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

#ifdef INCLUDE_MAIN

IN VO Solve()
{
  CIN( int , K , N , M );
  CIN_A( int , K , A );
  CIN_A( int , N , B );
  Map<int,int> A_hind{};
  FOR( k , 0 , K ){
    A_hind[A[k]]++;
  }
  using path_type = tuple<int,ll,ll>;
  vector<vector<path_type>> E( N + 2 );
  FOR_ITR( A_hind ){
    E[0].push_back( { itr->first , 0 , itr->second } );
  }
  FOREQ( i , 1 , N ){
    E[i].push_back( { N + 1 , 0 , B[i-1] } );
  }
  FOR( j , 0 , M ){
    CIN( ll , uj , vj , wj );
    E[uj].push_back( { vj , wj , K } );
    E[vj].push_back( { uj , wj , K } );
  }
  Graph graph{ N + 2 , Get( E ) };
  MinimumCostFlow mcf{ graph , 1LL , 1LL<<62 };
  auto [answer,flow] = mcf.GetFlow( 0 , N + 1 , K , false );
  RETURN( answer );
}
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/Geometry/Graph/Dijkstra/Potentialised/MinimumCostFlow/a_Body.hpp"
#else
TE <TY U,TY MONOID,TY SEMIGROUP>CL VirtualSemirng{PU:VI U Sum(CO U& u0,CO U& u1)= 0;VI CO U& Zero()CO NE = 0;VI U Product(CO U& u0,CO U& u1)= 0;VI MONOID& AdditiveMonoid()NE = 0;VI SEMIGROUP& MultiplicativeSemigroup()NE = 0;US type = U;};TE <TY U,TY MONOID,TY SEMIGROUP>CL AbstractSemirng:VI PU VirtualSemirng<U,MONOID,SEMIGROUP>{PU:MONOID m_R0;SEMIGROUP m_R1;IN AbstractSemirng(MONOID R0,SEMIGROUP R1);IN U Sum(CO U& u0,CO U& u1);IN CO U& Zero()CO NE;IN U Product(CO U& u0,CO U& u1);IN MONOID& AdditiveMonoid()NE;IN SEMIGROUP& MultiplicativeSemigroup()NE;};TE <TY U>CL Semirng:PU AbstractSemirng<U,AdditiveMonoid<U>,MultiplicativeMagma<U>>{PU:IN Semirng();};
TE <TY U,TY MONOID,TY SEMIGROUP> IN AbstractSemirng<U,MONOID,SEMIGROUP>::AbstractSemirng(MONOID R0,SEMIGROUP R1):m_R0(MO(R0)),m_R1(MO(R1)){}TE <TY U> IN Semirng<U>::Semirng():AbstractSemirng<U,AdditiveMonoid<U>,MultiplicativeMagma<U>>(AdditiveMonoid<U>(),MultiplicativeMagma()){}TE <TY U,TY MONOID,TY SEMIGROUP> IN U AbstractSemirng<U,MONOID,SEMIGROUP>::Sum(CO U& u0,CO U& u1){RE m_R0.Sum(u0,u1);}TE <TY U,TY MONOID,TY SEMIGROUP> IN CO U& AbstractSemirng<U,MONOID,SEMIGROUP>::Zero()CO NE{RE m_R0.Zero();}TE <TY U,TY MONOID,TY SEMIGROUP> IN U AbstractSemirng<U,MONOID,SEMIGROUP>::Product(CO U& u0,CO U& u1){RE m_R1.Product(u0,u1);}TE <TY U,TY MONOID,TY SEMIGROUP> IN MONOID& AbstractSemirng<U,MONOID,SEMIGROUP>::AdditiveMonoid()NE{RE m_R0;}TE <TY U,TY MONOID,TY SEMIGROUP> IN SEMIGROUP& AbstractSemirng<U,MONOID,SEMIGROUP>::MultiplicativeSemigroup()NE{RE m_R1;}

