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

VO Solve()
{

  CIN( int , N , M );
  using weight_type = ll;
  using path_type = Tuple<int,weight_type>;
  using edge_type = Tuple<int,int,weight_type>;
  CIN_A( edge_type , 0 , M , UVW );
  CIN( string , S );
  vector<vector<path_type>> e( N );
  FOR( m , 0 , M ){
    auto& [u,v,w] = UVW[m]; --u; --v;
    e[u] <<= {v,w};
  }
  /* 状態を1次元分追加 */
  using T = T2<int>;
  int K = 2;
  SET_HW( N , K );
  auto edge = [&]( const T& v , const char& c ){
    auto& [i,k] = v;
    vector<Pair<T,weight_type>> a{};
    if( k + 1 < K ){
      if( S[i] == c ){
        a <<= {T{i,k+1},0};
      }
    }
    RUN( e[i] , [j,w] ){
      a <<= {T{j,k},w};
    }
    return a;
  };
  auto edge1 = [&]( const T& v ){ return edge( v , 'K' ); };
  GridGraph graph1{ edge1 };
  auto edge2 = [&]( const T& v ){ return edge( v , 'P' ); };
  GridGraph graph2{ edge2 };
  BranchedDijkstra dijk1{ graph1 };
  auto d1 = dijk1.GetDistance( vector<tuple<T,ll,bool>>{{T{0,0},0,0}} , 1 );
  WHAT( d1 );
  vector<tuple<T,ll,int>> t_start{};
  FOR( i , 0 , N ){
    if( S[i] == 'C' && !d1[EnumHW_inv({i,1})].empty() ){
      t_start <<= tuple{T{i,0},d1[EnumHW_inv({i,1})][0][O],i};
    }
  }
  WHAT( t_start );
  BranchedDijkstra dijk2{ graph2 };
  auto d2 = dijk2.GetDistance( t_start , 2 );
  WHAT( d2 );
  ll a = dijk2.Infty();
  FOR( i , 0 , N ){
    if( S[i] == 'C' ){
      RUN( d2[EnumHW_inv({i,1})] , [u,s] ){
        if( s != i ){
          SetMin( a , u );
        }
      }
    }
  }
  RETURN( a == dijk2.Infty() ? -1 : a );
}
REPEAT_MAIN(1);

#else /* INCLUDE_MAIN */
#ifdef INCLUDE_SUB


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

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

  CERR( "全てのケースを確認しました。" );
}

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

  }
  CERR( "全てのケースを確認しました。" );
}

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

/* VVV 常設でないライブラリは以下に挿入する。*/

template <typename GRAPH , typename U , typename COMM_SEMIGRP , typename T = inner_t<GRAPH>>
class AbstractBranchedDijkstra :
  public PointedSet<U>
{

private:
  GRAPH& m_G;
  // コンストラクタに引数が必要なMultiplicativeMonoidなどはstaticメンバ関数による
  // 参照返しがしにくく、コンストラクタの返り値である右辺値を受け取ることを許容したいので
  // 左辺値参照にはしない。
  COMM_SEMIGRP m_M;

public:
  inline AbstractBranchedDijkstra( GRAPH& G , COMM_SEMIGRP M , const U& infty );

  // TUPLEはtupleなど。Sはoperator<()が全順序である型。
  // 各点への最短径路長を始辺の色ごとに上位length個まで求める。
  template <template <typename...> typename TUPLE , typename S> vector<vector<Pair<U,S>>> GetDistance( const vector<TUPLE<T,U,S>>& t_start , const int& length );
  // S=bool、length=0で通常のダイクストラ法の代替も可能。

};

template <typename GRAPH , typename T = inner_t<GRAPH>>
class BranchedDijkstra :
  public AbstractBranchedDijkstra<GRAPH,ll,AdditiveMonoid<>,T>
{

public:
  // llをテンプレートにしてllにキャストするとデフォルト引数を取る時に型推論できないことに注意。
  inline BranchedDijkstra( GRAPH& G , const ll& infty = 1e18 );

};

template <typename GRAPH , typename U , typename COMM_SEMIGRP , typename T> inline AbstractBranchedDijkstra<GRAPH,U,COMM_SEMIGRP,T>::AbstractBranchedDijkstra( GRAPH& G , COMM_SEMIGRP M , const U& infty ) : PointedSet<U>( infty ) , m_G( G ) , m_M( move( M ) ) { static_assert( is_same_v<T,inner_t<GRAPH>> ); }
template <typename GRAPH , typename T> inline BranchedDijkstra<GRAPH,T>::BranchedDijkstra( GRAPH& G ,  const ll& infty ) : AbstractBranchedDijkstra<GRAPH,ll,AdditiveMonoid<>,T>( G , AdditiveMonoid<>() , infty ) {}

template <typename GRAPH , typename U , typename COMM_SEMIGRP , typename T> template <template <typename...> typename TUPLE , typename S>
vector<vector<Pair<U,S>>> AbstractBranchedDijkstra<GRAPH,U,COMM_SEMIGRP,T>::GetDistance( const vector<TUPLE<T,U,S>>& t_start , const int& length )
{

  const int& size = m_G.size();
  const U& infty = this->Infty();
  vector weight( size , vector<Pair<U,S>>{} );
  vector<Set<S>> fixed( size );
  set<tuple<U,S,int>> vertex{};

  auto set = [&]( const int& i , const U& u , const S& s ){

    assert( u < infty );
    vertex <<= {u,s,i};
    auto& wi = weight[i];
    int L = wi.size() , l = -1;
    
    for( int j = 0 ; j < L ; j++ ){

      auto& [uij,sij] = wi[j];

       if( sij == s ){

         SetMin( uij  , u );
         l = j;
         break;

      }

    }

    if( l == -1 ){

      wi <<= {u,s};
      l = L++;

    }

    while( l > 0 && wi[l] < wi[l-1] ){

      swap( wi[l] , wi[l-1] );
      l--;

    }

    if( L > length ){

      pop( wi );

    }

  };

  for( auto& [t,u,s] : t_start ){

    auto&& i = m_G.Enumeration_inv( t );
    set( i , u , s );

  }
  
  while( ! vertex.empty() ){

    auto [ui,s,i] = pop_min( vertex );

    if( len( fixed[i] ) == length || In( s , fixed[i] ) ){

      continue;

    }
    
    fixed[i] <<= s;

    for( auto&& edge_ij : m_G.Edge( m_G.Enumeration( i ) ) ){

      auto&& j = m_G.Enumeration_inv( get<0>( edge_ij ) );
      
      if( len( fixed[j] ) < length && ! In( s , fixed[j] ) ){

        set( j , m_M.Product( ui , get<1>( edge_ij ) ) , s );

      }

    }

  }

  return weight;

