO AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::Add(CO T& t,CO U& u){U& a_t = m_a[m_G.Enumeration_inv(t)];a_t = m_M.Sum(MO(a_t),u);}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP>VO AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::SubTreeAdd(CO T& t_start,CO VE<T>& t_finals,CO U& u){if(u == m_M.Zero()){RE;}m_updated = true;U& m_lazy_addition_t_start = m_lazy_addition[m_G.Enumeration_inv(t_start)];m_lazy_addition_t_start = m_M.Sum(MO(m_lazy_addition_t_start),u);CO U u_inv = m_M.Inverse(u);for(auto& t_final:t_finals){auto&& edge = m_G.Edge(t_final);for(auto& t_outside:edge){U& m_lazy_addition_t_outside = m_lazy_addition[m_G.Enumeration_inv(t_outside)];m_lazy_addition_t_outside = m_M.Sum(MO(m_lazy_addition_t_outside),u_inv);}}RE;}TE <TY U> IN VO DifferenceSequence<U>::InitialSegmentAdd(CRI t_start,CO U& u){TH->SubTreeAdd(t_start,{},u);}TE <TY U> IN VO DifferenceSequence<U>::FinalSegmentAdd(CRI t_final,CO U& u){IntervalAdd(0,t_final,u);}TE <TY U> IN VO DifferenceSequence<U>::IntervalAdd(CRI t_start,CRI t_final,CO U& u){TH->SubTreeAdd(t_start,{t_final},u);}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP>AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>& AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::OP+=(CO VE<U>& a){CRI SZ = m_G.SZ();AS(SZ == int(a.SZ()));for(int i = 0;i < SZ;i++){m_a[i]= m_M.Sum(MO(m_a[i]),a[i]);}RE *TH;}TE <TY T,TY FOREST,TY PREV,TY U,TY GROUP>VO AbstractDifferenceSequence<T,FOREST,PREV,U,GROUP>::Update(){if(!m_updated){RE;}CRI SZ = m_G.SZ();for(int i = 0;i < SZ;i++){auto& m_lazy_addition_i = m_lazy_addition[i];auto&& j = m_G.Enumeration_inv(m_Prev(m_G.Enumeration(i)));if(j != i){AS(j < i);auto& m_lazy_addition_j = m_lazy_addition[j];m_lazy_addition_i = m_M.Sum(MO(m_lazy_addition_j),m_lazy_addition_i);}}CO U& zero = m_M.Zero();for(int i = 0;i < SZ;i++){auto& m_lazy_addition_i = m_lazy_addition[i];U& m_ai = m_a[i];m_ai = m_M.Sum(MO(m_ai),m_lazy_addition_i);m_lazy_addition_i = zero;}m_updated = false;RE;}IN int PrevOfLinearGraph::OP()(CRI t_start)CO NE{RE max(t_start - 1,0);}
#endif