TE <TY U,TY GROUP,TY MONOID>CL VirtualRing:VI PU VirtualSemirng<U,GROUP,MONOID>{PU:VI U Inverse(CO U& u)= 0;VI CO U& One()CO NE = 0;IN GROUP& AdditiveGroup()NE;IN MONOID& MultiplicativeMonoid()NE;};TE <TY U,TY GROUP,TY MONOID>CL AbstractRing:VI PU VirtualRing<U,GROUP,MONOID>,PU AbstractSemirng<U,GROUP,MONOID>{PU:IN AbstractRing(GROUP R0,MONOID R1);IN U Inverse(CO U& u);IN CO U& One()CO NE;};TE <TY U = ll>CL Ring:VI PU AbstractRing<U,AdditiveGroup<U>,MultiplicativeMonoid<U>>{PU:IN Ring(CO U& one_U);};
TE <TY U,TY GROUP,TY MONOID> IN AbstractRing<U,GROUP,MONOID>::AbstractRing(GROUP R0,MONOID R1):AbstractSemirng<U,GROUP,MONOID>(MO(R0),MO(R1)){}TE <TY U> IN Ring<U>::Ring(CO U& one_U):AbstractRing<U,AdditiveGroup<U>,MultiplicativeMonoid<U>>(AdditiveGroup<U>(),MultiplicativeMonoid<U>(one_U)){}TE <TY U,TY GROUP,TY MONOID> IN U AbstractRing<U,GROUP,MONOID>::Inverse(CO U& u){RE TH->m_R0.Inverse(u);}TE <TY U,TY GROUP,TY MONOID> IN CO U& AbstractRing<U,GROUP,MONOID>::One()CO NE{RE TH->m_R1.One();}TE <TY U,TY GROUP,TY MONOID> IN GROUP& VirtualRing<U,GROUP,MONOID>::AdditiveGroup()NE{RE TH->AdditiveMonoid();}TE <TY U,TY GROUP,TY MONOID> IN MONOID& VirtualRing<U,GROUP,MONOID>::MultiplicativeMonoid()NE{RE TH->MultiplicativeSemigroup();}

