#ifdef DEBUG
  #define _GLIBCXX_DEBUG
  #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr ); signal( SIGABRT , &AlertAbort )
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , DEBUG_VALUE )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define COUT( ... ) VariadicCout( cout << "出力： " , __VA_ARGS__ ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << endl
  #define COUT_A( A , N ) cout << "出力： "; OUTPUT_ARRAY( cout , A , N ) << endl
  #define CERR_ITR( A ) OUTPUT_ITR( cerr , A ) << endl
  #define COUT_ITR( A ) cout << "出力： "; OUTPUT_ITR( cout , A ) << endl
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); assert( ( MIN ) <= A && A <= ( MAX ) )
  #define AUTO_CHECK bool auto_checked = true; AutoCheck( auto_checked ); if( auto_checked ){ return 0; };
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr )
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , VALUE )
  #define CERR( ... ) 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << "\n"
  #define CERR_A( A , N ) 
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout , A , N ) << "\n"
  #define CERR_ITR( A ) 
  #define COUT_ITR( A ) OUTPUT_ITR( cout , A ) << "\n"
  #define ASSERT( A , MIN , MAX ) assert( ( MIN ) <= A && A <= ( MAX ) )
  #define AUTO_CHECK
#endif
#include <bits/stdc++.h>
using namespace std;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
using ld = long double;
using lld = __float128;
template <typename INT> using T2 = pair<INT,INT>;
template <typename INT> using T3 = tuple<INT,INT,INT>;
template <typename INT> using T4 = tuple<INT,INT,INT,INT>;
using path = pair<int,ll>;
// #define RANDOM_TEST
#if defined( DEBUG ) && defined( RANDOM_TEST )
  ll GetRand( const ll& Rand_min , const ll& Rand_max );
 #define SET_ASSERT( A , MIN , MAX ) CERR( #A , " = " , ( A = GetRand( MIN , MAX ) ) )
 #define RETURN( ANSWER ) if( ( ANSWER ) == guchoku ){ CERR( ANSWER , "==" , guchoku ); goto END_MAIN; } else { CERR( ANSWER , "!=" , guchoku ); QUIT; }
#else
  #define SET_ASSERT( A , MIN , MAX ) cin >> A; ASSERT( A , MIN , MAX )
  #define RETURN( ANSWER ) COUT( ANSWER ); QUIT
#endif
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define TYPE_OF( VAR ) decay_t<decltype( VAR )>
#define CEXPR( LL , BOUND , VALUE ) constexpr LL BOUND = VALUE
#define CIN( LL , ... ) LL __VA_ARGS__; VariadicCin( cin , __VA_ARGS__ )
#define CIN_ASSERT( A , MIN , MAX ) TYPE_OF( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A( LL , A , N ) LL A[N]; FOR( VARIABLE_FOR_CIN_A , 0 , N ){ cin >> A[VARIABLE_FOR_CIN_A]; }
#define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
#define GETLINE( ... ) GETLINE_SEPARATE( " " , ... )
#define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( TYPE_OF( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( TYPE_OF( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( TYPE_OF( INITIAL ) VAR = INITIAL ; VAR >= FINAL ; VAR -- )
#define AUTO_ITR( ARRAY ) auto itr_ ## ARRAY = ARRAY .begin() , end_ ## ARRAY = ARRAY .end()
#define FOR_ITR( ARRAY ) for( AUTO_ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT_ ## HOW_MANY_TIMES , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS )
#define OUTPUT_ARRAY( OS , A , N ) FOR( VARIABLE_FOR_OUTPUT_ARRAY , 0 , N ){ OS << A[VARIABLE_FOR_OUTPUT_ARRAY] << (VARIABLE_FOR_OUTPUT_ARRAY==N-1?"":" "); } OS
#define OUTPUT_ITR( OS , A ) { auto ITERATOR_FOR_OUTPUT_ITR = A.begin() , END_FOR_OUTPUT_ITR = A.end(); bool VARIABLE_FOR_OUTPUT_ITR = ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR; while( VARIABLE_FOR_OUTPUT_ITR ){ OS << *ITERATOR_FOR_COUT_ITR; ( VARIABLE_FOR_OUTPUT_ITR = ++ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR ) ? OS : OS << " "; } } OS
#define QUIT goto END_MAIN
#define TEST_CASE_NUM( BOUND ) DEXPR( int , bound_T , BOUND , min( BOUND , 100 ) ); int T = 1; if constexpr( bound_T > 1 ){ SET_ASSERT( T , 1 , bound_T ); }
#define START_MAIN REPEAT( T ){ { if constexpr( bound_T > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_T , ":" ); }
#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 FINISH_MAIN QUIT; } END_MAIN: CERR( "" ); }