#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/AffineSpace/CumulativeProduct/a_Body.hpp"
#else
#define SFINAE_FOR_CP_BS enable_if_t<is_invocable_r_v<bool,F,U,int>>*
TE <TY U>CL VirtualCumulativeProduct{PU:VI U PathProduct(CRI i,CRI j)= 0;VI int Parent(CRI i)= 0;VI int LCA(CRI i,CRI j)= 0;};TE <TY U,TY GROUP>CL PathProductImplementation:VI PU VirtualCumulativeProduct<U>{PU:GROUP m_M;int m_SZ;VE<U> m_right;VE<U> m_left;IN PathProductImplementation(GROUP M,CRI SZ);IN U PathProduct(CRI i_start,CRI i_final);};TE <TY U,TY GROUP>CL AbstractCumulativeProduct:PU PathProductImplementation<U,GROUP>{PU:VE<U> m_a;IN AbstractCumulativeProduct(GROUP M,CRI SZ = 0);IN AbstractCumulativeProduct(GROUP M,VE<U> a);IN VO Initialise(CO VE<U>& a);IN VO Set(CRI i,CO U& u);IN VO RightMultiply(CRI i,CO U& u);IN VO LeftMultiply(CRI i,U u);IN CO U& RightInitialSegmentProduct(int i_final);IN CO U& LeftInitialSegmentProduct(int i_final);IN U RightIntervalProduct(CRI i_start,int i_final);IN U LeftIntervalProduct(CRI i_start,int i_final);ll CountRightIntervalProductInverseImage(CO U& u);ll CountLeftIntervalProductInverseImage(CO U& u);TE <TY F,SFINAE_FOR_CP_BS = nullptr> int RightBinarySearch(CO F& f);TE <TY F,SFINAE_FOR_CP_BS = nullptr> int LeftBinarySearch(CO F& f);TE <TY F,SFINAE_FOR_CP_BS = nullptr> int RightBinarySearch(CRI i_start,CO F& f);TE <TY F,SFINAE_FOR_CP_BS = nullptr> int LeftBinarySearch(CRI i_start,CO F& f);int RightBinarySearch(CO U& u);int LeftBinarySearch(CO U& u);int RightBinarySearch(CRI i_start,CO U& u);int LeftBinarySearch(CRI i_start,CO U& u);IN int Parent(CRI i);IN int LCA(CRI i,CRI j);};TE <TY GROUP,TY...Args> AbstractCumulativeProduct(GROUP M,CO Args&... args)-> AbstractCumulativeProduct<inner_t<GROUP>,GROUP>;TE <TY U>CL CumulativeSum:PU AbstractCumulativeProduct<U,AdditiveGroup<U>>{PU:IN CumulativeSum(CRI SZ = 0);IN CumulativeSum(VE<U> a);IN VO Add(CRI i,CO U& u);IN CO U& InitialSegmentSum(int i_final);IN U IntervalSum(CRI i_start,int i_final);ll CountIntervalSumInverseImage(CO U& u = U());TE <TY...Args> int BinarySearch(CO Args&... args);};
TE <TY U,TY GROUP> IN PathProductImplementation<U,GROUP>::PathProductImplementation(GROUP M,CRI SZ):m_M(MO(M)),m_SZ(SZ),m_right(m_SZ,m_M.One()),m_left(m_right){}TE <TY U,TY GROUP> IN AbstractCumulativeProduct<U,GROUP>::AbstractCumulativeProduct(GROUP M,CRI SZ):AbstractCumulativeProduct(M,VE<U>(SZ,M.One())){}TE <TY U,TY GROUP> IN AbstractCumulativeProduct<U,GROUP>::AbstractCumulativeProduct(GROUP M,VE<U> a):PathProductImplementation<U,GROUP>(MO(M),a.SZ()),m_a(MO(a)){if(!m_a.empty()){TH->m_right[0]= TH->m_left[0]= m_a[0];for(int i = 1;i < TH->m_SZ;i++){TH->m_right[i]= TH->m_M.Product(TH->m_right[i-1],m_a[i]);TH->m_left[i]= TH->m_M.Product(m_a[i],TH->m_left[i-1]);}}}TE <TY U> IN CumulativeSum<U>::CumulativeSum(CRI SZ):CumulativeSum(VE<U>(SZ)){}TE <TY U> IN CumulativeSum<U>::CumulativeSum(VE<U> a):AbstractCumulativeProduct<U,AdditiveGroup<U>>(AdditiveGroup<U>(),MO(a)){}TE <TY U,TY GROUP> IN VO AbstractCumulativeProduct<U,GROUP>::Initialise(CO VE<U>& a){*TH = AbstractCumulativeProduct(MO(TH->m_M),a);}TE <TY U,TY GROUP> IN VO AbstractCumulativeProduct<U,GROUP>::Set(CRI i,CO U& u){AS(0 <= i && i < TH->_SZ);m_a[i]= u;if(i == 0){TH->m_right[0]= TH->m_left[0]= m_a[0];}for(int j = max(i,1);j < TH->m_SZ;j++){TH->m_right[j]= TH->m_M.Product(TH->m_right[j-1],m_a[j]);TH->m_left[j]= TH->m_M.Product(m_a[j],TH->m_left[j-1]);}}TE <TY U,TY GROUP> IN VO AbstractCumulativeProduct<U,GROUP>::RightMultiply(CRI i,CO U& u){Set(i,TH->m_M.Product(MO(m_a[i]),u));}TE <TY U,TY GROUP> IN VO AbstractCumulativeProduct<U,GROUP>::LeftMultiply(CRI i,U u){Set(i,TH->m_M.Product(MO(u),m_a[i]));}TE <TY U> IN VO CumulativeSum<U>::Add(CRI i,CO U& u){TH->RightMultiply(i,u);}TE <TY U,TY GROUP> IN U PathProductImplementation<U,GROUP>::PathProduct(CRI i_start,CRI i_final){AS(0 <= i_start && i_start < m_SZ && 0 <= i_final && i_final < m_SZ);CO int k = TH->LCA(i_start,i_final);RE m_M.Product(m_M.Product(m_left[i_start],m_M.Inverse(m_left[k])),k == 0?m_right[i_final]:m_M.Product(m_M.Inverse(m_right[TH->Parent(k)]),m_right[i_final]));}TE <TY U,TY GROUP> IN CO