}

/* AAA 常設でないライブラリは以上に挿入する。*/

#define INCLUDE_SUB
#include __FILE__
#else /* INCLUDE_LIBRARY */
#ifdef DEBUG
  #define _GLIBCXX_DEBUG
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , test_case_num_bound , BOUND ); int test_case_num = 1; if CE( test_case_num_bound > 1 ){ FINISH_MAIN
  #ifdef USE_GETLINE
    #define SET_SEPARATE( SEPARATOR , ... ) VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
    #define SET( ... ) SET_SEPARATE( '\n' , __VA_ARGS__ )
    #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; SET_SEPARATE( SEPARATOR , __VA_ARGS__ )
    #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
    #define FINISH_MAIN GETLINE( test_case_num_str ); test_case_num = stoi( test_case_num_str ); ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } }
  #else
    #define SET( ... ) VariadicCin( cin , __VA_ARGS__ )
    #define CIN( LL , ... ) LL __VA_ARGS__; SET( __VA_ARGS__ )
    #define SET_A( I , N , ... ) VariadicResize( N + I , __VA_ARGS__ ); FOR( VARIABLE_FOR_SET_A , 0 , N ){ VariadicSet( cin , VARIABLE_FOR_SET_A + I , __VA_ARGS__ ); }
    #define CIN_A( LL , I , N , ... ) VE<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
    #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) VE VAR( N0 + I0 , VE<LL>( N1 + I1 ) ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , VAR[VARIABLE_FOR_CIN_AA + I0] ); }
    #define FINISH_MAIN SET_ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } }
  #endif
  #define SET_ASSERT( A , MIN , MAX ) SET( A ); ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ )
  #define CERR( ... ) 
  #define CERRNS( ... ) 
  #define COUT_A( I , N , A ) CoutArray( cout , I , N , A ) << ENDL
  #define CERR_A( I , N , A ) 
  #define WHAT( ... ) 
  #define TLE( CONDITION ) if( !( CONDITION ) ){ cerr << "TLE" << endl; ll TLE_VAR = 1; while( TLE_VAR != 0 ){ ( TLE_VAR += 2 ) %= int( 1e9 ); } cerr << TLE_VAR << endl; }
  #define MLE( CONDITION ) if( !( CONDITION ) ){ cerr << "MLE" << endl; vector<vector<ll>> MLE_VAR{}; REPEAT( 1e6 ){ MLE_VAR.push_back( vector<ll>( 1e6 ) ); } cerr << MLE_VAR << endl; }
  #define OLE( CONDITION ) if( !( CONDITION ) ){ cerr << "OLE" << endl; REPEAT( 1e8 ){ COUT( "OLE" ); } }
#endif
#ifdef REACTIVE
  #ifndef DEBUG
    #define LOCAL( ... ) 
    #define RSET( A , ... ) SET( A )
  #endif
  #define RCIN( LL , A , ... ) LL A; RSET( A , __VA_ARGS__ )
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now(); double loop_average_time = 0.0 , loop_start_time = loop_average_time , current_time = loop_start_time; int loop_count = current_time; assert( loop_count == 0 )
#define CURRENT_TIME ( current_time = static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 ) )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME , loop_count == 0 ? loop_start_time = current_time : loop_average_time = ( current_time - loop_start_time ) / loop_count , ++loop_count , current_time < TL_MS - loop_average_time * 2 - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( I , N , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A + I] , MIN , MAX ); }
#define SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_AA0 , 0 , N0 ){ FOR( VARIABLE_FOR_SET_AA1 , 0 , N1 ){ SET_ASSERT( A[VARIABLE_FOR_SET_AA0 + I0][VARIABLE_FOR_SET_AA1 + I1] , MIN , MAX ); } }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); SET_A_ASSERT( I , N , A , MIN , MAX )
#define CIN_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) vector A( N0 + I0 , vector<decldecay_t( MAX )>( N1 + I1 ) ); SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX )
#define PR1( A1 , ... ) A1
#define PR2( A1 , A2 , ... ) A2
#define PR3( A1 , A2 , A3 , ... ) A3
#define FOR_( VAR , INITIAL , FINAL , UPPER , COMP , INCR ) for( decldecay_t( UPPER ) VAR = INITIAL ; VAR COMP ( FINAL ) ; VAR INCR )
#define FOR( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , < , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQ( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , <= , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQINV( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , INITIAL , + 1 > , PR3( __VA_ARGS__ , -= PR2( __VA_ARGS__ , ? ) , -- ) )
#define ITR( ARRAY ) auto begin_ ## ARRAY = ARRAY .BE() , itr_ ## ARRAY = begin_ ## ARRAY , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define RUN( ARRAY , ... ) for( auto&& __VA_ARGS__ : ARRAY )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS ); cerr << fixed << setprecision( DECIMAL_DIGITS )
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ , false ); auto answer = Answer( __VA_ARGS__ , false ); bool match = naive == answer; CERR( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive ==" , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ CERR( "出力の不一致が検出されました。" ); RE; }
#define CHECK( ... ) auto answer = Answer( __VA_ARGS__ , false ); CERR( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Answer == " , answer )

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

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

/* VVV 常設ライブラリは以下に挿入する。*/
#ifdef DEBUG
  #include "C:/Users/user/Documents/Programming/Contest/Template/Local/a_Body.hpp"
  US MP = DMod;
#else
/* BinarySearch (2KB)*/
CE bool reactive =
#ifdef REACTIVE
true;
#else
false;
#endif
#define BS(AN,MINIMUM,MAXIMUM,EXPRESSION,DESIRED_INEQUALITY,CO_TARGET,INEQUALITY_FOR_CHECK,UPDATE_U,UPDATE_L,UPDATE_AN,EXTERNAL)ST_AS(! is_same<decldecay_t(CO_TARGET),uint>::value && ! is_same<decldecay_t(CO_TARGET),ull>::value);ll AN = MINIMUM;{ll AN ## _L = MINIMUM;ll AN ## _R = MAXIMUM;AN = UPDATE_AN;ll EXPRESSION_BS;CO ll CO_TARGET_BS =(CO_TARGET);ll DIFFERENCE_BS;WH(AN ## _L < AN ## _R){DIFFERENCE_BS =(EXPRESSION_BS =(EXPRESSION))- CO_TARGET_BS;if(DIFFERENCE_BS INEQUALITY_FOR_CHECK 0){AN ## _R = UPDATE_U;}else{AN ## _L = UPDATE_L;}AN = UPDATE_AN;}if(AN ## _L > AN ## _R || !(reactive ||(EXPRESSION)DESIRED_INEQUALITY CO_TARGET_BS)){AN = EXTERNAL;}}
#define MIN_GEQ(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,>=,CO_TARGET,>=,AN,AN + 1,Mid(AN ## _L,AN ## _R),(MAXIMUM)+ 1)
#define MAX_LEQ(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,<=,CO_TARGET,>,AN - 1,AN,Mid(AN ## _L + 1,AN ## _R),(MINIMUM)- 1)
#define MAX_GEQ(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,>=,CO_TARGET,<,AN - 1,AN,Mid(AN ## _L + 1,AN ## _R),(MINIMUM)- 1)
#define MIN_LEQ(AN,MINIMUM,MAXIMUM,EXPRESSION,CO_TARGET)BS(AN,MINIMUM,MAXIMUM,EXPRESSION,<=,CO_TARGET,<=,AN,AN + 1,Mid(AN ## _L,AN ## _R),(MAXIMUM)+ 1)
TE <TY INT> IN CE INT Mid(CO INT& l,CO INT& r){RE l +((r - l)>> 1);}

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

/* Set (2KB)*/
#define DC_OF_HASH(...)struct hash<__VA_ARGS__>{IN size_t OP()(CO __VA_ARGS__& n)CO;};
CL is_ordered{PU:is_ordered()= delete;TE <TY T> ST CE auto Check(CO T& t)-> decltype(t < t,true_type());ST CE false_type Check(...);TE <TY T> ST CE CO bool value = is_same_v< decltype(Check(declval<T>())),true_type >;};
TE <TY T>US Set = conditional_t<is_COructible_v<unordered_set<T>>,unordered_set<T>,conditional_t<is_ordered::value<T>,set<T>,VO>>;

#define DF_OF_POP_FOR_SET(SET)TE <TY T> IN T pop_max(SET& S){AS(!S.empty());auto IT = --S.EN();T AN = *IT;S.erase(IT);RE AN;}TE <TY T> IN T pop_min(SET& S){AS(!S.empty());auto IT = S.BE();T AN = *IT;S.erase(IT);RE AN;}TE <TY T> IN SET& OP<<=(SET& S,T t){S.insert(MO(t));RE S;}TE <TY T,TY U> IN SET& OP<<=(SET& S,U&& u){S.insert(T{forward<U>(u)});RE S;}TE <TY T> IN SET& OP>>=(SET& S,CO T& t){auto IT = S.find(t);if(IT != S.EN()){S.erase(IT);}RE S;}TE <TY T,TY U> IN SET& OP>>=(SET& S,CO U& u){RE S >>= T{u};}TE <TY T> IN CO T& Get(CO SET& S,int i){auto BE = S.BE(),EN = S.EN();auto& IT = i < 0?(++i,--EN):BE;WH(i > 0 && IT != EN){--i;++IT;}WH(i < 0 && IT != BE){++i;--IT;}AS(i == 0);RE *IT;}
#define DF_OF_UNION_FOR_SET(SET)TE <TY T> IN SET& OP|=(SET& S0,SET S1){S0.merge(MO(S1));RE S0;}TE <TY T> IN SET OP|(SET S0,SET S1){RE MO(S0.SZ()< S1.SZ()?S1 |= MO(S0):S0 |= MO(S1));}
TE <TY SET,TY T> IN TY SET::const_iterator MaximumLeq(CO SET& S,CO T& t){auto IT = S.upper_bound(t);RE IT == S.BE()?S.EN():--IT;}TE <TY SET,TY T> IN TY SET::const_iterator MaximumLt(CO SET& S,CO T& t){auto IT = S.lower_bound(t);RE IT == S.BE()?S.EN():--IT;}TE <TY SET,TY T> IN TY SET::const_iterator MinimumGeq(CO SET& S,CO T& t){RE S.lower_bound(t);}TE <TY SET,TY T> IN TY SET::const_iterator MinimumGt(CO SET& S,CO T& t){RE S.upper_bound(t);}TE <TY SET,TY ITERATOR> IN VO EraseBack(SET& S,ITERATOR& IT){IT = S.erase(IT);}TE <TY SET,TY ITERATOR> IN VO EraseFront(SET& S,ITERATOR& IT){IT = S.erase(IT);IT == S.BE()?IT = S.EN():--IT;}TE <TE <TY...> TY SET,TY T,TY...Args> IN bool In(CO T& t,CO SET<T,Args...>& S){RE S.count(t)> 0;}DF_OF_POP_FOR_SET(set<T>);DF_OF_POP_FOR_SET(unordered_set<T>);DF_OF_POP_FOR_SET(multiset<T>);DF_OF_POP_FOR_SET(unordered_multiset<T>);DF_OF_UNION_FOR_SET(set<T>);DF_OF_UNION_FOR_SET(unordered_set<T>);DF_OF_UNION_FOR_SET(multiset<T>);DF_OF_UNION_FOR_SET(unordered_multiset<T>);DF_OF_UNION_FOR_SET(VE<T>);DF_OF_UNION_FOR_SET(LI<T>);