// 入出力用関数
template <class Traits> inline basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is ) { return is; }
template <class Traits , typename Arg , typename... ARGS> inline basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is , Arg& arg , ARGS&... args ) { return VariadicCin( is >> arg , args... ); }
template <class Traits> inline basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , const char& separator ) { return is; }
template <class Traits , typename Arg , typename... ARGS> inline basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , const char& separator , Arg& arg , ARGS&... args ) { return VariadicGetline( getline( is , arg , separator ) , separator , args... ); }
template <class Traits , typename Arg> inline basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , const Arg& arg ) { return os << arg; }
template <class Traits , typename Arg1 , typename Arg2 , typename... ARGS> inline basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , const Arg1& arg1 , const Arg2& arg2 , const ARGS&... args ) { return VariadicCout( os << arg1 << " " , arg2 , args... ); }

// 算術用関数
template <typename T> inline T Residue( const T& a , const T& p ){ return a >= 0 ? a % p : p - 1 - ( ( - ( a + 1 ) ) % p ); }
inline ll MIN( const ll& a , const ll& b ){ return min( a , b ); }
inline ull MIN( const ull& a , const ull& b ){ return min( a , b ); }
inline ll MAX( const ll& a , const ll& b ){ return max( a , b ); }
inline ull MAX( const ull& a , const ull& b ){ return max( a , b ); }

#define POWER( ANSWER , ARGUMENT , EXPONENT )				\
  static_assert( ! is_same<TYPE_OF( ARGUMENT ),int>::value && ! is_same<TYPE_OF( ARGUMENT ),uint>::value ); \
  TYPE_OF( ARGUMENT ) ANSWER{ 1 };					\
  {									\
    TYPE_OF( ARGUMENT ) ARGUMENT_FOR_SQUARE_FOR_POWER = ( ARGUMENT );	\
    TYPE_OF( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT );	\
    while( EXPONENT_FOR_SQUARE_FOR_POWER != 0 ){			\
      if( EXPONENT_FOR_SQUARE_FOR_POWER % 2 == 1 ){			\
	ANSWER *= ARGUMENT_FOR_SQUARE_FOR_POWER;			\
      }									\
      ARGUMENT_FOR_SQUARE_FOR_POWER *= ARGUMENT_FOR_SQUARE_FOR_POWER;	\
      EXPONENT_FOR_SQUARE_FOR_POWER /= 2;				\
    }									\
  }									\

#define POWER_MOD( ANSWER , ARGUMENT , EXPONENT , MODULO )		\
  ll ANSWER{ 1 };							\
  {									\
    ll ARGUMENT_FOR_SQUARE_FOR_POWER = ( ( MODULO ) + ( ( ARGUMENT ) % ( MODULO ) ) ) % ( MODULO ); \
    TYPE_OF( EXPONENT ) EXPONENT_FOR_SQUARE_FOR_POWER = ( EXPONENT );	\
    while( EXPONENT_FOR_SQUARE_FOR_POWER != 0 ){			\
      if( EXPONENT_FOR_SQUARE_FOR_POWER % 2 == 1 ){			\
	ANSWER = ( ANSWER * ARGUMENT_FOR_SQUARE_FOR_POWER ) % ( MODULO ); \
      }									\
      ARGUMENT_FOR_SQUARE_FOR_POWER = ( ARGUMENT_FOR_SQUARE_FOR_POWER * ARGUMENT_FOR_SQUARE_FOR_POWER ) % ( MODULO ); \
      EXPONENT_FOR_SQUARE_FOR_POWER /= 2;				\
    }									\
  }									\