#define DIJKSTRA_PREP(INITIALISE_PREV)CO U& one = m_M.One();AS(one < infty);auto&& i_start = m_G.Enumeration_inv(t_start);AS(0 <= i_start && i_start < SZ);INITIALISE_PREV;
#define DIJKSTRA_BODY_1(SET_PREV)if(path_LE == -1){path_LE = SZ - 1;}weight[i_start]= one;int i = i_start;for(int num = 0;num < path_LE;num++){CO U& weight_i = weight[i];fixed[i]= true;auto&& edge_i = m_G.Edge(m_G.Enumeration(i));for(auto IT = edge_i.BE(),EN = edge_i.end();IT != EN;IT++){auto&& j = m_G.Enumeration_inv(IT->first);if(!fixed[j]){CO U& edge_ij = get<1>(*IT);U temp = m_M.Product(weight_i,edge_ij);AS(temp < infty);U& weight_j = weight[j];if(temp < weight_j){SET_PREV;weight_j = MO(temp);}}}U temp = infty;for(int j = 0;j < SZ;j++){if(!fixed[j]){U& weight_j = weight[j];if(weight_j < temp){temp = weight_j;i = j;}}}}
#define DIJKSTRA_BODY_2(CHECK_FINAL,SET_PREV)AS(path_LE == -1);set<pair<U,int>> vertex{};vertex.insert(pair<U,int>(weight[i_start]= one,i_start));WH(! vertex.empty()){auto BE = vertex.BE();auto[weight_i,i]= *BE;CHECK_FINAL;fixed[i]= true;vertex.erase(BE);auto&& edge_i = m_G.Edge(m_G.Enumeration(i));VE<pair<U,int>> changed_vertex{};for(auto IT = edge_i.BE(),EN = edge_i.end();IT != EN;IT++){auto&& j = m_G.Enumeration_inv(IT->first);if(!fixed[j]){CO U& edge_ij = get<1>(*IT);U temp = m_M.Product(weight_i,edge_ij);AS(temp < infty);U& weight_j = weight[j];if(temp < weight_j){if(weight_j != infty){vertex.erase(pair<U,int>(weight_j,j));}SET_PREV;changed_vertex.push_back(pair<U,int>(weight_j = MO(temp),j));}}}for(auto& v:changed_vertex){vertex.insert(v);}}
#define DIJKSTRA_BODY(INITIALISE_PREV,CHECK_FINAL,SET_PREV)DIJKSTRA_PREP(INITIALISE_PREV);if(many_edges){DIJKSTRA_BODY_1(SET_PREV);}else{DIJKSTRA_BODY_2(CHECK_FINAL,SET_PREV);}
TE <TY T,TY GRAPH,TY U,TY COMM_MONOID>CL AbstractDijkstra:PU PointedSet<U>{PU:GRAPH& m_G;COMM_MONOID m_M;IN AbstractDijkstra(GRAPH& G,COMM_MONOID M,CO U& infty);U GetDistance(CO inner_t<GRAPH>& t_start,CO inner_t<GRAPH>& t_final,CO bool& many_edges = true,int path_LE = -1);VE<U> GetDistance(CO inner_t<GRAPH>& t_start,CO bool& many_edges = true,int path_LE = -1);VO SetDistance(VE<U>& weight,VE<bool>& fixed,CO inner_t<GRAPH>& t_start,CO bool& many_edges = true,int path_LE = -1);pair<U,LI<inner_t<GRAPH>>> GetPath(CO inner_t<GRAPH>& t_start,CO inner_t<GRAPH>& t_final,CO bool& many_edges = true,int path_LE = -1);TE <TE <TY...> TY V> pair<VE<U>,VE<LI<inner_t<GRAPH>>>> GetPath(CO inner_t<GRAPH>& t_start,CO V<inner_t<GRAPH>>& t_finals,CO bool& many_edges = true,int path_LE = -1);pair<VE<U>,VE<LI<inner_t<GRAPH>>>> GetPath(CO inner_t<GRAPH>& t_start,CO bool& many_edges = true,int path_LE = -1);};TE <TY GRAPH,TY U,TY COMM_MONOID> AbstractDijkstra(GRAPH& G,COMM_MONOID M,CO U& infty)-> AbstractDijkstra<inner_t<GRAPH>,GRAPH,U,COMM_MONOID>;TE <TY T,TY GRAPH>CL Dijkstra:PU AbstractDijkstra<T,GRAPH,ll,AdditiveMonoid<>>{PU:IN Dijkstra(GRAPH& G);};TE <TY GRAPH> Dijkstra(GRAPH& G)-> Dijkstra<inner_t<GRAPH>,GRAPH>;