/* Tuple (6KB)*/
#define DF_OF_AR_FOR_TUPLE(OPR)TE <TY T,TY U,TE <TY...> TY PAIR> IN auto OP OPR ## =(PAIR<T,U>& t0,CO PAIR<T,U>& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = get<0>(t1);get<1>(t0)OPR ## = get<1>(t1);RE t0;}TE <TY T,TY U,TY V,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V>& t0,CO TUPLE<T,U,V>& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = get<0>(t1);get<1>(t0)OPR ## = get<1>(t1);get<2>(t0)OPR ## = get<2>(t1);RE t0;}TE <TY T,TY U,TY V,TY W,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V,W>& t0,CO TUPLE<T,U,V,W>& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = get<0>(t1);get<1>(t0)OPR ## = get<1>(t1);get<2>(t0)OPR ## = get<2>(t1);get<3>(t0)OPR ## = get<3>(t1);RE t0;}TE <TY ARG,TY T,TY U,TE <TY...> TY PAIR> IN auto OP OPR ## =(PAIR<T,U>& t0,CO ARG& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = t1;get<1>(t0)OPR ## = t1;RE t0;}TE <TY ARG,TY T,TY U,TY V,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V>& t0,CO ARG& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = t1;get<1>(t0)OPR ## = t1;get<2>(t0)OPR ## = t1;RE t0;}TE <TY ARG,TY T,TY U,TY V,TY W,TE <TY...> TY TUPLE> IN auto OP OPR ## =(TUPLE<T,U,V,W>& t0,CO ARG& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = t1;get<1>(t0)OPR ## = t1;get<2>(t0)OPR ## = t1;get<3>(t0)OPR ## = t1;RE t0;}TE <TE <TY...> TY TUPLE,TY...ARGS,TY ARG> IN auto OP OPR(CO TUPLE<ARGS...>& t0,CO ARG& t1)-> decldecay_t((get<0>(t0),t0)){auto t = t0;RE MO(t OPR ## = t1);}
#define DF_OF_INCREMENT_FOR_TUPLE(INCR)TE <TY T,TY U,TE <TY...> TY PAIR> IN auto OP INCR(PAIR<T,U>& t)-> decltype((get<0>(t),t))&{INCR get<0>(t);INCR get<1>(t);RE t;}TE <TY T,TY U,TY V,TE <TY...> TY TUPLE> IN auto OP INCR(TUPLE<T,U,V>& t)-> decltype((get<0>(t),t))&{INCR get<0>(t);INCR get<1>(t);INCR get<2>(t);RE t;}TE <TY T,TY U,TY V,TY W,TE <TY...> TY TUPLE> IN auto OP INCR(TUPLE<T,U,V,W>& t)-> decltype((get<0>(t),t))&{INCR get<0>(t);INCR get<1>(t);INCR get<2>(t);INCR get<3>(t);RE t;}
TE <CL Traits,TY T> IN IS& OP>>(IS& is,tuple<T>& arg){RE is >> get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY...> TY V> IN auto OP>>(IS& is,V<T,U>& arg)-> decltype((get<0>(arg),is))&{RE is >> get<0>(arg)>> get<1>(arg);}TE <CL Traits,TY T,TY U,TY V> IN IS& OP>>(IS& is,tuple<T,U,V>& arg){RE is >> get<0>(arg)>> get<1>(arg)>> get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W> IN IS& OP>>(IS& is,tuple<T,U,V,W>& arg){RE is >> get<0>(arg)>> get<1>(arg)>> get<2>(arg)>> get<3>(arg);}TE <CL Traits,TY T> IN OS& OP<<(OS& os,CO tuple<T>& arg){RE os << get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY...> TY V> IN auto OP<<(OS& os,CO V<T,U>& arg)-> decltype((get<0>(arg),os))&{RE os << get<0>(arg)<< " " << get<1>(arg);}TE <CL Traits,TY T,TY U,TY V> IN OS& OP<<(OS& os,CO tuple<T,U,V>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W> IN OS& OP<<(OS& os,CO tuple<T,U,V,W>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg)<< " " << get<3>(arg);}DF_OF_AR_FOR_TUPLE(+);TE <TY T,TY U,TE <TY...> TY V> IN auto OP-(CO V<T,U>& t)-> decldecay_t((get<0>(t),t)){RE{-get<0>(t),-get<1>(t)};}TE <TY T,TY U,TY V> IN tuple<T,U,V> OP-(CO tuple<T,U,V>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t)};}TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W> OP-(CO tuple<T,U,V,W>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t),-get<3>(t)};}DF_OF_AR_FOR_TUPLE(-);DF_OF_AR_FOR_TUPLE(*);DF_OF_AR_FOR_TUPLE(/);DF_OF_AR_FOR_TUPLE(%);DF_OF_INCREMENT_FOR_TUPLE(++);DF_OF_INCREMENT_FOR_TUPLE(--);

TE <int n>CL TupleAccessIndex{};TE <TY...Types>CL Tuple:PU tuple<Types...>{PU:IN Tuple(Types&&... args);TE <TY...Args> IN Tuple(Args&&... args);TE <int n> IN auto& OP[](CO TupleAccessIndex<n>& i)NE;TE <int n> IN CO auto& OP[](CO TupleAccessIndex<n>& i)CO NE;};TE <TY...Types>CL tuple_size<Tuple<Types...>>:PU tuple_size<tuple<Types...>>{};TE <size_t n,TY...Types>CL tuple_element<n,Tuple<Types...>>:PU tuple_element<n,tuple<Types...>>{};
TE <TY T,TY U> US Pair = Tuple<T,U>;TE <TY INT> US T2 = Pair<INT,INT>;TE <TY INT> US T3 = Tuple<INT,INT,INT>;TE <TY INT> US T4 = Tuple<INT,INT,INT,INT>;
CE TupleAccessIndex<0> O{};CE TupleAccessIndex<1> I{};CE TupleAccessIndex<2> II{};CE TupleAccessIndex<3> III{};
TE <TY...Types> IN Tuple<Types...>::Tuple(Types&&... args):tuple<Types...>(MO(args)...){}TE <TY...Types> TE <TY...Args> IN Tuple<Types...>::Tuple(Args&&... args):tuple<Types...>(forward<Args>(args)...){}TE <TY...Types> TE <int n> IN auto& Tuple<Types...>::OP[](CO TupleAccessIndex<n>& i)NE{RE get<n>(*TH);}TE <TY...Types> TE <int n> IN CO auto& Tuple<Types...>::OP[](CO TupleAccessIndex<n>& i)CO NE{RE get<n>(*TH);}TE <TY RET,TE <TY...> TY PAIR,TY INT> T2<RET> cast(CO PAIR<INT,INT>& t){RE{get<0>(t),get<1>(t)};}TE <TY RET,TY INT> T3<RET> cast(CO tuple<INT,INT,INT>& t){RE{get<0>(t),get<1>(t),get<2>(t)};}TE <TY RET,TY INT> T4<RET> cast(CO tuple<INT,INT,INT,INT>& t){RE{get<0>(t),get<1>(t),get<2>(t),get<3>(t)};}