#define FACTORIAL_MOD( ANSWER , ANSWER_INV , INVERSE , MAX_INDEX , CONSTEXPR_LENGTH , MODULO ) \
  static ll ANSWER[CONSTEXPR_LENGTH];					\
  static ll ANSWER_INV[CONSTEXPR_LENGTH];				\
  static ll INVERSE[CONSTEXPR_LENGTH];					\
  {									\
    ll VARIABLE_FOR_PRODUCT_FOR_FACTORIAL = 1;				\
    ANSWER[0] = VARIABLE_FOR_PRODUCT_FOR_FACTORIAL;			\
    FOREQ( i , 1 , MAX_INDEX ){						\
      ANSWER[i] = ( VARIABLE_FOR_PRODUCT_FOR_FACTORIAL *= i ) %= ( MODULO ); \
    }									\
    ANSWER_INV[0] = ANSWER_INV[1] = INVERSE[1] = VARIABLE_FOR_PRODUCT_FOR_FACTORIAL = 1; \
    FOREQ( i , 2 , MAX_INDEX ){						\
      ANSWER_INV[i] = ( VARIABLE_FOR_PRODUCT_FOR_FACTORIAL *= INVERSE[i] = ( MODULO ) - ( ( ( ( MODULO ) / i ) * INVERSE[ ( MODULO ) % i ] ) % ( MODULO ) ) ) %= ( MODULO ); \
    }									\
  }									\

// 二分探索テンプレート
// EXPRESSIONがANSWERの広義単調関数の時、EXPRESSION >= TARGETの整数解を格納。
#define BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , DESIRED_INEQUALITY , TARGET , INEQUALITY_FOR_CHECK , UPDATE_U , UPDATE_L , UPDATE_ANSWER ) \
  static_assert( ! is_same<TYPE_OF( TARGET ),uint>::value && ! is_same<TYPE_OF( TARGET ),ull>::value ); \
  ll ANSWER = MINIMUM;							\
  if( MINIMUM <= MAXIMUM ){						\
    ll VARIABLE_FOR_BINARY_SEARCH_L = MINIMUM;				\
    ll VARIABLE_FOR_BINARY_SEARCH_U = MAXIMUM;				\
    ANSWER = ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2; \
    ll VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH;			\
    while( VARIABLE_FOR_BINARY_SEARCH_L != VARIABLE_FOR_BINARY_SEARCH_U ){ \
      VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH = ( EXPRESSION ) - ( TARGET ); \
      CERR( "二分探索中： " << VARIABLE_FOR_BINARY_SEARCH_L << "<=" << ANSWER << "<=" << VARIABLE_FOR_BINARY_SEARCH_U << ":" << EXPRESSION << "-" << TARGET << "=" << VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH ); \
      if( VARIABLE_FOR_DIFFERENCE_FOR_BINARY_SEARCH INEQUALITY_FOR_CHECK 0 ){	\
	VARIABLE_FOR_BINARY_SEARCH_U = UPDATE_U;			\
      } else {								\
	VARIABLE_FOR_BINARY_SEARCH_L = UPDATE_L;			\
      }									\
      ANSWER = UPDATE_ANSWER;						\
    }									\
    CERR( "二分探索終了： " << VARIABLE_FOR_BINARY_SEARCH_L << "<=" << ANSWER << "<=" << VARIABLE_FOR_BINARY_SEARCH_U << ":" << EXPRESSION << ( EXPRESSION > TARGET ? ">" : EXPRESSION < TARGET ? "<" : "=" ) << TARGET ); \
    if( EXPRESSION DESIRED_INEQUALITY TARGET ){				\
      CERR( "二分探索成功" );						\
    } else {								\
      CERR( "二分探索失敗" );						\
      ANSWER = MAXIMUM + 1;						\
    }									\
  } else {								\
    CERR( "二分探索失敗： " << MINIMUM << ">" << MAXIMUM );		\
    ANSWER = MAXIMUM + 1;						\
  }									\