U& AbstractCumulativeProduct<U,GROUP>::RightInitialSegmentProduct(int i_final){i_final = min(i_final,TH->m_SZ - 1);RE 0 <= i_final?TH->m_right[i_final]:TH->m_M.One();}TE <TY U,TY GROUP> IN CO U& AbstractCumulativeProduct<U,GROUP>::LeftInitialSegmentProduct(int i_final){i_final = min(i_final,TH->m_SZ - 1);RE 0 <= i_final?TH->m_left[i_final]:TH->m_M.One();;}TE <TY U> IN CO U& CumulativeSum<U>::InitialSegmentSum(int i_final){RE TH->RightInitialSegmentProduct(MO(i_final));}TE <TY U,TY GROUP> IN U AbstractCumulativeProduct<U,GROUP>::RightIntervalProduct(CRI i_start,int i_final){i_final = min(i_final,TH->m_SZ - 1);RE i_start <= i_final?i_start <= 0?TH->m_right[i_final]:TH->m_M.Product(TH->m_M.Inverse(TH->m_right[i_start-1]),TH->m_right[i_final]):TH->m_M.One();}TE <TY U,TY GROUP> IN U AbstractCumulativeProduct<U,GROUP>::LeftIntervalProduct(CRI i_start,int i_final){i_final = min(i_final,TH->m_SZ - 1);RE i_start <= i_final?i_start <= 0?TH->m_left[i_final]:TH->m_M.Product(TH->m_left[i_final],TH->m_M.Inverse(TH->m_left[i_start - 1])):TH->m_M.One();}TE <TY U> IN U CumulativeSum<U>::IntervalSum(CRI i_start,int i_final){RE TH->RightIntervalProduct(i_start,MO(i_final));}TE <TY U,TY GROUP> ll AbstractCumulativeProduct<U,GROUP>::CountRightIntervalProductInverseImage(CO U& u){Map<U,ll> f{};f[u]++;ll AN = 0;for(int i = 0;i < TH->m_SZ;i++){CO U& m_right_i = TH->m_right[i];f.count(m_right_i)== 1?AN += f[m_right_i]:AN;f[TH->m_M.Product(m_right_i,u)]++;}RE AN;}TE <TY U,TY GROUP> ll AbstractCumulativeProduct<U,GROUP>::CountLeftIntervalProductInverseImage(CO U& u){Map<U,ll> f{};f[u]++;ll AN = 0;for(int i = 0;i < TH->m_SZ;i++){CO U& m_left_i = TH->m_left[i];f.count(m_left_i)== 1?AN += f[m_left_i]:AN;f[TH->m_M.Product(u,m_left_i)]++;}RE AN;}TE <TY U> ll CumulativeSum<U>::CountIntervalSumInverseImage(CO U& u){RE TH->CountRightIntervalProductInverseImage(u);}TE <TY U,TY GROUP> TE <TY F,SFINAE_FOR_CP_BS>int AbstractCumulativeProduct<U,GROUP>::RightBinarySearch(CO F& f){if(f(m_a[0],0)){RE 0;}int l = 0;int r = TH->m_SZ;WH(l + 1 < r){int m =(l + r)/ 2;(f(RightInitialSegmentProduct(m),m)?r:l)= m;}RE r;}TE <TY U,TY GROUP> TE <TY F,SFINAE_FOR_CP_BS>int AbstractCumulativeProduct<U,GROUP>::LeftBinarySearch(CO F& f){if(f(m_a[0],0)){RE 0;}int l = 0;int r = TH->m_SZ;WH(l + 1 < r){int m =(l + r)/ 2;(f(LeftInitialSegmentProduct(m),m)?r:l)= m;}RE r;}TE <TY U,TY GROUP> TE <TY F,SFINAE_FOR_CP_BS>int AbstractCumulativeProduct<U,GROUP>::RightBinarySearch(CRI i_start,CO F& f){if(f(m_a[i_start],i_start)){RE i_start;}int l = i_start;int r = TH->m_SZ;WH(l + 1 < r){int m =(l + r)/ 2;(f(RightIntervalProduct(i_start,m),m)?r:l)= m;}RE r;}TE <TY U,TY GROUP> TE <TY F,SFINAE_FOR_CP_BS>int AbstractCumulativeProduct<U,GROUP>::LeftBinarySearch(CRI i_start,CO F& f){if(f(m_a[i_start],i_start)){RE i_start;}int l = i_start;int r = TH->m_SZ;WH(l + 1 < r){int m =(l + r)/ 2;(f(LeftIntervalProduct(i_start,m),m)?r:l)= m;}RE r;}TE <TY U,TY GROUP> IN int AbstractCumulativeProduct<U,GROUP>::RightBinarySearch(CO U& u){RE RightBinarySearch([&](CO U& prod,CRI i){RE !(prod < u);});}TE <TY U,TY GROUP> IN int AbstractCumulativeProduct<U,GROUP>::LeftBinarySearch(CO U& u){RE LeftBinarySearch([&](CO U& prod,CRI i){RE !(prod < u);});}TE <TY U,TY GROUP> IN int AbstractCumulativeProduct<U,GROUP>::RightBinarySearch(CRI i_start,CO U& u){RE RightBinarySearch(i_start,[&](CO U& prod,CRI i){RE !(prod < u);});}TE <TY U,TY GROUP> IN int AbstractCumulativeProduct<U,GROUP>::LeftBinarySearch(CRI i_start,CO U& u){RE LeftBinarySearch(i_start,[&](CO U& prod,CRI i){RE !(prod < u);});}TE <TY U> TE <TY...Args> int CumulativeSum<U>::BinarySearch(CO Args&... args){RE TH->RightBinarySearch(args...);}TE <TY U,TY GROUP> IN int AbstractCumulativeProduct<U,GROUP>::Parent(CRI i){RE i - 1;}TE <TY U,TY GROUP> IN int AbstractCumulativeProduct<U,GROUP>::LCA(CRI i,CRI j){RE min(i,j);}
#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