#define DF_OF_HASH_FOR_TUPLE(PAIR)TE <TY T,TY U> IN size_t hash<PAIR<T,U>>::OP()(CO PAIR<T,U>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<T> h0;ST CO hash<U> h1;RE(h0(get<0>(n))* seed)^ h1(get<1>(n));}
TE <TY T> DC_OF_HASH(tuple<T>);TE <TY T,TY U> DC_OF_HASH(pair<T,U>);TE <TY T,TY U> DC_OF_HASH(tuple<T,U>);TE <TY T,TY U,TY V> DC_OF_HASH(tuple<T,U,V>);TE <TY T,TY U,TY V,TY W> DC_OF_HASH(tuple<T,U,V,W>);
TE <TY T> IN size_t hash<tuple<T>>::OP()(CO tuple<T>& n)CO{ST CO hash<T> h;RE h(get<0>(n));}DF_OF_HASH_FOR_TUPLE(pair);DF_OF_HASH_FOR_TUPLE(tuple);TE <TY T,TY U,TY V> IN size_t hash<tuple<T,U,V>>::OP()(CO tuple<T,U,V>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<pair<T,U>> h01;ST CO hash<V> h2;RE(h01({get<0>(n),get<1>(n)})* seed)^ h2(get<2>(n));}TE <TY T,TY U,TY V,TY W> IN size_t hash<tuple<T,U,V,W>>::OP()(CO tuple<T,U,V,W>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<pair<T,U>> h01;ST CO hash<pair<V,W>> h23;RE(h01({get<0>(n),get<1>(n)})* seed)^ h23({get<2>(n),get<3>(n)});}

/* Vector (3KB)*/
#define DF_OF_COUT_FOR_VE(V)TE <CL Traits,TY Arg> IN OS& OP<<(OS& os,CO V<Arg>& arg){auto BE = arg.BE(),EN = arg.EN();auto IT = BE;WH(IT != EN){(IT == BE?os:os << " ")<< *IT;IT++;}RE os;}
DF_OF_COUT_FOR_VE(VE);DF_OF_COUT_FOR_VE(LI);DF_OF_COUT_FOR_VE(set);DF_OF_COUT_FOR_VE(unordered_set);DF_OF_COUT_FOR_VE(multiset);IN VO VariadicResize(CRI SZ){}TE <TY Arg,TY... ARGS> IN VO VariadicResize(CRI SZ,Arg& arg,ARGS&... args){arg.resize(SZ);VariadicResize(SZ,args...);}

#define DF_OF_AR_FOR_VE(V,OPR)TE <TY T> IN V<T>& OP OPR ## =(V<T>& a0,CO V<T>& a1){AS(a0.SZ()<= a1.SZ());auto IT0 = a0.BE(),EN0 = a0.EN();auto IT1 = a1.BE();WH(IT0 != EN0){*(IT0++)OPR ## = *(IT1++);}RE a0;}TE <TY T> IN V<T>& OP OPR ## =(V<T>& a,CO T& t){for(auto& x:a){x OPR## = t;}RE a;}TE <TY T,TY U> IN V<T> OP OPR(V<T> a,CO U& u){RE MO(a OPR ## = u);}
#define DF_OF_INCREMENT_FOR_VE(V,INCR)TE <TY T> IN V<T>& OP INCR(V<T>& a){for(auto& i:a){INCR i;}RE a;}
#define DF_OF_SHIFT_FOR_VE(V)TE <TY T> IN V<T>& OP<<=(V<T>& a,T t){a.push_back(MO(t));RE a;}TE <TY T,TY U> IN V<T>& OP<<=(V<T>& a,U&& u){RE a <<= T{forward<U>(u)};}TE <TY T> IN T pop(V<T>& a){AS(!a.empty());T AN = MO(a.back());a.pop_back();RE AN;}
#define DF_OF_ARS_FOR_VE(V)DF_OF_AR_FOR_VE(V,+);DF_OF_AR_FOR_VE(V,-);DF_OF_AR_FOR_VE(V,*);DF_OF_AR_FOR_VE(V,/);DF_OF_AR_FOR_VE(V,%);DF_OF_INCREMENT_FOR_VE(V,++);DF_OF_INCREMENT_FOR_VE(V,--);TE <TY T> IN V<T> OP-(V<T> a){RE MO(a *= T(-1));}TE <TY T> IN V<T> OP*(CO T& t,V<T> v){RE MO(v *= t);}DF_OF_SHIFT_FOR_VE(V);
DF_OF_ARS_FOR_VE(VE);DF_OF_ARS_FOR_VE(LI);DF_OF_SHIFT_FOR_VE(basic_string);
TE <TY V> IN auto Get(V& a){RE[&](CRI i = 0)-> CO decldecay_t(a[0])&{RE a[i];};}TE <TY T> IN VE<T> id(CRI SZ){VE<T> AN(SZ);for(int i = 0;i < SZ;i++){AN[i]= i;}RE AN;}TE <TY V> IN VO Sort(V& a,CO bool& reversed = false){US T = decldecay_t(a[0]);if(reversed){ST auto comp =[](CO T& t0,CO T& t1){RE t1 < t0;};sort(a.BE(),a.EN(),comp);}else{sort(a.BE(),a.EN());}}TE <TY V0,TY V1> IN VO Sort(V0& a,V1& b,CO bool& reversed = false){CO int SZ = a.SZ();AS(SZ == int(b.SZ()));VE<pair<decldecay_t(a[0]),decldecay_t(b[0])>> v(SZ);for(int i = 0;i < SZ;i++){v[i]={MO(a[i]),MO(b[i])};}Sort(v,reversed);for(int i = 0;i < SZ;i++){a[i]= MO(v[i].first);b[i]= MO(v[i].second);}}TE <TY V> IN pair<VE<int>,VE<int>> IndexSort(CO V& a,CO bool& reversed = false){CO int SZ = a.SZ();auto index = id<int>(SZ),ord = index;sort(index.BE(),index.EN(),[&](CRI i,CRI j){CO pair ti{a[i],i},tj{a[j],j};RE reversed?tj < ti:ti < tj;});for(int i = 0;i < SZ;i++){ord[index[i]]= i;}RE{MO(index),MO(ord)};}TE <TY V> IN int len(CO V& a){RE a.SZ();}TE <TY V> IN VO Reverse(V& a){CO int SZ = len(a),half = SZ / 2;for(int i = 0;i < half;i++){swap(a[i],a[SZ-1-i]);}};TE <TY V> IN V Reversed(V a){Reverse(a);RE MO(a);}TE <TY RET,TE <TY...> TY V,TY T> IN V<RET> cast(CO V<T>& a){V<RET> AN{};for(auto& x:a){AN <<= a;}}

/* Map (1KB)*/
#define DF_OF_AR_FOR_MAP(MAP,OPR)TE <TY T,TY U> IN MAP<T,U>& OP OPR ## =(MAP<T,U>& a,CO pair<T,U>& v){a[v.first]OPR ## = v.second;RE a;}TE <TY T,TY U> IN MAP<T,U>& OP OPR ## =(MAP<T,U>& a0,CO MAP<T,U>& a1){for(auto&[t,u]:a1){a0[t]OPR ## = u;}RE a0;}TE <TY T,TY U,TY ARG> IN MAP<T,U> OP OPR(MAP<T,U> a,CO ARG& arg){RE MO(a OPR ## = arg);}
#define DF_OF_ARS_FOR_MAP(MAP)DF_OF_AR_FOR_MAP(MAP,+);DF_OF_AR_FOR_MAP(MAP,-);DF_OF_AR_FOR_MAP(MAP,*);DF_OF_AR_FOR_MAP(MAP,/);DF_OF_AR_FOR_MAP(MAP,%);
TE <TY T,TY U>US Map = conditional_t<is_COructible_v<unordered_map<T,int>>,unordered_map<T,U>,conditional_t<is_ordered::value<T>,map<T,U>,VO>>;
DF_OF_ARS_FOR_MAP(map);DF_OF_ARS_FOR_MAP(unordered_map);

