#ifdef DEBUG
  #define _GLIBCXX_DEBUG
  #define REPEAT_MAIN( BOUND ) START_MAIN; signal( SIGABRT , &AlertAbort ); AutoCheck( exec_mode ); if( exec_mode == debug_mode || exec_mode == library_search_mode ){ return 0; } else if( exec_mode == experiment_mode ){ Experiment(); return 0; } else if( exec_mode == small_test_mode ){ SmallTest(); return 0; }; DEXPR( int , bound_test_case_num , BOUND , min( BOUND , 100 ) ); int test_case_num = 1; if( exec_mode == solve_mode ){ if constexpr( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } } else if( exec_mode == random_test_mode ){ CERR( "ランダムテストを行う回数を指定してください。" ); cin >> test_case_num; } FINISH_MAIN
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , DEBUG_VALUE )
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); assert( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) if( exec_mode == solve_mode ){ cin >> A; ASSERT( A , MIN , MAX ); } else if( exec_mode == random_test_mode ){ CERR( #A , " = " , ( A = GetRand( MIN , MAX ) ) ); } else { assert( false ); }
  #define SOLVE_ONLY static_assert( __FUNCTION__[0] == 'S' )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define COUT( ... ) VariadicCout( cout << "出力： " , __VA_ARGS__ ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << endl
  #define COUT_A( A , N ) cout << "出力： "; OUTPUT_ARRAY( cout , A , N ) << endl
  #define CERR_ITR( A ) OUTPUT_ITR( cerr , A ) << endl
  #define COUT_ITR( A ) cout << "出力： "; OUTPUT_ITR( cout , A ) << endl
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if constexpr( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } FINISH_MAIN
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , VALUE )
  #define ASSERT( A , MIN , MAX ) assert( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) cin >> A; ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #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"
#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 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 constexpr( 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 TYPE_OF( VAR ) decay_t<decltype( VAR )>
#define CEXPR( LL , BOUND , VALUE ) constexpr LL BOUND = VALUE
#define CIN( LL , ... ) SOLVE_ONLY; LL __VA_ARGS__; VariadicCin( cin , __VA_ARGS__ )
#define CIN_ASSERT( A , MIN , MAX ) TYPE_OF( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define SET_A( A , N ) SOLVE_ONLY; FOR( VARIABLE_FOR_CIN_A , 0 , N ){ cin >> A[VARIABLE_FOR_CIN_A]; }
#define CIN_A( LL , A , N ) LL A[N]; SET_A( A , N );
#define GETLINE_SEPARATE( SEPARATOR , ... ) SOLVE_ONLY; string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
#define GETLINE( ... ) SOLVE_ONLY; GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
#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 RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); return
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ ); auto answer = Answer( __VA_ARGS__ ); bool match = naive == answer; COUT( #__VA_ARGS__ , ":" , naive , match ? "==" : "!=" , answer ); if( !match ){ return; }

// 入出力用
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
template <typename T> inline T Residue( const T& a , const T& p ){ return a >= 0 ? a % p : p - 1 - ( ( - ( a + 1 ) ) % p ); }

#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
template <typename T> inline T Add( const T& t0 , const T& t1 ) { return t0 + t1; }
template <typename T> inline T XorAdd( 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 AddInv( const T& t ) { return -t; }
template <typename T> inline T Id( const T& v ) { return v; }
template <typename T> inline T Min( const T& a , const T& b ){ return a < b ? a : b; }
template <typename T> inline T Max( const T& a , const T& b ){ return a < b ? b : a; }

// グリッド問題用
int H , W , H_minus , W_minus , HW;
vector<vector<bool> > non_wall;
inline T2<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<path> ( &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 , vector<vector<bool> >& non_wall , const char& walkable = '.'  , const char& unwalkable = '#' ){non_wall.push_back(vector<bool>(W));auto& non_wall_i=non_wall[i];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> list<PATH> E( const int& i );
template <typename PATH> vector<list<PATH> > e;