// 単調増加の時にEXPRESSION >= TARGETの最小解を格納。
#define BS1( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , TARGET , >= , ANSWER , ANSWER + 1 , ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \

// 単調増加の時にEXPRESSION <= TARGETの最大解を格納。
#define BS2( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , TARGET , > , ANSWER - 1 , ANSWER , ( VARIABLE_FOR_BINARY_SEARCH_L + 1 + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \

// 単調減少の時にEXPRESSION >= TARGETの最大解を格納。
#define BS3( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , TARGET , < , ANSWER - 1 , ANSWER , ( VARIABLE_FOR_BINARY_SEARCH_L + 1 + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \

// 単調減少の時にEXPRESSION <= TARGETの最小解を格納。
#define BS4( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , TARGET )		\
  BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , TARGET , <= , ANSWER , ANSWER + 1 , ( VARIABLE_FOR_BINARY_SEARCH_L + VARIABLE_FOR_BINARY_SEARCH_U ) / 2 ) \

// t以下の値が存在すればその最大値のiterator、存在しなければend()を返す。
template <typename T> inline typename set<T>::iterator MaximumLeq( set<T>& S , const T& t ) { const auto end = S.end(); if( S.empty() ){ return end; } auto itr = S.upper_bound( t ); return itr == end ? S.find( *( S.rbegin() ) ) : itr == S.begin() ? end : --itr; }
// t未満の値が存在すればその最大値のiterator、存在しなければend()を返す。
template <typename T> inline typename set<T>::iterator MaximumLt( set<T>& S , const T& t ) { const auto end = S.end(); if( S.empty() ){ return end; } auto itr = S.lower_bound( t ); return itr == end ? S.find( *( S.rbegin() ) ) : itr == S.begin() ? end : --itr; }
// t以上の値が存在すればその最小値のiterator、存在しなければend()を返す。
template <typename T> inline typename set<T>::iterator MinimumGeq( set<T>& S , const T& t ) { return S.lower_bound( t ); }
// tより大きい値が存在すればその最小値のiterator、存在しなければend()を返す。
template <typename T> inline typename set<T>::iterator MinimumGt( set<T>& S , const T& t ) { return S.upper_bound( t ); }

// データ構造用関数
template <typename T> inline T add( const T& t0 , const T& t1 ) { return t0 + t1; }
template <typename T> inline T xor_add( const T& t0 , const T& t1 ){ return t0 ^ t1; }
template <typename T> inline T multiply( const T& t0 , const T& t1 ) { return t0 * t1; }
template <typename T> inline const T& zero() { static const T z = 0; return z; }
template <typename T> inline const T& one() { static const T o = 1; return o; }\
template <typename T> inline T add_inv( const T& t ) { return -t; }
template <typename T> inline T id( const T& v ) { return v; }