/* StdStream (2KB)*/
TE <CL Traits> IN IS& VariadicCin(IS& is){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicCin(IS& is,Arg& arg,ARGS&... args){RE VariadicCin(is >> arg,args...);}TE <CL Traits> IN IS& VariadicSet(IS& is,CRI i){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicSet(IS& is,CRI i,Arg& arg,ARGS&... args){RE VariadicSet(is >> arg[i],i,args...);}TE <CL Traits> IN IS& VariadicGetline(IS& is,CO char& separator){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicGetline(IS& is,CO char& separator,Arg& arg,ARGS&... args){RE VariadicGetline(getline(is,arg,separator),separator,args...);}TE <CL Traits,TY Arg> IN OS& VariadicCout(OS& os,Arg&& arg){RE os << forward<Arg>(arg);}TE <CL Traits,TY Arg1,TY Arg2,TY... ARGS> IN OS& VariadicCout(OS& os,Arg1&& arg1,Arg2&& arg2,ARGS&&... args){RE VariadicCout(os << forward<Arg1>(arg1)<< " ",forward<Arg2>(arg2),forward<ARGS>(args)...);}TE <CL Traits,TY Arg> IN OS& VariadicCoutNonSep(OS& os,Arg&& arg){RE os << forward<Arg>(arg);}TE <CL Traits,TY Arg1,TY Arg2,TY... ARGS> IN OS& VariadicCoutNonSep(OS& os,Arg1&& arg1,Arg2&& arg2,ARGS&&... args){RE VariadicCoutNonSep(os << forward<Arg1>(arg1),forward<Arg2>(arg2),forward<ARGS>(args)...);}TE <CL Traits,TY ARRAY> IN OS& CoutArray(OS& os,CRI i_start,CRI i_ulim,ARRAY&& a){for(int i = i_start;i < i_ulim;i++){(i == i_start?os:(os << " "))<< a[i];}RE os;}

/* Module (6KB)*/
#define DC_OF_CPOINT(POINT)IN CO U& POINT()CO NE
#define DC_OF_POINT(POINT)IN U& POINT()NE
#define DF_OF_CPOINT(POINT)TE <TY U> IN CO U& VirtualPointedSet<U>::POINT()CO NE{RE Point();}
#define DF_OF_POINT(POINT)TE <TY U> IN U& VirtualPointedSet<U>::POINT()NE{RE Point();}
TE <TY U>CL UnderlyingSet{PU:US type = U;};TE <TY U>CL VirtualPointedSet:VI PU UnderlyingSet<U>{PU:VI CO U& Point()CO NE = 0;VI U& Point()NE = 0;DC_OF_CPOINT(Unit);DC_OF_CPOINT(Zero);DC_OF_CPOINT(One);DC_OF_CPOINT(Infty);DC_OF_POINT(init);DC_OF_POINT(root);};TE <TY U>CL PointedSet:VI PU VirtualPointedSet<U>{PU:U m_b_U;IN PointedSet(U b_u = U());IN CO U& Point()CO NE;IN U& Point()NE;};TE <TY U>CL VirtualNSet:VI PU UnderlyingSet<U>{PU:VI U Transfer(CO U& u)= 0;IN U Inverse(CO U& u);};TE <TY U,TY F_U>CL AbstractNSet:VI PU VirtualNSet<U>{PU:F_U m_f_U;IN AbstractNSet(F_U f_U);IN AbstractNSet<U,F_U>& OP=(CO AbstractNSet&)NE;IN U Transfer(CO U& u);};TE <TY U>CL VirtualMagma:VI PU UnderlyingSet<U>{PU:VI U Product(U u0,CO U& u1)= 0;IN U Sum(U u0,CO U& u1);};TE <TY U = ll>CL AdditiveMagma:VI PU VirtualMagma<U>{PU:IN U Product(U u0,CO U& u1);};TE <TY U = ll>CL MultiplicativeMagma:VI PU VirtualMagma<U>{PU:IN U Product(U u0,CO U& u1);};TE <TY U,TY M_U>CL AbstractMagma:VI PU VirtualMagma<U>{PU:M_U m_m_U;IN AbstractMagma(M_U m_U);IN AbstractMagma<U,M_U>& OP=(CO AbstractMagma<U,M_U>&)NE;IN U Product(U u0,CO U& u1);};
TE <TY U> IN PointedSet<U>::PointedSet(U b_U):m_b_U(MO(b_U)){}TE <TY U> IN CO U& PointedSet<U>::Point()CO NE{RE m_b_U;}TE <TY U> IN U& PointedSet<U>::Point()NE{RE m_b_U;}DF_OF_CPOINT(Unit);DF_OF_CPOINT(Zero);DF_OF_CPOINT(One);DF_OF_CPOINT(Infty);DF_OF_POINT(init);DF_OF_POINT(root);TE <TY U,TY F_U> IN AbstractNSet<U,F_U>::AbstractNSet(F_U f_U):m_f_U(MO(f_U)){ST_AS(is_invocable_r_v<U,F_U,U>);}TE <TY U,TY F_U> IN AbstractNSet<U,F_U>& AbstractNSet<U,F_U>::operator=(CO AbstractNSet<U,F_U>&)NE{RE *TH;}TE <TY U,TY F_U> IN U AbstractNSet<U,F_U>::Transfer(CO U& u){RE m_f_U(u);}TE <TY U> IN U VirtualNSet<U>::Inverse(CO U& u){RE Transfer(u);}TE <TY U,TY M_U> IN AbstractMagma<U,M_U>::AbstractMagma(M_U m_U):m_m_U(MO(m_U)){ST_AS(is_invocable_r_v<U,M_U,U,U>);}TE <TY U,TY M_U> IN AbstractMagma<U,M_U>& AbstractMagma<U,M_U>::OP=(CO AbstractMagma<U,M_U>&)NE{RE *TH;}TE <TY U> IN U AdditiveMagma<U>::Product(U u0,CO U& u1){RE MO(u0 += u1);}TE <TY U> IN U MultiplicativeMagma<U>::Product(U u0,CO U& u1){RE MO(u0 *= u1);}TE <TY U,TY M_U> IN U AbstractMagma<U,M_U>::Product(U u0,CO U& u1){RE m_m_U(MO(u0),u1);}TE <TY U> IN U VirtualMagma<U>::Sum(U u0,CO U& u1){RE Product(MO(u0),u1);}

TE <TY U>CL VirtualMonoid:VI PU VirtualMagma<U>,VI PU VirtualPointedSet<U>{};TE <TY U = ll>CL AdditiveMonoid:VI PU VirtualMonoid<U>,PU AdditiveMagma<U>,PU PointedSet<U>{};TE <TY U = ll>CL MultiplicativeMonoid:VI PU VirtualMonoid<U>,PU MultiplicativeMagma<U>,PU PointedSet<U>{PU:IN MultiplicativeMonoid(U e_U);};TE <TY U,TY M_U>CL AbstractMonoid:VI PU VirtualMonoid<U>,PU AbstractMagma<U,M_U>,PU PointedSet<U>{PU:IN AbstractMonoid(M_U m_U,U e_U);};
TE <TY U> IN MultiplicativeMonoid<U>::MultiplicativeMonoid(U e_U):PointedSet<U>(MO(e_U)){}TE <TY U,TY M_U> IN AbstractMonoid<U,M_U>::AbstractMonoid(M_U m_U,U e_U):AbstractMagma<U,M_U>(MO(m_U)),PointedSet<U>(MO(e_U)){}

TE <TY U>CL VirtualGroup:VI PU VirtualMonoid<U>,VI PU VirtualPointedSet<U>,VI PU VirtualNSet<U>{};TE <TY U = ll>CL AdditiveGroup:VI PU VirtualGroup<U>,PU AdditiveMonoid<U>{PU:IN U Transfer(CO U& u);};TE <TY U,TY M_U,TY I_U>CL AbstractGroup:VI PU VirtualGroup<U>,PU AbstractMonoid<U,M_U>,PU AbstractNSet<U,I_U>{PU:IN AbstractGroup(M_U m_U,U e_U,I_U i_U);};
TE <TY U,TY M_U,TY I_U> IN AbstractGroup<U,M_U,I_U>::AbstractGroup(M_U m_U,U e_U,I_U i_U):AbstractMonoid<U,M_U>(MO(m_U),MO(e_U)),AbstractNSet<U,I_U>(MO(i_U)){}TE <TY U> IN U AdditiveGroup<U>::Transfer(CO U& u){RE -u;}