  #define REPEAT_MAIN( BOUND ) START_MAIN; signal( SIGABRT , &AlertAbort ); AutoCheck( exec_mode , use_getline ); CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if( exec_mode == solve_mode ){ if CE( bound_test_case_num > 1 ){ CERR( "テストケースの個数を入力してください。" ); SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } } else { if( exec_mode == experiment_mode ){ Experiment(); } else if( exec_mode == small_test_mode ){ SmallTest(); } else if( exec_mode == random_test_mode ){ CERR( "ランダムテストを行う回数を指定してください。" ); SET_LL( test_case_num ); REPEAT( test_case_num ){ RandomTest(); } } RE 0; } FINISH_MAIN
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE2 )
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) if( exec_mode == solve_mode ){ SET_LL( A ); ASSERT( A , MIN , MAX ); } else if( exec_mode == random_test_mode ){ CERR( #A , " = " , ( A = GetRand( MIN , MAX ) ) ); } else { AS( false ); }
  #define SOLVE_ONLY ST_AS( __FUNCTION__[0] == 'S' )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define COUT( ... ) VariadicCout( cout << "出力： " , __VA_ARGS__ ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << endl
  #define COUT_A( A , N ) cout << "出力： "; OUTPUT_ARRAY( cout , A , N ) << endl
  #define CERR_ITR( A ) OUTPUT_ITR( cerr , A ) << endl
  #define COUT_ITR( A ) cout << "出力： "; OUTPUT_ITR( cout , A ) << endl
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  // #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if CE( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } FINISH_MAIN
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) SET_LL( A ); ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #define CERR( ... ) 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define CERR_A( A , N ) 
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout , A , N ) << ENDL
  #define CERR_ITR( A ) 
  #define COUT_ITR( A ) OUTPUT_ITR( cout , A ) << ENDL
#endif
#ifdef REACTIVE
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#ifdef USE_GETLINE
  #define SET_LL( A ) { GETLINE( A ## _str ); A = stoll( A ## _str ); }
  #define GETLINE_SEPARATE( SEPARATOR , ... ) SOLVE_ONLY; string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) SOLVE_ONLY; GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
#else
  #define SET_LL( A ) cin >> A
  #define CIN( LL , ... ) SOLVE_ONLY; LL __VA_ARGS__; VariadicCin( cin , __VA_ARGS__ )
  #define SET_A( A , N ) SOLVE_ONLY; FOR( VARIABLE_FOR_SET_A , 0 , N ){ cin >> A[VARIABLE_FOR_SET_A]; }
  #define CIN_A( LL , A , N ) VE<LL> A( N ); SET_A( A , N );
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define FINISH_MAIN REPEAT( test_case_num ){ if CE( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } Solve(); CERR( "" ); } }
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now()
#define CURRENT_TIME static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME < TL_MS - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( A , N , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A] , MIN , MAX ); }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( A , N , MIN , MAX ) vector<decldecay_t( MAX )> A( N ); SET_A_ASSERT( A , N , MIN , MAX )
#define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( decldecay_t( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( decldecay_t( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( decldecay_t( INITIAL ) VAR = INITIAL ; VAR + 1 > FINAL ; VAR -- )
#define AUTO_ITR( ARRAY ) auto itr_ ## ARRAY = ARRAY .BE() , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( AUTO_ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT_ ## HOW_MANY_TIMES , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS )
#define OUTPUT_ARRAY( OS , A , N ) FOR( VARIABLE_FOR_OUTPUT_ARRAY , 0 , N ){ OS << A[VARIABLE_FOR_OUTPUT_ARRAY] << (VARIABLE_FOR_OUTPUT_ARRAY==N-1?"":" "); } OS
#define OUTPUT_ITR( OS , A ) { auto ITERATOR_FOR_OUTPUT_ITR = A.BE() , EN_FOR_OUTPUT_ITR = A.EN(); bool VARIABLE_FOR_OUTPUT_ITR = ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR; WH( VARIABLE_FOR_OUTPUT_ITR ){ OS << *ITERATOR_FOR_COUT_ITR; ( VARIABLE_FOR_OUTPUT_ITR = ++ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR ) ? OS : OS << " "; } } OS
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ ); auto answer = Answer( __VA_ARGS__ ); bool match = naive == answer; COUT( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive == " , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ RE; }