// デバッグ用
#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( int& exec_mode );
  inline void Solve();
  inline void Experiment();
  inline void SmallTest();
  inline void RandomTest();
  ll GetRand( const ll& Rand_min , const ll& Rand_max );
  int exec_mode;
  CEXPR( int , solve_mode , 0 );
  CEXPR( int , debug_mode , 1 );
  CEXPR( int , library_search_mode , 2 );
  CEXPR( int , experiment_mode , 3 );
  CEXPR( int , small_test_mode , 4 );
  CEXPR( int , random_test_mode , 5 );
#endif

// 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。

#define DIJKSTRA_BODY( SET_FOUND , SET_WEIGHT , UPDATE_FOUND , CHECK_FOUND , INITIALISE_PREV , SET_PREV ) \
  static const U& unit = Unit();					\
  assert( unit != m_found && unit < m_infty );				\
  const int i_start = e_inv( t_start );					\
  set<pair<U,int> > vertex{};						\
  SET_FOUND;								\
  SET_WEIGHT;								\
  vertex.insert( pair<U,int>( weight[i_start] = unit , i_start ) );	\
  INITIALISE_PREV;							\
									\
  if( i_start != i_final ){						\
  									\
    while( ! vertex.empty() ){						\
									\
      auto itr_vertex = vertex.begin();					\
      const pair<U,int> v = *itr_vertex;				\
      const int& i = v.second;						\
									\
      if( i == i_final ){						\
									\
	break;								\
									\
      }									\
									\
      const U& u = v.first;						\
      UPDATE_FOUND;							\
      vertex.erase( itr_vertex );					\
      const list<pair<T,U> > edge_i = E( e( i ) );			\
      list<pair<U,int> > changed_vertex{};				\
									\
      for( auto itr_edge_i = edge_i.begin() , end_edge_i = edge_i.end() ; itr_edge_i != end_edge_i ; itr_edge_i++ ){ \
									\
	const int& j = e_inv( itr_edge_i->first );			\
	U& weight_j = weight[j];					\
      									\
	if( CHECK_FOUND ){						\
									\
	  const U& edge_ij = itr_edge_i->second;			\
	  const U temp = Addition( u , edge_ij );			\
	  assert( edge_ij != m_found && temp != m_found && !( temp < edge_ij ) && temp < m_infty ); \
									\
	  if( weight_j > temp ){					\
									\
	    if( weight_j != m_infty ){					\
									\
	      vertex.erase( pair<U,int>( weight_j , j ) );		\
									\
	    }								\
									\
	    SET_PREV;							\
	    changed_vertex.push_back( pair<U,int>( weight_j = temp , j ) ); \
									\
	  }								\
									\
	}								\
									\
      }									\
									\
      for( auto itr_changed = changed_vertex.begin() , end_changed = changed_vertex.end() ; itr_changed != end_changed ; itr_changed++ ){ \
									\
	vertex.insert( *itr_changed );					\
									\
      }									\
									\
    }									\
									\
  }									\
  
// メモリが不足する場合はEの定義を前計算しないでその都度計算させること。
template <typename T , typename U , list<pair<T,U> > E(const T&) , int size_max>
class DijkstraBody
{

private:
  int m_size;
  U m_infty;
  U m_found;

public:
  inline DijkstraBody( const int& size , const U& infty , const U& found );
  // 経路が存在しない場合の返り値はm_infty
  U Solve( const T& t_start , const T& t_final );
  U Solve( const T& t_start , const T& t_final , list<T>& path );
  void Solve( const T& t_start , vector<U>& weight );
  void Solve( const T& t_start , vector<U>& weight , list<T> ( &path )[size_max] );
  
  const U& Infty() const;
  
private:
  virtual const U& Unit() const = 0;
  virtual U Addition( const U& , const U& ) const = 0;
  virtual T e( const int& i ) = 0;
  virtual int e_inv( const T& t ) = 0;
  virtual void Reset() = 0;

};

// 入力の範囲内で要件
// (1) Eの値の各成分の第2成分が0以上である。
// (2) 2^{31}-1がEの値の各成分の第2成分size_max個以下の和で表せるいかなる数よりも大きい。
// (6) Vの各要素u,vに対し、辺u->vが複数存在する場合は重みが最小のものが前にpushされている。
// が成り立つ場合にのみサポート。