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

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

/* Grid (3KB)*/
#define SET_GRID H_minus = H - 1;W_minus = W - 1;HW = ll(H)* W
#define SET_HW(h,w)H = h;W = w;SET_GRID
#define CIN_HW SET(H,W);SET_GRID
TE <TY E>CL GridGraph:PU EnumerationGraph<T2<int>,T2<int>(&)(CRI),int(&)(CO T2<int>&),E>{PU:IN GridGraph(E e);};int H,W,H_minus,W_minus;ll HW;VE<string> grid;char walkable = '.';CO string direction="URDL";bool grid_edge_i_plus = true;bool grid_edge_j_plus = true;bool grid_edge_i_minus = true;bool grid_edge_j_minus = true;
IN T2<int> EnumHW(CRI v){RE{v / W,v % W};}IN int EnumHW_inv(CO T2<int>& ij){auto&[i,j]= ij;RE i * W + j;}TE <TY E> IN GridGraph<E>::GridGraph(E e):EnumerationGraph<T2<int>,T2<int>(&)(CRI),int(&)(CO T2<int>&),E>(HW,EnumHW,EnumHW_inv,MO(e)){AS(HW >> 31 == 0 && H * W == HW);}VE<T2<int>> EdgeOnGrid(CO T2<int>& v){VE<T2<int>> AN{};auto&[i,j]= v;if(grid[i][j]== walkable){if(grid_edge_i_minus && i > 0 && grid[i-1][j]== walkable){AN.push_back({i-1,j});}if(grid_edge_i_plus && i+1 < H && grid[i+1][j]== walkable){AN.push_back({i+1,j});}if(grid_edge_j_minus && j > 0 && grid[i][j-1]== walkable){AN.push_back({i,j-1});}if(grid_edge_j_plus && j+1 < W && grid[i][j+1]== walkable){AN.push_back({i,j+1});}}RE AN;}VE<Pair<T2<int>,ll>> WEdgeOnGrid(CO T2<int>& v){VE<Pair<T2<int>,ll>> AN{};auto&[i,j]= v;if(grid[i][j]== walkable){if(grid_edge_i_minus && i>0 && grid[i-1][j]== walkable){AN.push_back({{i-1,j},1});}if(grid_edge_i_plus && i+1 < H && grid[i+1][j]== walkable){AN.push_back({{i+1,j},1});}if(grid_edge_j_minus && j>0 && grid[i][j-1]== walkable){AN.push_back({{i,j-1},1});}if(grid_edge_i_plus && j+1 < W && grid[i][j+1]== walkable){AN.push_back({{i,j+1},1});}}RE AN;}IN VO SetWallStringOnGrid(){grid.resize(H);for(int i = 0;i < H;i++){SET(grid[i]);AS(int(grid[i].SZ())== W);}}IN int DirectionNumberOnGrid(CRI i,CRI j,CRI k,CRI h,CO bool& xy_axis = false){RE xy_axis?i < k?1:i > k?3:j < h?0:j > h?2:-1:i < k?2:i > k?0:j < h?1:j > h?3:-1;}IN int DirectionNumberOnGrid(CO T2<int>& v,CO T2<int>& w,CO bool& xy_axis = false){auto&[i,j]= v;auto&[k,h]= w;RE DirectionNumberOnGrid(i,j,k,h);}IN int DirectionNumberOnGrid(CRI v,CRI w,CO bool& xy_axis = false){RE DirectionNumberOnGrid(EnumHW(v),EnumHW(w));}IN int DirectionNumberOnGrid(CO char& c){RE c == 'U'?0:c == 'R'?1:c == 'D'?2:c == 'L'?3:-1;}IN int ReverseDirectionNumberOnGrid(CRI n){AS(0 <= n && n < 4);RE n ^ 2;}IN T2<int> DirectionVEOnGrid(CO char& c,CO bool& xy_axis = false){CO int n = DirectionNumberOnGrid(c);AS(n != -1);RE T2<int>{xy_axis?n == 1?1:n == 3?-1:0:n == 0?-1:n == 2?1:0,xy_axis?n == 0?1:n == 2?-1:0:n == 1?1:n == 3?-1:0};}

/* DynamicModulo (9KB)*/
CEXPR(uint,P,998244353);
#define RP Represent
#define DeRP Derepresent

TE <TY INT1,TY INT2> CE INT1 Residue(INT1 n,CO INT2& M)NE{RE MO(n < 0?((((++n)*= -1)%= M)*= -1)+= M - 1:n < M?n:n %= M);}
TE <TY INT> IN INT ModularInverse(CO INT& base,ll c){AS(base > 0);ll a[2]={0,1 % base};INT b[2]={base,INT((c %= base)< 0?c += base:c)};WH(b[1]!= 0){CO INT q = b[0]/ b[1];(a[0]-= q * a[1]% base)< 0?a[0]+= base:a[0];b[0]-= q * b[1];swap(a[0],a[1]);swap(b[0],b[1]);}AS(b[0]== 1 &&(a[0]* c - 1)% base == 0);RE a[0];}

TE <int NUM> CL DMods;TE <int NUM>CL COantsForDMods{PU:COantsForDMods()= delete;ST uint g_M;ST CE CO uint g_memory_bound = 2e6;ST uint g_memory_le;ST uint g_M_minus;ST bool g_M_is_prime;};
TE <int NUM> uint COantsForDMods<NUM>::g_M = P;TE <int NUM> uint COantsForDMods<NUM>::g_memory_le = g_memory_bound;TE <int NUM> uint COantsForDMods<NUM>::g_M_minus = P-1;TE <int NUM> bool COantsForDMods<NUM>::g_M_is_prime = true;