TE <TY T,TY GRAPH,TY U,TY COMM_MONOID> IN AbstractDijkstra<T,GRAPH,U,COMM_MONOID>::AbstractDijkstra(GRAPH& G,COMM_MONOID M,CO U& infty):PointedSet<U>(infty),m_G(G),m_M(MO(M)){ST_AS(! is_same_v<U,int>);}TE <TY T,TY GRAPH> IN Dijkstra<T,GRAPH>::Dijkstra(GRAPH& G):AbstractDijkstra<T,GRAPH,ll,AdditiveMonoid<>>(G,AdditiveMonoid<>(),4611686018427387904){}TE <TY T,TY GRAPH,TY U,TY COMM_MONOID>U AbstractDijkstra<T,GRAPH,U,COMM_MONOID>::GetDistance(CO inner_t<GRAPH>& t_start,CO inner_t<GRAPH>& t_final,CO bool& many_edges,int path_LE){CRI SZ = m_G.SZ();CO U& infty = TH->Infty();VE weight(SZ,infty);VE<bool> fixed(SZ);auto&& i_final = m_G.Enumeration_inv(t_final);DIJKSTRA_BODY(,if(i == i_final){break;},);U AN{MO(weight[i_final])};RE AN;}TE <TY T,TY GRAPH,TY U,TY COMM_MONOID>VE<U> AbstractDijkstra<T,GRAPH,U,COMM_MONOID>::GetDistance(CO inner_t<GRAPH>& t_start,CO bool& many_edges,int path_LE){CRI SZ = m_G.SZ();CO U& infty = TH->Infty();VE weight(SZ,infty);VE<bool> fixed(SZ);DIJKSTRA_BODY(,,);RE weight;}TE <TY T,TY GRAPH,TY U,TY COMM_MONOID>VO AbstractDijkstra<T,GRAPH,U,COMM_MONOID>::SetDistance(VE<U>& weight,VE<bool>& fixed,CO inner_t<GRAPH>& t_start,CO bool& many_edges,int path_LE){CRI SZ = m_G.SZ();CO U& infty = TH->Infty();AS(int(weight.SZ())== SZ);AS(int(fixed.SZ())== SZ);DIJKSTRA_BODY(,,);RE;}TE <TY T,TY GRAPH,TY U,TY COMM_MONOID>pair<U,LI<inner_t<GRAPH>>> AbstractDijkstra<T,GRAPH,U,COMM_MONOID>::GetPath(CO inner_t<GRAPH>& t_start,CO inner_t<GRAPH>& t_final,CO bool& many_edges,int path_LE){CRI SZ = m_G.SZ();CO U& infty = TH->Infty();VE weight(SZ,infty);VE<bool> fixed(SZ);auto&& i_final = m_G.Enumeration_inv(t_final);DIJKSTRA_BODY(VE<int> prev(SZ),if(i == i_final){break;},prev[j]= i);int i = i_final;LI<inner_t<GRAPH>> path{};path.push_back(t_final);if(weight[i]!= infty){WH(i != i_start){i = prev[i];path.push_front(m_G.Enumeration(i));}}U AN{MO(weight[i_final])};RE{MO(AN),MO(path)};}TE <TY T,TY GRAPH,TY U,TY COMM_MONOID> TE <TE <TY...> TY V>pair<VE<U>,VE<LI<inner_t<GRAPH>>>> AbstractDijkstra<T,GRAPH,U,COMM_MONOID>::GetPath(CO inner_t<GRAPH>& t_start,CO V<inner_t<GRAPH>>& t_finals,CO bool& many_edges,int path_LE){CRI SZ = m_G.SZ();CO U& infty = TH->Infty();VE weight(SZ,infty);VE<bool> fixed(SZ);DIJKSTRA_BODY(VE<int> prev(SZ),,prev[j]= i);CO int path_SZ = t_finals.SZ();VE<LI<inner_t<GRAPH>>> path;path.reserve(path_SZ);for(auto IT = t_finals.BE(),EN = t_finals.end();IT != EN;IT++){LI<inner_t<GRAPH>> path_j{};CO inner_t<GRAPH>& t_final = *IT;path_j.push_back(t_final);int i = m_G.Enumeration_inv(t_final);if(weight[i]!= infty){WH(i != i_start){i = prev[i];path_j.push_front(m_G.Enumeration(i));}}path.push_back(path_j);}RE{MO(weight),MO(path)};}TE <TY T,TY GRAPH,TY U,TY COMM_MONOID>pair<VE<U>,VE<LI<inner_t<GRAPH>>>> AbstractDijkstra<T,GRAPH,U,COMM_MONOID>::GetPath(CO inner_t<GRAPH>& t_start,CO bool& many_edges,int path_LE){CRI SZ = m_G.SZ();VE<inner_t<GRAPH>> t_finals(SZ);for(int i = 0;i < SZ;i++){t_finals[i]= i;}RE GetPath(t_start,t_finals,many_edges,path_LE);}