// 単一始点単一終点最短経路探索／経路復元なしO((size+|E|)log size)
// 単一始点単一終点最短経路探索／経路復元ありO((size+|E|)log size)
// 単一始点全点最短経路探索／経路復元なしO((size+|E|)log size)
// 単一始点全点最短経路探索／経路復元ありO(size^2 + |E| log size)
template <list<pair<int,ll> > E(const int&) , int size_max>
class Dijkstra :
  public DijkstraBody<int,ll,E,size_max>
{

public:
  inline Dijkstra( const int& size );
  
private:
  inline const ll& Unit() const;
  inline ll Addition( const ll& , const ll& ) const;
  inline int e( const int& i );
  inline int e_inv( const int& t );
  inline void Reset();

};

// 入力の範囲内で要件
// (1) Eの値の各成分の第2成分がe_T()以上である。
// (2) inftyがEの値の各成分の第2成分size_max個以下の和で表せるいかなる項よりも大きい。
// (3) foundがEの値の各成分の第2成分size_max個以下の和で表せず、inftyとも異なる。
// (4) (U,m_U:U^2->U,e_U:1->U)がbool operator<(const U&,const U&)に関して全順序モノイドである。
// (6) Vの各要素u,vに対し、辺u->vが複数存在する場合は重みが最小のものが前にpushされている。
// が成り立つ場合にのみサポート。

// 単一始点単一終点最短経路探索／経路復元なしO((size+|E|)(log size)^2)
// 単一始点単一終点最短経路探索／経路復元ありO((size+|E|)(log size)^2)
// 単一始点全点最短経路探索／経路復元なしO((size+|E|)(log size)^2)
// 単一始点全点最短経路探索／経路復元ありO(size^2 log size + |E|(log size)^2)
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max>
class MemorisationDijkstra :
  public DijkstraBody<T,U,E,size_max>
{

private:
  int m_length;
  map<T,int> m_memory;
  vector<T> m_memory_inv;

public:
  inline MemorisationDijkstra( const int& size , const U& infty = 9223372036854775807 , const U& found = -1 );
  
private:
  inline const U& Unit() const;
  inline U Addition( const U& , const U& ) const;
  inline T e( const int& i );
  inline int e_inv( const T& t );
  inline void Reset();

};

// 入力の範囲内で要件
// (1) Eの値の各成分の第2成分がe_T()以上である。
// (2) inftyがEの値の各成分の第2成分size_max個以下の和で表せるいかなる項よりも大きい。
// (3) foundがEの値の各成分の第2成分size_max個以下の和で表せず、inftyとも異なる。
// (4) (U,m_U:U^2->U,e_U:1->U)がbool operator<(const U&,const U&)に関して全順序モノイドである。
// (5) (enum_T,enum_T_inv)が互いに逆写像である。
// (6) Vの各要素u,vに対し、辺u->vが複数存在する場合は重みが最小のものが前にpushされている。
// が成り立つ場合にのみサポート。

// 単一始点単一終点最短経路探索／経路復元なしO((size+|E|)log size)
// 単一始点単一終点最短経路探索／経路復元ありO((size+|E|)log size)
// 単一始点全点最短経路探索／経路復元なしO((size+|E|)log size)
// 単一始点全点最短経路探索／経路復元ありO(size^2 + |E| log size)
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max , T enum_T(const int&) , int enum_T_inv(const T&)>
class EnumerationDijkstra :
  public DijkstraBody<T,U,E,size_max>
{

public:
  inline EnumerationDijkstra( const int& size , const U& infty = 9223372036854775807 , const U& found = -1 );
  
private:
  inline const U& Unit() const;
  inline U Addition( const U& , const U& ) const;
  inline T e( const int& i );
  inline int e_inv( const T& t );
  inline void Reset();