#define SFINAE_FOR_DMOD enable_if_t<is_COructible_v<uint,decay_t<T>>>*
#define DC_OF_CM_FOR_DMOD(OPR)IN bool OP OPR(CO DMods<NUM>& n)CO NE
#define DC_OF_AR_FOR_DMOD(OPR,EX)IN DMods<NUM> OP OPR(DMods<NUM> n)CO EX;
#define DF_OF_CM_FOR_DMOD(OPR)TE <int NUM> IN bool DMods<NUM>::OP OPR(CO DMods<NUM>& n)CO NE{RE m_n OPR n.m_n;}
#define DF_OF_AR_FOR_DMOD(OPR,EX,LEFT,OPR2)TE <int NUM> IN DMods<NUM> DMods<NUM>::OP OPR(DMods<NUM> n)CO EX{RE MO(LEFT OPR2 ## = *TH);}TE <int NUM,TY T,SFINAE_FOR_DMOD = nullptr> IN DMods<NUM> OP OPR(T n0,CO DMods<NUM>& n1)EX{RE MO(DMods<NUM>(MO(n0))OPR ## = n1);}
TE <int NUM>CL DMods{PU:uint m_n;IN DMods()NE;IN DMods(CO DMods<NUM>& n)NE;IN DMods(DMods<NUM>&& n)NE;TE <TY T,SFINAE_FOR_DMOD = nullptr> IN DMods(T n)NE;IN DMods<NUM>& OP=(DMods<NUM> n)NE;IN DMods<NUM>& OP+=(CO DMods<NUM>& n)NE;IN DMods<NUM>& OP-=(CO DMods<NUM>& n)NE;IN DMods<NUM>& OP*=(CO DMods<NUM>& n)NE;IN DMods<NUM>& OP/=(DMods<NUM> n);IN DMods<NUM>& OP^=(ll EX);IN DMods<NUM>& OP<<=(ll n);IN DMods<NUM>& OP>>=(ll n);IN DMods<NUM>& OP++()NE;IN DMods<NUM> OP++(int)NE;IN DMods<NUM>& OP--()NE;IN DMods<NUM> OP--(int)NE;DC_OF_CM_FOR_DMOD(==);DC_OF_CM_FOR_DMOD(!=);DC_OF_CM_FOR_DMOD(<);DC_OF_CM_FOR_DMOD(<=);DC_OF_CM_FOR_DMOD(>);DC_OF_CM_FOR_DMOD(>=);DC_OF_AR_FOR_DMOD(+,NE);DC_OF_AR_FOR_DMOD(-,NE);DC_OF_AR_FOR_DMOD(*,NE);DC_OF_AR_FOR_DMOD(/,);IN DMods<NUM> OP^(ll EX)CO;IN DMods<NUM> OP<<(ll n)CO;IN DMods<NUM> OP>>(ll n)CO;IN DMods<NUM> OP-()CO NE;IN VO swap(DMods<NUM>& n)NE;IN CRUI RP()CO NE;ST IN DMods<NUM> DeRP(uint n)NE;ST IN CO DMods<NUM>& Factorial(CRL n);ST IN CO DMods<NUM>& FactorialInverse(CRL n);ST IN DMods<NUM> Combination(CRL n,CRL i);ST IN CO DMods<NUM>& zero()NE;ST IN CO DMods<NUM>& one()NE;ST IN CRUI GetModulo()NE;ST IN VO SetModulo(CRUI M,CO bool& M_is_prime = false)NE;IN DMods<NUM>& SignInvert()NE;IN DMods<NUM>& Invert();IN DMods<NUM>& PPW(ll EX)NE;IN DMods<NUM>& NNPW(ll EX)NE;ST IN CO DMods<NUM>& Inverse(CRI n);ST IN CO DMods<NUM>& TwoPower(CRI n);ST IN CO DMods<NUM>& TwoPowerInverse(CRI n);US COants = COantsForDMods<NUM>;};
US DMod = DMods<0>;
TE <int NUM> IN DMods<NUM>::DMods()NE:m_n(){}TE <int NUM> IN DMods<NUM>::DMods(CO DMods<NUM>& n)NE:m_n(n.m_n){}TE <int NUM> IN DMods<NUM>::DMods(DMods<NUM>&& n)NE:m_n(MO(n.m_n)){}TE <int NUM> TE <TY T,SFINAE_FOR_DMOD> IN DMods<NUM>::DMods(T n)NE:m_n(Residue(MO(n),COants::g_M)){}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP=(DMods<NUM> n)NE{m_n = MO(n.m_n);RE *TH;}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP+=(CO DMods<NUM>& n)NE{(m_n += n.m_n)< COants::g_M?m_n:m_n -= COants::g_M;RE *TH;}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP-=(CO DMods<NUM>& n)NE{m_n < n.m_n?(m_n += COants::g_M)-= n.m_n:m_n -= n.m_n;RE *TH;}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP*=(CO DMods<NUM>& n)NE{m_n = Residue(MO(ull(m_n)* n.m_n),COants::g_M);RE *TH;}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP/=(DMods<NUM> n){RE OP*=(n.Invert());}TE <int NUM> IN DMods<NUM>& DMods<NUM>::PPW(ll EX)NE{DMods<NUM> pw{*TH};EX--;WH(EX != 0){(EX & 1)== 1?*TH *= pw:*TH;EX >>= 1;pw *= pw;}RE *TH;}TE <int NUM> IN DMods<NUM>& DMods<NUM>::NNPW(ll EX)NE{RE EX == 0?(m_n = 1,*TH):PPW(MO(EX));}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP^=(ll EX){if(EX < 0){m_n = ModularInverse(COants::g_M,MO(m_n));EX *= -1;}RE NNPW(MO(EX));}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP<<=(ll n){RE *TH *=(n < 0 && -n < int(COants::g_memory_le))?TwoPowerInverse(- int(n)):(n >= 0 && n < int(COants::g_memory_le))?TwoPower(int(n)):DMods<NUM>(2)^= MO(n);}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP>>=(ll n){RE *TH <<= MO(n *= -1);}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP++()NE{m_n < COants::g_M_minus?++m_n:m_n = 0;RE *TH;}TE <int NUM> IN DMods<NUM> DMods<NUM>::OP++(int)NE{DMods<NUM> n{*TH};OP++();RE n;}TE <int NUM> IN DMods<NUM>& DMods<NUM>::OP--()NE{m_n == 0?m_n = COants::g_M_minus:--m_n;RE *TH;}TE <int NUM> IN DMods<NUM> DMods<NUM>::OP--(int)NE{DMods<NUM> n{*TH};OP--();RE n;}DF_OF_CM_FOR_DMOD(==);DF_OF_CM_FOR_DMOD(!=);DF_OF_CM_FOR_DMOD(>);DF_OF_CM_FOR_DMOD(>=);DF_OF_CM_FOR_DMOD(<);DF_OF_CM_FOR_DMOD(<=);DF_OF_AR_FOR_DMOD(+,NE,n,+);DF_OF_AR_FOR_DMOD(-,NE,n.SignInvert(),+);DF_OF_AR_FOR_DMOD(*,NE,n,*);DF_OF_AR_FOR_DMOD(/,,n.Invert(),*);TE <int NUM> IN DMods<NUM> DMods<NUM>::OP^(ll EX)CO{RE MO(DMods<NUM>(*TH)^= MO(EX));}TE <int NUM> IN DMods<NUM> DMods<NUM>::OP<<(ll n)CO{RE MO(DMods<NUM>(*TH)<<= MO(n));}TE <int NUM> IN DMods<NUM> DMods<NUM>::OP>>(ll n)CO{RE MO(DMods<NUM>(*TH)>>= MO(n));}TE <int NUM> IN DMods<NUM> DMods<NUM>::OP-()CO NE{RE MO(DMods<NUM>(*TH).SignInvert());}TE <int NUM> IN DMods<NUM>& DMods<NUM>::SignInvert()NE{m_n > 0?m_n = COants::g_M - m_n:m_n;RE *TH;}TE <int NUM> IN DMods<NUM>& DMods<NUM>::Invert(){m_n = COants::g_M_is_prime && m_n < COants::g_memory_le?Inverse(int(m_n)).m_n:ModularInverse(COants::g_M,MO(m_n));RE *TH;}TE <int NUM> IN VO DMods<NUM>::swap(DMods<NUM>& n)NE{std::swap(m_n,n.m_n);}TE <int NUM> IN CO DMods<NUM>& DMods<NUM>::Inverse(CRI n){if(COants::g_M == 1){RE zero();}AS(COants::g_M_is_prime && 0 < n && n < int(COants::g_memory_le));ST VE<DMods<NUM>> memory ={zero(),one()};ST int le_curr = 2;WH(le_curr <= n){memory.push_back(DeRP(COants::g_M - memory[COants::g_M % le_curr].m_n * ull(COants::g_M / le_curr)% COants::g_M));le_curr++;}RE memory[n];}TE <int NUM> IN CO DMods<NUM>& DMods<NUM>::TwoPower(CRI n){if(COants::g_M == 1){RE zero();}AS(0 <= n && n < int(COants::g_memory_le));ST VE<DMods<NUM>> memory ={one()};ST int le_curr = 1;WH(le_curr <= n){memory.push_back(memory.back()+ memory.back());le_curr++;}RE memory[n];}TE <int NUM> IN CO DMods<NUM>& DMods<NUM>::TwoPowerInverse(CRI n){if(COants::g_M == 1){RE zero();}AS(0 <= n && n < int(COants::g_memory_le));ST VE<DMods<NUM>> memory ={one()};ST int le_curr = 1;WH(le_curr <= n){auto& m = memory.back().m_n;memory.push_back(DeRP(((m & 1)== 0?m:m + COants::g_M)>> 1));le_curr++;}RE memory[n];}TE <int NUM> IN CO DMods<NUM>& DMods<NUM>::Factorial(CRL n){AS(0 <= n);if(ll(COants::g_M)<= n){RE zero();}ST VE<DMods<NUM>> memory ={one(),one()};ST int le_curr = 2;WH(le_curr <= n && memory.back().m_n != 0){memory.push_back(memory.back()* DeRP(le_curr));le_curr++;}RE le_curr <= n?memory.back():memory[n];}TE <int NUM> IN CO DMods<NUM>& DMods<NUM>::FactorialInverse(CRL n){AS(0 <= n && n < COants::g_M);ST VE<DMods<NUM>> memory ={one(),one()};ST int le_curr = 2;WH(le_curr <= n){memory.push_back(memory[le_curr - 1]* Inverse(le_curr));le_curr++;}RE memory[n];}TE <int NUM> IN DMods<NUM> DMods<NUM>::Combination(CRL n,CRL i){RE 0 <= i && i <= n?Factorial(n)* FactorialInverse(i)* FactorialInverse(n - i):zero();}TE <int NUM> IN CRUI DMods<NUM>::RP()CO NE{RE m_n;}TE <int NUM> IN DMods<NUM> DMods<NUM>::DeRP(uint n)NE{DMods<NUM> n_copy{};n_copy.m_n = MO(n);RE n_copy;}TE <int NUM> IN CO DMods<NUM>& DMods<NUM>::zero()NE{ST CO DMods<NUM> z{};RE z;}TE <int NUM> IN CO DMods<NUM>& DMods<NUM>::one()NE{ST CO DMods<NUM> o{1};RE o;}TE <int NUM> IN CRUI DMods<NUM>::GetModulo()NE{RE COants::g_M;}TE <int NUM> IN VO DMods<NUM>::SetModulo(CRUI M,CO bool& M_is_prime)NE{COants::g_M = M;COants::g_memory_le = M < COants::g_memory_bound?M:COants::g_memory_bound;COants::g_M_minus = M - 1;COants::g_M_is_prime = M_is_prime;}TE <int NUM> IN DMods<NUM> Inverse(CO DMods<NUM>& n){RE MO(DMods<NUM>(n).Invert());}TE <int NUM> IN DMods<NUM> Power(DMods<NUM> n,ll EX){RE MO(n ^= MO(EX));}TE <int NUM> IN VO swap(DMods<NUM>& n0,DMods<NUM>& n1)NE{n0.swap(n1);}TE <int NUM,CL Traits> IN IS& OP>>(IS& is,DMods<NUM>& n){ll m;is >> m;n = m;RE is;}TE <int NUM,CL Traits> IN OS& OP<<(OS& os,CO DMods<NUM>& n){RE os << n.RP();}