// グリッド問題用関数
int H , W , H_minus , W_minus , HW;
inline pair<int,int> EnumHW( const int& v ) { return { v / W , v % W }; }
inline int EnumHW_inv( const int& h , const int& w ) { return h * W + w; }
const string direction[4] = {"U","R","D","L"};
// (i,j)->(k,h)の方向番号を取得
inline int DirectionNumberOnGrid( const int& i , const int& j , const int& k , const int& h ){return i<k?2:i>k?0:j<h?1:j>h?3:(assert(false),-1);}
// v->wの方向番号を取得
inline int DirectionNumberOnGrid( const int& v , const int& w ){auto [i,j]=EnumHW(v);auto [k,h]=EnumHW(w);return DirectionNumberOnGrid(i,j,k,h);}
// 方向番号の反転U<->D、R<->L
inline int ReverseDirectionNumberOnGrid( const int& n ){assert(0<=n&&n<4);return(n+2)%4;}
inline void SetEdgeOnGrid( const string& Si , const int& i , list<int> ( &e )[] , const char& walkable = '.' ){FOR(j,0,W){if(Si[j]==walkable){int v = EnumHW_inv(i,j);if(i>0){e[EnumHW_inv(i-1,j)].push_back(v);}if(i+1<H){e[EnumHW_inv(i+1,j)].push_back(v);}if(j>0){e[EnumHW_inv(i,j-1)].push_back(v);}if(j+1<W){e[EnumHW_inv(i,j+1)].push_back(v);}}}}
inline void SetEdgeOnGrid( const string& Si , const int& i , list<pair<int,ll> > ( &e )[] , const char& walkable = '.' ){FOR(j,0,W){if(Si[j]==walkable){const int v=EnumHW_inv(i,j);if(i>0){e[EnumHW_inv(i-1,j)].push_back({v,1});}if(i+1<H){e[EnumHW_inv(i+1,j)].push_back({v,1});}if(j>0){e[EnumHW_inv(i,j-1)].push_back({v,1});}if(j+1<W){e[EnumHW_inv(i,j+1)].push_back({v,1});}}}}
inline void SetWallOnGrid( const string& Si , const int& i , bool ( &non_wall_i )[] , const char& walkable = '.'  , const char& unwalkable = '#' ){FOR(j,0,W){non_wall_i[j]=Si[j]==walkable?true:(assert(Si[j]==unwalkable),false);}}

// グラフ用関数
template <typename path_type> list<path_type> E( const int& i ); // 本体をmain()の後に定義
template <typename path_type> vector<list<path_type> > e;

// デバッグ用関数
#ifdef DEBUG
  inline void AlertAbort( int n ) { CERR( "abort関数が呼ばれました。assertマクロのメッセージが出力されていない場合はオーバーフローの有無を確認をしてください。" ); }
  void AutoCheck( bool& auto_checked );
#endif
void Jikken();
template <typename RET , typename... ARGS> RET Guchoku( ARGS... args );

// 圧縮用
#define TE template
#define TY typename
#define US using
#define ST static
#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 MO move
#define TH this
#define CRI CO int&
#define CRUI CO uint&
#define CRL CO ll&

/*

C-x 3 C-x o C-x C-fによるファイル操作用

BIT:
c:/Users/user/Documents/Programming/Mathematics/SetTheory/DirectProduct/AffineSpace/BIT/compress.txt

BFS:
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/BreadthFirstSearch/compress.txt

DFS on Tree:
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/DepththFirstSearch/Tree/compress.txt

Divisor:
c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Prime/Divisor/compress.txt

Mod:
c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Mod/ConstexprModulo/compress.txt

Polynomial
c:/Users/user/Documents/Programming/Mathematics/Polynomial/compress.txt

*/

// VVV ライブラリは以下に挿入する。

template <typename T , int exponent_lim>
class Power_constexpr
{
private:
  T m_val[exponent_lim];

public:
  inline constexpr Power_constexpr( const T& t , const T& init = T( 1 ) );

  constexpr const T& operator[]( const int& i ) const;
  
};

template <typename T , int exponent_lim> inline constexpr Power_constexpr<T,exponent_lim>::Power_constexpr( const T& t , const T& init ) : m_val{ init } { for( int exponent = 1 ; exponent < exponent_lim ; exponent++ ){ ( m_val[exponent] = m_val[exponent-1] ) *= t; } }

template <typename T , int exponent_lim> inline constexpr const T& Power_constexpr<T,exponent_lim>::operator[]( const int& i ) const { assert( i < exponent_lim ); return m_val[i]; }