};

template <typename T , typename U , list<pair<T,U> > E(const T&) , int size_max> inline DijkstraBody<T,U,E,size_max>::DijkstraBody( const int& size , const U& infty , const U& found ) : m_size( size ) , m_infty( infty ) , m_found( found ) { static_assert( ! is_same<U,int>::value ); }
template <list<pair<int,ll> > E(const int&) , int size_max> inline Dijkstra<E,size_max>::Dijkstra( const int& size ) : DijkstraBody<int,ll,E,size_max>( size , 9223372036854775807 , -1 ) {}
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max> inline MemorisationDijkstra<T,U,m_U,e_U,E,size_max>::MemorisationDijkstra( const int& size , const U& infty , const U& found ) : DijkstraBody<T,U,E,size_max>( size , infty , found ) , m_length() , m_memory() , m_memory_inv() {}
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max , T enum_T(const int&) , int enum_T_inv(const T&)> inline EnumerationDijkstra<T,U,m_U,e_U,E,size_max,enum_T,enum_T_inv>::EnumerationDijkstra( const int& size , const U& infty , const U& found ) : DijkstraBody<T,U,E,size_max>( size , infty , found ) {}

template <typename T , typename U , list<pair<T,U> > E(const T&) , int size_max>
U DijkstraBody<T,U,E,size_max>::Solve( const T& t_start , const T& t_final )
{

  const int i_final = e_inv( t_final );					\
  DIJKSTRA_BODY( , vector<U> weight( m_size , m_infty ) , weight[i] = m_found , weight_j != m_found , , );
  Reset();
  return weight[i_final];

}

template <typename T , typename U , list<pair<T,U> > E(const T&) , int size_max>
U DijkstraBody<T,U,E,size_max>::Solve( const T& t_start , const T& t_final , list<T>& path )
{

  const int i_final = e_inv( t_final );					\
  DIJKSTRA_BODY( , vector<U> weight( m_size , m_infty ) , weight[i] = m_found , weight_j != m_found , vector<int> prev( m_size ) , prev[j] = i );
  int i = i_final;

  while( i != i_start ){

    path.push_front( e( i ) );
    i = prev[i];

  }

  path.push_front( t_start );
  Reset();
  return weight[i_final];

}

template <typename T , typename U , list<pair<T,U> > E(const T&) , int size_max>
void DijkstraBody<T,U,E,size_max>::Solve( const T& t_start , vector<U>& weight )
{

  constexpr const int i_final = -1;
  DIJKSTRA_BODY( vector<bool> found( m_size ) , weight = vector<U>( m_size , m_infty ) , found[i] = true , !found[j] , , );
  Reset();
  return;

}

template <typename T , typename U , list<pair<T,U> > E(const T&) , int size_max>
void DijkstraBody<T,U,E,size_max>::Solve( const T& t_start , vector<U>& weight , list<T> ( &path )[size_max] )
{

  constexpr const int i_final = -1;
  DIJKSTRA_BODY( vector<bool> found( m_size ) , weight = vector<U>( m_size , m_infty ) , found[i] = true , !found[j] , vector<int> prev( m_size ) , prev[j] = i );

  for( int j = 0 ; j < m_size ; j++ ){

    list<T>& path_j = path[j];
    int i = j;

    while( i != i_start ){

      path_j.push_front( e( i ) );
      i = prev[i];

    }

    path_j.push_front( t_start );

  }

  Reset();
  return;

}

template <typename T , typename U , list<pair<T,U> > E(const T&) , int size_max> const U& DijkstraBody<T,U,E,size_max>::Infty() const { return m_infty; }

template <list<pair<int,ll> > E(const int&) , int size_max> inline const ll& Dijkstra<E,size_max>::Unit() const { static const ll unit = 0; return unit; }
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max> inline const U& MemorisationDijkstra<T,U,m_U,e_U,E,size_max>::Unit() const { return e_U(); }
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max , T enum_T(const int&) , int enum_T_inv(const T&)> inline const U& EnumerationDijkstra<T,U,m_U,e_U,E,size_max,enum_T,enum_T_inv>::Unit() const { return e_U(); }

template <list<pair<int,ll> > E(const int&) , int size_max> inline ll Dijkstra<E,size_max>::Addition( const ll& u0 , const ll& u1 ) const { return u0 + u1; }
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max> inline U MemorisationDijkstra<T,U,m_U,e_U,E,size_max>::Addition( const U& u0 , const U& u1 ) const { return m_U( u0 , u1 ); }
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max , T enum_T(const int&) , int enum_T_inv(const T&)> inline U EnumerationDijkstra<T,U,m_U,e_U,E,size_max,enum_T,enum_T_inv>::Addition( const U& u0 , const U& u1 ) const { return m_U( u0 , u1 ); }