IN int GCD(){RE 0;}TE <TY INT1,TY...INT2>INT1 GCD(INT1 b_0,INT2... args){b_0 < 0?b_0 = -b_0:b_0;INT1 b_1 = GCD(MO(args)...);WH(b_1 != 0){swap(b_0 %= b_1,b_1);}RE MO(b_0);}TE <TY INT,TE <TY...> TY V>INT GCD(V<INT> a){INT AN = 0;for(auto& b:a){AN = GCD(MO(AN),MO(b));}RE AN;}

#define DF_OF_HASH_FOR_MOD(MOD)IN size_t hash<MOD>::OP()(CO MOD& n)CO{ST CO hash<decldecay_t(n.RP())> h;RE h(n.RP());}
TE <int NUM> DC_OF_HASH(DMods<NUM>);TE <int NUM> DF_OF_HASH_FOR_MOD( DMods<NUM> );
US MP = DMod;

/* Iteration (3KB) */
#define SPECIALSATION_OF_AR_PROGRESSION_SUM(TYPE)TE <> IN TYPE ArithmeticProgressionSum(CO TYPE& l,CO TYPE& r,CO TYPE& d){RE SpecialisedArithmeticProgressionSum(l,r,d);}
 TE <TY T,TY U,TE <TY...> TY V,TY OPR> T LeftConnectiveProd(T t,CO V<U>& f,OPR opr){for(auto& u:f){t = opr(MO(t),u);}RE MO(t);}TE <TY T,TY U,TE <TY...> TY V> IN T Sum(CO V<U>& f){RE LeftConnectiveProd(T{0},f,[](T t0,CO U& u1){RE MO(t0 += u1);});}TE <TY T,TY U,TE <TY...> TY V> IN T Prod(CO V<U>& f){RE LeftConnectiveProd(T{1},f,[](T t0,CO U& u1){RE MO(t0 *= u1);});}TE <TY T> IN T& SetMax(T& t){RE t;}TE <TY T,TY U,TY... Args> IN T& SetMax(T& t0,CO U& u1,CO Args&... args){RE SetMax(t0 < u1?t0 = u1:t0,args...);}TE <TY T> IN T& SetMin(T& t){RE t;}TE <TY T,TY U,TY... Args> IN T& SetMin(T& t0,CO U& u1,CO Args&... args){RE SetMin(u1 < t0?t0 = u1:t0,args...);}TE <TY T> IN CO T& Max(CO VE<T>& f){RE *max_element(f.BE(),f.EN());}TE <TY T,TE <TY...> TY SET> IN CO T& Max(CO SET<T>& f){RE *--f.EN();}TE <TY T,TY U,TY...Args> IN T Max(T t0,CO U& t1,CO Args&... args){RE MO(SetMax(t0,t1,args...));}TE <TY T> IN CO T& Min(CO VE<T>& f){RE *min_element(f.BE(),f.EN());}TE <TY T,TE <TY...> TY SET> IN CO T& Min(CO SET<T>& f){RE *f.BE();}TE <TY T,TY U,TY...Args> IN T Min(T t0,CO U& t1,CO Args&... args){RE MO(SetMin(t0,t1,args...));}TE <TY T,TY UINT>T Power(CO T& t,CO UINT& EX,T init = 1){RE EX > 1?Power(t * t,EX >> 1,MO(EX & 1?init *= t:init)):MO(EX > 0?init *= t:(AS(EX == 0),init));}TE <TY T> IN T PowerMemorisation(CO T& t,CRI EX){AS(EX >= 0);ST Map<T,VE<T>> memory{};auto& AN = memory[t];if(AN.empty()){AN.push_back(1);}WH(int(AN.SZ())<= EX){AN.push_back(AN.back()* t);}RE AN[EX];}TE <TY INT> IN INT ArithmeticProgressionSum(CO INT& l,CO INT& r,CO INT& d = 1){RE(l + r)*((r - l)/ d + 1)/ 2;}TE <TY INT> IN INT SpecialisedArithmeticProgressionSum(CO INT& l,CO INT& r,CO INT& d){AS(l - 1 <= r);CO INT c =(r - l)/ d;RE l - 1 == r?0:(c & 1)== 0?(c + 1)*(l + d *(c >> 1)):((c + 1)>> 1)*((l << 1)+ d * c);}
SPECIALSATION_OF_AR_PROGRESSION_SUM(int);
SPECIALSATION_OF_AR_PROGRESSION_SUM(uint);
SPECIALSATION_OF_AR_PROGRESSION_SUM(ll);
SPECIALSATION_OF_AR_PROGRESSION_SUM(ull);
TE <TY INT> IN INT ArithmeticProgressionSum(CO INT& r){RE ArithmeticProgressionSum(INT{},r);}TE <TY INT> IN INT SquareSum(CO INT& r){RE r *(r + 1)*(2 * r + 1)/ 6;}TE <TY T,TY UINT> IN T GeometricProgressionSum(T rate,UINT EX_max,CO T& init = 1){T rate_minus = rate - 1;RE rate_minus == 0?init * ++EX_max:(Power(MO(rate),MO(++EX_max))- 1)/ MO(rate_minus)* init;}TE <TY T,TY UINT>T GeometricProgressionLinearCombinationSum(VE<T> rate,VE<UINT> EX_max,CO VE<T>& init){CO int SZ = init.SZ();AS(int(rate.SZ())== SZ && int(EX_max.SZ())== SZ);T AN{};for(int i = 0;i < SZ;i++){AN += GeometricProgressionSum(MO(rate[i]),MO(EX_max[i]),init[i]);}RE AN;}

/* Sqrt (1KB) */
TE <TY INT>INT RoundDownSqrt(CO INT& n){ST_AS(is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull>);AS(n >= 0);if(n <= 1){RE n;}CE INT r_max = is_same_v<INT,int>?46341:is_same_v<INT,uint>?65536:is_same_v<INT,ll>?3037000500:4294967296;INT l = 1,r = min(r_max,n);WH(l < r - 1){CO INT m =(l + r)>> 1;(m <= n / m?l:r)= m;}RE l;}TE <TY INT>INT RoundUpSqrt(CO INT& n){ST_AS(is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull>);AS(n >= 0);if(n <= 2){RE n;}CE INT r_max = is_same_v<INT,int>?46341:is_same_v<INT,uint>?65536:is_same_v<INT,ll>?3037000500:4294967296;CO INT n_minus = n - 1;INT l = 1,r = min(r_max,n);WH(l + 1 < r){CO INT m =(l + r)>> 1;(m <= n_minus / m?l:r)= m;}RE r;}TE <TY INT> bool IsSquare(CO INT& n){CO INT r = RoundDownSqrt(n);RE n == r * r;}

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

/* string (1KB)*/
TE <TY INT> IN char IntToChar(CO INT& i,CO char& c = 'a'){RE c + i;}TE <TY INT = int> IN INT CharToInt(CO char& i){RE i -(i < 'a'?'A':'a');}TE <TY INT>string ArrayToString(CO VE<INT>& A,CO char& c = 'a'){CO int N = A.SZ();string S(N,c);for(int i = 0;i < N;i++){S[i]= IntToChar<INT>(A[i],c);}RE S;}TE <TY INT = int>VE<INT> StringToArray(CO string& S){CO int N = S.SZ();VE<int> A(N);for(int i = 0;i < N;i++){A[i]= CharToInt<INT>(S[i]);}RE A;}TE <TY INT>string ArrayToParenthesisString(CO VE<INT>& A){CO int N = A.SZ();string S(N,'(');for(int i = 0;i < N;i++){AS(0 <= A[i]&& A[i]<= 1);S[i]= "()"[A[i]];}RE S;}TE <TY INT = int>VE<INT> ParenthesisStringToArray(CO string& S){CO int N = S.SZ();VE<int> A(N);for(int i = 0;i < N;i++){A[i]= S[i]- '(';}RE A;}
#endif
/* AAA 常設ライブラリは以上に挿入する。*/

#define INCLUDE_LIBRARY
#include __FILE__
#endif /* INCLUDE_LIBRARY */
#endif /* INCLUDE_SUB */
#endif /* INCLUDE_MAIN */