#define DC_OF_FIRST_SEARCH(BREADTH)TE <int V_max> CL BREADTH ## FirstSearch_Body{PU:int m_V;int m_init;LI<int> m_next;bool m_found[V_max];int m_prev[V_max];IN BREADTH ## FirstSearch_Body(CRI V);IN BREADTH ## FirstSearch_Body(CRI V,CRI init);IN VO Reset(CRI init);IN VO Shift(CRI init);IN CRI SZ()CO;IN CRI init()CO;IN bool& found(CRI i);IN CRI prev(CRI i)CO;int Next();virtual LI<int> e(CRI t)= 0;};TE <int V_max,LI<int> E(CRI)> CL BREADTH ## FirstSearch:PU BREADTH ## FirstSearch_Body<V_max>{PU:TE<TY... Args> IN BREADTH ## FirstSearch(CO Args&... args);IN LI<int> e(CRI t);};TE <int V_max,LI<int> E(CRI)> VO BREADTH ## FirstConnectedComponent(CRI V,int(&vertex)[V_max],int& count);
#define DF_OF_FIRST_SEARCH(BREADTH,PUSH)TE <int V_max> IN BREADTH ## FirstSearch_Body<V_max>::BREADTH ## FirstSearch_Body(CRI V):m_V(V),m_init(),m_next(),m_found(),m_prev(){assert(m_V <= V_max);for(int i = 0;i < m_V;i++){m_prev[i] = -1;}}TE <int V_max> IN BREADTH ## FirstSearch_Body<V_max>::BREADTH ## FirstSearch_Body(CRI V,CRI init):BREADTH ## FirstSearch_Body(V){m_init = init;m_next.push_back(m_init);m_found[m_init] = true;}TE <int V_max,LI<int> E(CRI)> TE <TY... Args> IN BREADTH ## FirstSearch<V_max,E>::BREADTH ## FirstSearch(CO Args&... args):BREADTH ## FirstSearch_Body<V_max>(args...){}TE <int V_max> IN VO BREADTH ## FirstSearch_Body<V_max>::Reset(CRI init){m_init = init;assert(m_init < m_V);m_next.clear();m_next.push_back(m_init);for(int i = 0;i < m_V;i++){m_found[i] = i == m_init;m_prev[i] = -1;}}TE <int V_max> IN VO BREADTH ## FirstSearch_Body<V_max>::Shift(CRI init){m_init = init;assert(m_init < m_V);m_next.clear();if(! m_found[m_init]){m_next.push_back(m_init);m_found[m_init] = true;}}TE <int V_max> IN CRI BREADTH ## FirstSearch_Body<V_max>::SZ()CO{RE m_V;}TE <int V_max> IN CRI BREADTH ## FirstSearch_Body<V_max>::init()CO{RE m_init;}TE <int V_max> IN bool& BREADTH ## FirstSearch_Body<V_max>::found(CRI i){assert(i < m_V);RE m_found[i];}TE <int V_max> IN CRI BREADTH ## FirstSearch_Body<V_max>::prev(CRI i)CO{assert(i < m_V);RE m_prev[i];}TE <int V_max> int BREADTH ## FirstSearch_Body<V_max>::Next(){if(m_next.empty()){RE -1;}CO int i_curr = m_next.front();m_next.pop_front();LI<int> edge = e(i_curr);WH(! edge.empty()){CRI i = edge.front();bool& found_i = found(i);if(! found_i){m_next.PUSH(i);m_prev[i] = i_curr;found_i = true;}edge.pop_front();}RE i_curr;}TE <int V_max,LI<int> E(CRI)> IN LI <int> BREADTH ## FirstSearch<V_max,E>::e(CRI t){RE E(t);}TE <int V_max,LI<int> E(CRI)> VO BREADTH ## FirstConnectedComponentSearch(CRI V,int(&vertex)[V_max],int& count){BREADTH ## FirstSearch<V_max,E> bfs{V};count = 0;for(int i = 0;i < V;i++){vertex[i] = -1;}for(int i = 0;i < V;i++){if(vertex[i] == -1){bfs.Shift(i);int j = bfs.Next();WH(j != -1?vertex[j] == 0:false){vertex[j] = count;j = bfs.Next();}count++;}}RE;}
DC_OF_FIRST_SEARCH(Breadth);DF_OF_FIRST_SEARCH(Breadth,push_back);
TE <int V_max,LI<int> E(CRI)>VO SetDepth_Body(BreadthFirstSearch<V_max,E>& bfs,CRI V,CRI init,int ( &depth )[V_max]){for(int i = 1;i < V;i++){depth[i] = -1;}int i = bfs.Next();depth[i] = 0;WH((i = bfs.Next())!= -1){depth[i] = depth[bfs.prev( i )] + 1;}RE;}TE <int V_max,LI<int> E(CRI)>VO SetDepth(CRI V,CRI init,int ( &depth )[V_max]){BreadthFirstSearch<V_max,E> bfs{V,init};SetDepth_Body(bfs,V,init,depth);RE;}TE <int V_max,LI<int> E(CRI)>VO SetDepth(BreadthFirstSearch<V_max,E>& bfs,CRI V,CRI init,int ( &depth )[V_max]){bfs.Reset(init);SetDepth_Body(bfs,V,init,depth);RE;}