template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max> inline T MemorisationDijkstra<T,U,m_U,e_U,E,size_max>::e( const int& i ) { assert( i < m_length ); return m_memory_inv[i]; }
template <list<pair<int,ll> > E(const int&) , int size_max> inline int Dijkstra<E,size_max>::e( const int& i ) { return i; }
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max , T enum_T(const int&) , int enum_T_inv(const T&)> inline T EnumerationDijkstra<T,U,m_U,e_U,E,size_max,enum_T,enum_T_inv>::e( const int& i ) { return enum_T( i ); }

template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max> inline int MemorisationDijkstra<T,U,m_U,e_U,E,size_max>::e_inv( const T& t )
{

  using base = DijkstraBody<T,U,E,size_max>;
  
  if( m_memory.count( t ) == 0 ){

    assert( m_length < base::m_size );
    m_memory_inv.push_back( t );
    return m_memory[t] = m_length++;

  }
  
  return m_memory[t];

}

template <list<pair<int,ll> > E(const int&) , int size_max> inline int Dijkstra<E,size_max>::e_inv( const int& t ) { return t; }
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max , T enum_T(const int&) , int enum_T_inv(const T&)> inline int EnumerationDijkstra<T,U,m_U,e_U,E,size_max,enum_T,enum_T_inv>::e_inv( const T& t ) { return enum_T_inv( t ); }

template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max> inline void MemorisationDijkstra<T,U,m_U,e_U,E,size_max>::Reset() { m_length = 0; m_memory.clear(); m_memory_inv.clear(); }
template <list<pair<int,ll> > E(const int&) , int size_max> inline void Dijkstra<E,size_max>::Reset() {}
template <typename T , typename U , U m_U(const U&,const U&) , const U& e_U() , list<pair<T,U> > E(const T&) , int size_max , T enum_T(const int&) , int enum_T_inv(const T&)> inline void EnumerationDijkstra<T,U,m_U,e_U,E,size_max,enum_T,enum_T_inv>::Reset() {}

// AAA ライブラリは以上に挿入する。

// VVV テンプレート引数用の関数は以下に挿入する。

inline CEXPR( int , bound_N , 500 );
T2<int> v[bound_N + 2];
int N , D;
list<path> E( const int& t )
{
  T2<int>& vt = v[t];
  list<path> answer{};
  FOR( i , 0 , N ){
    T2<int>& vi = v[i];
    int x = abs( vt.first - vi.first );
    int y = abs( vt.second - vi.second );
    if( x < y ){
      swap( x , y );
    }
    answer.push_back( { i , ( x - 1 ) / D } );
  }
  return answer;
}

int k( const int& answer )
{
  D = answer;
  Dijkstra<E,bound_N+2> d{ N };
  return d.Solve( 0 , 1 );
}

// AAA テンプレート引数用の関数は以上に挿入する。

ll Naive( int N , int M , int K )
{
  ll answer = N + M + K;
  return answer;
}

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;
}

inline void Solve()
{
  SET_ASSERT( N , 1 , bound_N );
  CEXPR( int , bound , 100000 ); // 0が5個
  CIN_ASSERT( K , 0 , bound );
  N += 2;
  FOR( i , 0 , N ){
    CIN_ASSERT( x , 0 , bound );
    CIN_ASSERT( y , 0 , bound );
    v[i] = { x + y , x - y };
  }
  BS4( answer , 1 , bound , k( answer ) , K );  
  RETURN( answer );
}

inline void 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];
  // }
}

inline void SmallTest()
{
  // CEXPR( int , bound , 10 );
  // FOREQ( N , 0 , bound ){
  //   FOREQ( M , 0 , bound ){
  //     FOREQ( K , 0 , bound ){
  //   	COMPARE( N , M , K );
  //     }
  //   }
  //   // COMPARE( N );
  // }
}

REPEAT_MAIN(1);