#define POTENTIALISED_DIJKSTRA_BODY(GET,WEIGHT)CO U& infty = TH->Infty();CO U& zero = m_M.Zero();auto edge =[&](CO T& t){CO U& potential_i = m_potential[m_G.Enumeration_inv(t)];AS(potential_i < infty);auto edge_i = m_G.Edge(t);VE<pair<T,U>> AN{};for(auto& e:edge_i){if(m_on(e)){CO auto& v_j = get<0>(e);U& w_j = get<1>(e);CO U& potential_j = m_potential[m_G.Enumeration_inv(v_j)];AS(w_j < infty && potential_j < infty);CO U potential_j_inv = m_M.Inverse(potential_j);w_j = m_M.Sum(m_M.Sum(w_j,potential_i),potential_j_inv);AS(!(w_j < zero)&& w_j < infty);AN.push_back({v_j,MO(w_j)});}}RE AN;};auto G = m_G.GetGraph(MO(edge));AbstractDijkstra d{G,m_M,infty};auto value = d.GET;CRI SZ = m_G.SZ();for(int i = 0;i < SZ;i++){auto& weight_i = WEIGHT[i];if(weight_i != infty){weight_i = m_M.Sum(weight_i,m_potential[i]);}}RE value;
TE <TY T,TY GRAPH,TY U,TY GROUP,TY On>CL AbstractPotentialisedDijkstra:PU PointedSet<U>{PU:GRAPH& m_G;GROUP m_M;T m_t_start;On m_on;VE<U> m_potential;IN AbstractPotentialisedDijkstra(GRAPH& G,GROUP M,CO T& t_start,CO U& infty,On on,VE<U> potential ={});IN CO VE<U>& Potential()CO NE;IN VO SetPotential(VE<U> potential);TE <TY...Args> VE<U> GetDistance(Args&&... args);TE <TY...Args> pair<VE<U>,VE<LI<T>>> GetPath(Args&&... args);};TE <TY T,TY GRAPH,TY On>CL PotentialisedDijkstra:PU AbstractPotentialisedDijkstra<T,GRAPH,ll,AdditiveGroup<>,On>{PU:TE <TY...Args> IN PotentialisedDijkstra(GRAPH& G,CO T& t_start,On on,Args&&... args);};
TE <TY T,TY GRAPH,TY U,TY GROUP,TY On> IN AbstractPotentialisedDijkstra<T,GRAPH,U,GROUP,On>::AbstractPotentialisedDijkstra(GRAPH& G,GROUP M,CO T& t_start,CO U& infty,On on,VE<U> potential):PointedSet<U>(infty),m_G(G),m_M(MO(M)),m_t_start(t_start),m_on(MO(on)),m_potential(potential){ST_AS(is_invocable_r_v<bool,On,decltype(declval<GRAPH>().Edge(declval<T>()).back())>);if(m_potential.empty()){m_potential = VE<U>(m_G.SZ(),m_M.Zero());}else{AS(int(m_potential.SZ())== m_G.SZ());}}TE <TY T,TY GRAPH,TY On> TE <TY...Args> IN PotentialisedDijkstra<T,GRAPH,On>::PotentialisedDijkstra(GRAPH& G,CO T& t_start,On on,Args&&... args):AbstractPotentialisedDijkstra<T,GRAPH,ll,AdditiveGroup<>,On>(G,AdditiveGroup<>(),t_start,4611686018427387904,MO(on),forward<decay_t<Args>>(args)...){}TE <TY T,TY GRAPH,TY U,TY GROUP,TY On> IN CO VE<U>& AbstractPotentialisedDijkstra<T,GRAPH,U,GROUP,On>::Potential()CO NE{RE m_potential;}TE <TY T,TY GRAPH,TY U,TY GROUP,TY On> IN VO AbstractPotentialisedDijkstra<T,GRAPH,U,GROUP,On>::SetPotential(VE<U> potential){AS(int(potential.SZ())== m_G.SZ());m_potential = MO(potential);}TE <TY T,TY GRAPH,TY U,TY GROUP,TY On> TE <TY...Args> VE<U> AbstractPotentialisedDijkstra<T,GRAPH,U,GROUP,On>::GetDistance(Args&&... args){POTENTIALISED_DIJKSTRA_BODY(GetDistance(m_t_start,forward<Args>(args)...),value);}TE <TY T,TY GRAPH,TY U,TY GROUP,TY On> TE <TY...Args> pair<VE<U>,VE<LI<T>>> AbstractPotentialisedDijkstra<T,GRAPH,U,GROUP,On>::GetPath(Args&&... args){POTENTIALISED_DIJKSTRA_BODY(GetPath(m_t_start,forward<Args>(args)...),get<0>(value));}