// AAA ライブラリは以上に挿入する。

DEXPR( int , bound_K , 8 , 3 );
int K = 0;
constexpr Power_constexpr<int,bound_K+1> power{ 5 };

template <typename path_type> list<path_type> E( const int& i )
{
  list<path_type> answer{};
  // list<path_type> answer = e<path_type>[i];
  // VVV 入力によらない処理は以下に挿入する。
  FOR( k , 0 , K ){
    const int& power_k = power[k];
    if( ( i / power_k ) % 5 > 0 ){
      answer.push_back( i - power_k );
    }
  }
  // AAA 入力によらない処理は以上に挿入する。
  return answer;
}

template <> ll Guchoku( ll N , ll M , ll K )
{
  ll answer = N + M + K;
  return answer;
}

// void Jikken()
// {
//   CEXPR( int , bound , 10 );
//   FOREQ( N , 0 , bound ){
//     // FOREQ( M , 0 , bound ){
//     //   FOREQ( K , 0 , bound ){
//     // 	COUT( N , M , K , Guchoku( N , M , K ) );
//     //   }
//     // }
//     cout << Guchoku<ll>( N ) << " \n"[N==bound];
//   }
// }

int main()
{
  UNTIE;
  AUTO_CHECK;
  // START_WATCH;
  TEST_CASE_NUM( 1 );
  START_MAIN;
  // Jikken();

  DEXPR( int , bound_N , 200000 , 20 );
  CIN_ASSERT( N , 1 , bound_N );
  SET_ASSERT( K , 1 , bound_K );
  assert( N <= power[K] );
  int S[bound_N];
  CEXPR( int , bound_Si , 100000000 );
  FOR( i , 0 , N ){
    CIN_ASSERT( Si , 0 , bound_Si );
    int Si_copy = 0;
    REPEAT( K ){
      ( Si_copy *= 5 ) += Si % 10;
      Si /= 10;
    }
    S[i] = Si_copy;
  }
  bool b[power[bound_K]] = {};
  FOR( i , 0 , N ){
    b[S[i]] = true;
  }
  int answer = 0;
  BreadthFirstSearch<power[bound_K],E> bfs{ power[bound_K] };
  FOR( i , 0 , N ){
    int& Si = S[i];
    bool& found_Si = bfs.found( Si );
    if( ! found_Si ){
      bfs.Shift( Si );
      int e = bfs.Next();
      while( ( e = bfs.Next() ) != -1 ){
	b[e] ? ++answer : answer;
      }
      found_Si = false;
    }
  }
  RETURN( answer );
  
  FINISH_MAIN;
}