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

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

// 入出力用
#ifndef DEBUG
#define DF_OF_COUT_FOR_VE(V)TE <CL Traits,TY Arg> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& 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;}
TE <CL Traits> IN basic_istream<char,Traits>& VariadicCin(basic_istream<char,Traits>& is){RE is;}
TE <CL Traits,TY Arg,TY... ARGS> IN basic_istream<char,Traits>& VariadicCin(basic_istream<char,Traits>& is,Arg& arg,ARGS&... args){RE VariadicCin(is >> arg,args...);}
TE <CL Traits> IN basic_istream<char,Traits>& VariadicGetline(basic_istream<char,Traits>& is,CO char& separator){RE is;}
TE <CL Traits,TY Arg,TY... ARGS> IN basic_istream<char,Traits>& VariadicGetline(basic_istream<char,Traits>& is,CO char& separator,Arg& arg,ARGS&... args){RE VariadicGetline(getline(is,arg,separator),separator,args...);}
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);
TE <CL Traits,TY Arg1,TY Arg2> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO pair<Arg1,Arg2>& arg){RE os << arg.first << " " << arg.second;}
TE <CL Traits,TY Arg> IN basic_ostream<char,Traits>& VariadicCout(basic_ostream<char,Traits>& os,CO Arg& arg){RE os << arg;}
TE <CL Traits,TY Arg1,TY Arg2,TY... ARGS> IN basic_ostream<char,Traits>& VariadicCout(basic_ostream<char,Traits>& os,CO Arg1& arg1,CO Arg2& arg2,CO ARGS&... args){RE VariadicCout(os << arg1 << " ",arg2,args...);}
#endif

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

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

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

// 尺取り法用
// VAR_TPA_LとVAR_TPA_RをINITで初期化し、VAR_TPA_RがCONTINUE_CONDITIONを満たす限り、
// 閉区間[VAR_TPA_L,VAR_TPA_R]が条件ON_CONDITIONを満たすか否かを判定し、
// trueになるかVAR_TAR_LがVAR_TAR_Rに追い付くまでVAR_TPA_Lの更新操作UPDATE_Lを繰り返し、
// その後VAR_TPA_Rの更新操作UPDATE_Rを行う。
// 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。
#define TPA( ANSWER , VAR_TPA , INIT , CONTINUE_CONDITION , UPDATE_L , UPDATE_R , ON_CONDITION , INFO ) \
  VE<tuple<decldecay_t( INIT ),decldecay_t( INIT ),decldecay_t( INFO )>> ANSWER{}; \
  {									\
    auto init_TPA = INIT;						\
    decldecay_t( ANSWER.front() ) ANSWER ## _temp = { init_TPA , init_TPA , INFO }; \
    auto ANSWER ## _prev = ANSWER ## _temp;				\
    auto& VAR_TPA ## _L = get<0>( ANSWER ## _temp );			\
    auto& VAR_TPA ## _R = get<1>( ANSWER ## _temp );			\
    auto& VAR_TPA ## _info = get<2>( ANSWER ## _temp );			\
    bool on_TPA_prev = false;						\
    WH( true ){								\
      bool continuing = CONTINUE_CONDITION;				\
      bool on_TPA = continuing && ( ON_CONDITION );			\
      CERR( continuing ? "尺取り中" : "尺取り終了" , "： [L,R] = [" , VAR_TPA ## _L , "," , VAR_TPA ## _R , "] ," , on_TPA_prev ? "on" : "off" , "->" , on_TPA ? "on" : "off" , ", info =" , VAR_TPA ## _info );	\
      if( on_TPA_prev && ! on_TPA ){					\
	ANSWER.push_back( ANSWER ## _prev );				\
      }									\
      if( continuing ){							\
	if( on_TPA || VAR_TPA ## _L == VAR_TPA ## _R ){			\
	  ANSWER ## _prev = ANSWER ## _temp;				\
	  UPDATE_R;							\
	} else {							\
	  UPDATE_L;							\
	}								\
      } else {								\
	break;								\
      }									\
      on_TPA_prev = on_TPA;						\
    }									\
  }									\