TE <TY T,TY GRAPH,TY U,TY RING>CL AbstractMinimumCostFlow{PU:GRAPH& m_G;RING m_R;U m_infty;VE<VE<tuple<int,U,U,int>>> m_full;VE<VE<tuple<T,U>>> m_flow;VE<VE<int>> m_edge_num;VE<VE<int>> m_edge_rev_num;IN AbstractMinimumCostFlow(GRAPH& G,RING R,CO U& infty);pair<U,VE<VE<tuple<T,U>>>> GetFlow(CO T& t_start,CO T& t_final,U f,CO bool& many_edges = true,int path_LE = -1);};TE <TY GRAPH,TY U,TY RING> AbstractMinimumCostFlow(GRAPH& G,RING R,CO U& infty)-> AbstractMinimumCostFlow<inner_t<GRAPH>,GRAPH,U,RING>;TE <TY T,TY GRAPH,TY U>CL MinimumCostFlow:PU AbstractMinimumCostFlow<T,GRAPH,U,Ring<U>>{PU:IN MinimumCostFlow(GRAPH& G,CO U& one_U,CO U& infty);};TE <TY GRAPH,TY U> MinimumCostFlow(GRAPH& G,CO U& one_U,CO U& infty)-> MinimumCostFlow<inner_t<GRAPH>,GRAPH,U>;
TE <TY T,TY GRAPH,TY U,TY RING> IN AbstractMinimumCostFlow<T,GRAPH,U,RING>::AbstractMinimumCostFlow(GRAPH& G,RING R,CO U& infty):m_G(G),m_R(MO(R)),m_infty(infty),m_full(),m_flow(),m_edge_num(),m_edge_rev_num(){CO U& zero = m_R.Zero();CRI SZ = m_G.SZ();m_full.resize(SZ);m_flow.resize(SZ);m_edge_num.resize(SZ,VE<int>(SZ,-1));m_edge_rev_num.resize(SZ,VE<int>(SZ,-1));for(int i = 0;i < SZ;i++){auto&& vi = m_G.Enumeration(i);for(auto&[vj,wj,fj]:m_G.Edge(vi)){AS(vi != vj && !(wj < zero)&& wj < m_infty && !(fj < zero)&& fj < m_infty);if(zero < fj){auto&& j = m_G.Enumeration_inv(vj);AS(m_edge_num[i][j]== -1);m_edge_num[i][j]= m_full[i].SZ();AS(m_edge_rev_num[j][i]== -1);m_edge_rev_num[j][i]= m_full[j].SZ();CO int flow_i_SZ = m_flow[i].SZ();m_full[i].push_back({j,wj,fj,flow_i_SZ});m_full[j].push_back({i,m_R.Inverse(wj),zero,flow_i_SZ});m_flow[i].push_back({vj,zero});if(m_edge_num[j][i]!= -1){swap(m_full[j][m_edge_num[j][i]],m_full[j][m_edge_rev_num[j][i]]);swap(m_edge_num[j][i],m_edge_rev_num[j][i]);}}}}}TE <TY T,TY GRAPH,TY U> IN MinimumCostFlow<T,GRAPH,U>::MinimumCostFlow(GRAPH& G,CO U& one_U,CO U& infty):AbstractMinimumCostFlow<T,GRAPH,U,Ring<U>>(G,Ring<U>(one_U),infty){}TE <TY T,TY GRAPH,TY U,TY RING>pair<U,VE<VE<tuple<T,U>>>> AbstractMinimumCostFlow<T,GRAPH,U,RING>::GetFlow(CO T& t_start,CO T& t_final,U f,CO bool& many_edges,int path_LE){CO U& zero = m_R.Zero();auto rest = m_full;auto flow = m_flow;auto edge =[&](CO T& t)-> CO VE<tuple<int,U,U,int>>&{RE rest[m_G.Enumeration_inv(t)];};auto on =[&](CO tuple<T,U,U,int>& e){RE zero < get<2>(e);};auto G = m_G.GetGraph(MO(edge));AbstractPotentialisedDijkstra pd{G,m_R.AdditiveGroup(),t_start,m_infty,MO(on)};auto&& i_start = m_G.Enumeration_inv(t_start);auto&& i_final = m_G.Enumeration_inv(t_final);CO VE<T> t_finals ={t_final};U w = zero;WH(zero < f){auto[weight,paths]= pd.GetPath(t_finals,many_edges,path_LE);CO U w_min = weight[i_final];pd.SetPotential(MO(weight));auto& path = paths.front();auto IT_path = path.BE(),IT_path_prev = IT_path,EN_path = path.EN();int i = i_start;U f_min = f;VE<tuple<U*,U*,U*,bool>> update{};update.reserve(path.SZ()- 1);WH(++IT_path != EN_path){auto&& j = m_G.Enumeration_inv(*IT_path);bool reversed = false;if(m_edge_rev_num[i][j]!= -1){auto&[j_copy,w_rev_ij,f_rev_ij,flow_num_ji]= rest[i][m_edge_rev_num[i][j]];if(zero < f_rev_ij){f_min = min(f_min,f_rev_ij);update.push_back({&f_rev_ij,&(get<2>(rest[j][m_edge_num[j][i]])),&(get<1>(flow[j][flow_num_ji])),reversed = true});}}if(!reversed){auto&[j_copy,w_ij,f_ij,flow_num_ij]= rest[i][m_edge_num[i][j]];f_min = min(f_min,f_ij);update.push_back({&f_ij,&(get<2>(rest[j][m_edge_rev_num[j][i]])),&(get<1>(flow[i][flow_num_ij])),reversed});}IT_path_prev = IT_path;i = j;}CO U f_min_inv = m_R.Inverse(f_min);f = m_R.Sum(f,f_min_inv);w = m_R.Sum(w,m_R.Product(f_min,w_min));for(auto&[p_f,p_f_rev,p_flow,reversed]:update){*p_f = m_R.Sum(MO(*p_f),f_min_inv);*p_f_rev = m_R.Sum(MO(*p_f_rev),f_min);*p_flow = m_R.Sum(MO(*p_flow),reversed?f_min_inv:f_min);}}RE{MO(w),MO(flow)};}
#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