#endif

// データ構造用
TE <TY T,TE <TY...> TY V> IN auto OP+(CO V<T>& a0,CO V<T>& a1)-> decldecay_t((declval<V<T>>().push_back(declval<T>()),a0)){if(a0.empty()){RE a1;}if(a1.empty()){RE a0;}AS(a0.SZ()== a1.SZ());V<T> AN{};for(auto IT0 = a0.BE(),IT1 = a1.BE(),EN0 = a0.EN();IT0 != EN0;IT0++,IT1++){AN.push_back(*IT0 + *IT1);}RE AN;}
TE <TY T,TY U> IN pair<T,U> OP+(CO pair<T,U>& t0,CO pair<T,U>& t1){RE{t0.first + t1.first,t0.second + t1.second};}
TE <TY T,TY U,TY V> IN tuple<T,U,V> OP+(CO tuple<T,U,V>& t0,CO tuple<T,U,V>& t1){RE{get<0>(t0)+ get<0>(t1),get<1>(t0)+ get<1>(t1),get<2>(t0)+ get<2>(t1)};}
TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W> OP+(CO tuple<T,U,V,W>& t0,CO tuple<T,U,V,W>& t1){RE{get<0>(t0)+ get<0>(t1),get<1>(t0)+ get<1>(t1),get<2>(t0)+ get<2>(t1),get<3>(t0)+ get<3>(t1)};}
TE <TY T> IN T Addition(CO T& t0,CO T& t1){RE t0 + t1;}
TE <TY T> IN T Xor(CO T& t0,CO T& t1){RE t0 ^ t1;}
TE <TY T> IN T MU(CO T& t0,CO T& t1){RE t0 * t1;}
TE <TY T> IN CO T& Zero(){ST CO T z{};RE z;}
TE <TY T> IN CO T& One(){ST CO T o = 1;RE o;}TE <TY T> IN T AdditionInv(CO T& t){RE -t;}
TE <TY T> IN T Id(CO T& v){RE v;}
TE <TY T> IN T Min(CO T& a,CO T& b){RE a < b?a:b;}
TE <TY T> IN T Max(CO T& a,CO T& b){RE a < b?b:a;}

// グラフ用
TE <TY T,TE <TY...> TY V> IN auto Get(CO V<T>& a){RE[&](CRI i = 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;}

// グリッド問題用
int H,W,H_minus,W_minus,HW;
VE<string> wall_str;VE<VE<bool> > non_wall;
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;}
CO string direction[4]={"U","R","D","L"};
IN int DirectionNumberOnGrid(CRI i,CRI j,CRI k,CRI h){RE i<k?2:i>k?0:j<h?1:j>h?3:(AS(false),-1);}
IN int DirectionNumberOnGrid(CRI v,CRI w){auto[i,j]=EnumHW(v);auto[k,h]=EnumHW(w);RE DirectionNumberOnGrid(i,j,k,h);}
IN int ReverseDirectionNumberOnGrid(CRI n){AS(0<=n&&n<4);RE(n+2)%4;}
IN VE<int> EdgeOnGrid(CRI v){VE<int>AN{};auto[i,j]=EnumHW(v);if(i>0&&wall_str[i-1][j]==walkable){AN.push_back(EnumHW_inv({i-1,j}));}if(i+1<H&&wall_str[i+1][j]==walkable){AN.push_back(EnumHW_inv({i+1,j}));}if(j>0&&wall_str[i][j-1]==walkable){AN.push_back(EnumHW_inv({i,j-1}));}if(j+1<W&&wall_str[i][j+1]==walkable){AN.push_back(EnumHW_inv({i,j+1}));}RE AN;}
IN VE<path> WeightedEdgeOnGrid(CRI v){VE<path>AN{};auto[i,j]=EnumHW(v);if(i>0&&wall_str[i-1][j]==walkable){AN.push_back({EnumHW_inv({i-1,j}),1});}if(i+1<H&&wall_str[i+1][j]==walkable){AN.push_back({EnumHW_inv({i+1,j}),1});}if(j>0&&wall_str[i][j-1]==walkable){AN.push_back({EnumHW_inv({i,j-1}),1});}if(j+1<W&&wall_str[i][j+1]==walkable){AN.push_back({EnumHW_inv({i,j+1}),1});}RE AN;}
IN VO SetWallStringOnGrid(CRI i,VE<string>& S){if(S.empty()){S.reSZ(H);}cin>>S[i];AS(int(S[i].SZ())==W);}
IN VO SetWallOnGrid(CRI i,VE<VE<bool>>& b){if(b.empty()){b.reSZ(H,VE<bool>(W));}auto&S_i=wall_str[i];auto&b_i=b[i];FOR(j,0,W){b_i[j]=S_i[j]==walkable?false:(AS(S_i[j]==unwalkable),true);}}

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

// VVV 常設ライブラリは以下に挿入する。
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Utility/Set/a_Body.hpp"
  #include "c:/Users/user/Documents/Programming/Mathematics/Algebra/Monoid/Group/a_Body.hpp"
  #include "c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/a_Body.hpp"
  #include "c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Mod/ConstexprModulo/Debug/a_Body.hpp"
  CEXPR(uint,P,998244353);US MP = Mod<P>;
#else
// Set (1KB)
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,TE <TY...> TY MOD>struct hash<MOD<T>>{IN size_t OP()(CO MOD<T>& n)CO;};TE <TY T1,TY T2,TE <TY...> TY PAIR>struct hash<PAIR<T1,T2>>{IN size_t OP()(CO PAIR<T1,T2>& n)CO;};TE <TY T1,TY T2,TY T3>struct hash<tuple<T1,T2,T3>>{IN size_t OP()(CO tuple<T1,T2,T3>& n)CO;};
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>>;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>>;
TE <TY T,TE <TY...> TY MOD> IN size_t hash<MOD<T>>::OP()(CO MOD<T>& n)CO{ST CO hash<T> h;RE h(n.RP());}TE <TY T0,TY T1,TE <TY...> TY PAIR> IN size_t hash<PAIR<T0,T1>>::OP()(CO PAIR<T0,T1>& n)CO{ST CO size_t seed = GetRand(1e3,1e8);ST CO hash<T0> h0;ST CO hash<T1> h1;RE(h0(get<0>(n))+ seed)^ h1(get<1>(n));}TE <TY T0,TY T1,TY T2> IN size_t hash<tuple<T0,T1,T2>>::OP()(CO tuple<T0,T1,T2>& n)CO{ST CO size_t seed = GetRand(1e3,1e8);ST CO hash<pair<T0,T1>> h01;ST CO hash<T2> h2;RE(h01({get<0>(n),get<1>(n)})+ seed)^ h2(get<2>(n));}

// Algebra (4KB)
#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 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 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 U AbstractNSet<U,F_U>::Transfer(CO U& u){RE m_f_U(u);}TE <TY U> IN U VirtualNSet<U>::Inverse(CO U& u){RE Transfer(u);}TE <TY U,TY M_U> IN AbstractMagma<U,M_U>::AbstractMagma(M_U m_U):m_m_U(MO(m_U)){ST_AS(is_invocable_r_v<U,M_U,U,U>);}TE <TY U> IN U AdditiveMagma<U>::Product(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;}

// 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;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,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 E> MemorisationGraph(CRI SZ,E edge)-> MemorisationGraph<decldecay_t(declval<E>()().back()),E>;TE <TY E> MemorisationGraph(CRI SZ,E edge)-> MemorisationGraph<decldecay_t(get<0>(declval<E>()().back())),E>;
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,E edge):EdgeImplimentation<T,T,CRI,E>(SZ,MO(edge)),m_LE(),m_memory(),m_memory_inv(){ST_AS(is_invocable_v<E> && 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 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));}

// ConstexprModulo (7KB)
CEXPR(uint,P,998244353);
#define RP Represent
#define DeRP Derepresent