#ifndef DEBUG
  #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( N , ... ) SOLVE_ONLY; VariadicResize( N , __VA_ARGS__ ); FOR( VARIABLE_FOR_SET_A , 0 , N ){ VariadicSet( cin , VARIABLE_FOR_SET_A , __VA_ARGS__ ); }
  #define CIN_A( LL , N , ... ) VE<LL> __VA_ARGS__; SET_A( N , __VA_ARGS__ );
#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( N , A , 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( N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N ); SET_A_ASSERT( N , A , 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 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 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>;

// 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 PositiveBaseModulo(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);}

// 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 )

// 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;						\
    }									\
  }									\

// 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 Sum(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 Product(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);}}

// 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/include/a_Body.hpp"
#else
// StdStream（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;}
TE <CL Traits,TY Arg1,TY Arg2> IN IS& OP>>(IS& is,pair<Arg1,Arg2>& arg){RE is >> arg.first >> arg.second;}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...);}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 OS& OP<<(OS& os,CO pair<Arg1,Arg2>& arg){RE os << arg.first << " " << arg.second;}TE <CL Traits,TY Arg> IN OS& VariadicCout(OS& os,CO Arg& arg){RE os << arg;}TE <CL Traits,TY Arg1,TY Arg2,TY... ARGS> IN OS& VariadicCout(OS& os,CO Arg1& arg1,CO Arg2& arg2,CO ARGS&... args){RE VariadicCout(os << arg1 << " ",arg2,args...);}

// Vector（1KB）
IN void VariadicResize(CRI SZ){}TE <TY Arg,TY... ARGS> IN void VariadicResize(CRI SZ,Arg& arg,ARGS&... args){arg.reSZ(SZ);VariadicResize(SZ,args...);}

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

// 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;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,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 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));}

// 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