TE <uint M,TY INT> CE INT RS(INT n)NE{RE MO(n < 0?((((++n)*= -1)%= M)*= -1)+= M - 1:n < INT(M)?n:n %= M);}TE <TY INT> CE INT& RSP(INT& n)NE{CE CO uint trunc =(1 << 23)- 1;INT n_u = n >> 23;n &= trunc;INT n_uq =(n_u / 7)/ 17;n_u -= n_uq * 119;n += n_u << 23;RE n < n_uq?n += P - n_uq:n -= n_uq;}
TE <uint M> CL Mod;TE <uint M>CL COantsForMod{PU:COantsForMod()= delete;ST CE CO uint g_memory_bound = 1e6;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_minus_1 = M - 2;ST CE int g_order_minus_1_neg = -g_order_minus_1;};
#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> 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> 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);TE <TY INT> CE Mod<M>& OP<<=(INT n);TE <TY INT> CE Mod<M>& OP>>=(INT 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(/,);TE <TY INT> CE Mod<M> OP^(INT EX)CO;TE <TY INT> CE Mod<M> OP<<(INT n)CO;TE <TY INT> CE Mod<M> OP>>(INT n)CO;CE Mod<M> OP-()CO NE;CE Mod<M>& SignInvert()NE;IN Mod<M>& Invert();TE <TY INT> CE Mod<M>& PW(INT EX);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>& Inverse(CRUI n);ST IN CO Mod<M>& Factorial(CRUI n);ST IN CO Mod<M>& FactorialInverse(CRUI n);ST IN Mod<M> Combination(CRUI n,CRUI i);ST IN CO Mod<M>& zero()NE;ST IN CO Mod<M>& one()NE;TE <TY INT> CE Mod<M>& PositivePW(INT EX)NE;TE <TY INT> CE Mod<M>& NonNegativePW(INT EX)NE;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> CE Mod<M>::Mod(T n)NE:m_n(RS<M>(MO(n))){ST_AS(is_COructible_v<uint,decay_t<T> >);}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:RSP(m_n_copy));RE *TH;}TE <uint M> IN Mod<M>& Mod<M>::OP/=(Mod<M> n){RE OP*=(n.Invert());}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::OP<<=(INT n){AS(n >= 0);RE *TH *= Mod<M>(2).NonNegativePW(MO(n));}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::OP>>=(INT n){AS(n >=0);WH(n-- > 0){((m_n & 1)== 0?m_n:m_n += M)>>= 1;}RE *TH;}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> TE <TY INT> CE Mod<M> Mod<M>::OP^(INT EX)CO{RE MO(Mod<M>(*TH).PW(MO(EX)));}TE <uint M> TE <TY INT> CE Mod<M> Mod<M>::OP<<(INT n)CO{RE MO(Mod<M>(*TH)<<= MO(n));}TE <uint M> TE <TY INT> CE Mod<M> Mod<M>::OP>>(INT n)CO{RE MO(Mod<M>(*TH)>>= MO(n));}TE <uint M> CE Mod<M> Mod<M>::OP-()CO NE{RE MO(Mod<M>(*TH).SignInvert());}TE <uint M> CE Mod<M>& Mod<M>::SignInvert()NE{m_n > 0?m_n = M - m_n:m_n;RE *TH;}TE <uint M> IN Mod<M>& Mod<M>::Invert(){AS(m_n != 0);uint m_n_neg;RE m_n < COants::g_memory_LE?(m_n = Inverse(m_n).m_n,*TH):((m_n_neg = M - m_n)< COants::g_memory_LE)?(m_n = M - Inverse(m_n_neg).m_n,*TH):NonNegativePW(COants::g_order_minus_1);}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::PositivePW(INT 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> TE <TY INT> CE Mod<M>& Mod<M>::NonNegativePW(INT EX)NE{RE EX == 0?(m_n = 1,*TH):PositivePW(MO(EX));}TE <uint M> TE <TY INT> CE Mod<M>& Mod<M>::PW(INT EX){bool neg = EX < 0;AS(!(neg && m_n == 0));RE neg?PositivePW(MO(EX *= COants::g_order_minus_1_neg)):NonNegativePW(MO(EX));}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(CRUI n){AS(n < COants::g_memory_LE);ST Mod<M> memory[COants::g_memory_LE]={zero(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr].m_n = 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>::Factorial(CRUI n){if(M <= n){RE zero();}AS(n < COants::g_memory_LE);ST Mod<M> memory[COants::g_memory_LE]={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){(memory[LE_curr]= memory[LE_curr - 1])*= LE_curr;LE_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::FactorialInverse(CRUI n){ST Mod<M> memory[COants::g_memory_LE]={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){(memory[LE_curr]= memory[LE_curr - 1])*= Inverse(LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN Mod<M> Mod<M>::Combination(CRUI n,CRUI i){RE 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> IN Mod<M> Inverse(CO Mod<M>& n){RE MO(Mod<M>(n).Invert());}TE <uint M,TY INT> CE Mod<M> PW(Mod<M> n,INT EX){RE MO(n.PW(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 basic_istream<char,Traits>& OP>>(basic_istream<char,Traits>& is,Mod<M>& n){ll m;is >> m;n = m;RE is;}TE <uint M,CL Traits> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO Mod<M>& n){RE os << n.RP();}
#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