#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 = ( ( ARGUMENT ) % ( MODULO ) ) % ( MODULO ); \
    ARGUMENT_FOR_SQUARE_FOR_POWER < 0 ? ARGUMENT_FOR_SQUARE_FOR_POWER += ( MODULO ) : ARGUMENT_FOR_SQUARE_FOR_POWER; \
    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 ) \
  ll ANSWER[CONSTEXPR_LENGTH];						\
  ll ANSWER_INV[CONSTEXPR_LENGTH];					\
  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;							\
  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;						\
  }									\
  if( VARIABLE_FOR_BINARY_SEARCH_L > VARIABLE_FOR_BINARY_SEARCH_U ){	\
    CERR( "二分探索失敗：" , VARIABLE_FOR_BINARY_SEARCH_L , ">" , VARIABLE_FOR_BINARY_SEARCH_U ); \
    ANSWER = MAXIMUM + 1;						\
  } else {								\
    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：" , #ANSWER , "=" , ANSWER );			\
    } else {								\
      CERR( "二分探索失敗：" , EXPRESSION , "<>"[EXPRESSION > TARGET],  TARGET ); \
      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。

template <typename T>
class CoordinateCompress
{

private:
  vector<T> m_a;
  map<T,int> m_enum;
  bool m_compressed;
  int m_size;

public:
  inline CoordinateCompress();
  template <typename U , int length_max> inline CoordinateCompress( const U ( &a )[length_max] , const int& length );
  template <typename U> inline CoordinateCompress( const vector<U>& a );

  inline void insert( const T& t );
  template <typename U , int length_max> inline void insert( const U ( &a )[length_max] , const int& length );
  template <typename U> inline void insert( const vector<U>& a );

  inline const T& GetSmallest( const int& i = 0 );
  inline const T& GetLargest( const int& i = 0 );
  inline int GetOrder( const T& t );
  inline const int& size();

  inline typename map<T,int>::iterator begin();
  inline typename map<T,int>::iterator end();

private:
  inline void Compress();
  
};

template <typename T> inline CoordinateCompress<T>::CoordinateCompress() : m_a() , m_enum() , m_compressed() , m_size() {}
template <typename T> template <typename U , int length_max> inline CoordinateCompress<T>::CoordinateCompress( const U ( &a )[length_max] , const int& length ) : CoordinateCompress() { insert( a , length ); }
template <typename T> template <typename U> inline CoordinateCompress<T>::CoordinateCompress( const vector<U>& a ) : CoordinateCompress() { insert( a ); }

template <typename T> inline void CoordinateCompress<T>::insert( const T& t ) { m_enum[t]; m_size = m_enum.size(); m_compressed = false; }
template <typename T> template <typename U , int length_max> inline void CoordinateCompress<T>::insert( const U ( &a )[length_max] , const int& length ) { assert( length <= length_max ); if( length > 0 ){ for( int i = 0 ; i < length ; i++ ){ m_enum[ a[i] ]; } m_size = m_enum.size(); m_compressed = false; } }
template <typename T> template <typename U> inline void CoordinateCompress<T>::insert( const vector<U>& a ) { const int length = a.size(); if( length > 0 ){ for( int i = 0 ; i < length ; i++ ){ m_enum[ a[i] ]; } m_size = m_enum.size(); m_compressed = false; } }

template <typename T> inline const T& CoordinateCompress<T>::GetSmallest( const int& i ) { if( ! m_compressed ){ Compress(); } assert( i < m_size ); return m_a[i]; }
template <typename T> inline const T& CoordinateCompress<T>::GetLargest( const int& i ) { if( ! m_compressed ){ Compress(); } assert( i < m_size ); return m_a[m_size - i - 1]; }
template <typename T> inline int CoordinateCompress<T>::GetOrder( const T& t ) { if( ! m_compressed ){ Compress(); } return m_enum.count( t ) == 1 ? m_enum[t] : -1; }
template <typename T> inline const int& CoordinateCompress<T>::size() { return m_size; }

template <typename T> inline typename map<T,int>::iterator CoordinateCompress<T>::begin() { return m_enum.begin(); }
template <typename T> inline typename map<T,int>::iterator CoordinateCompress<T>::end() { return m_enum.end(); }

template <typename T> inline void CoordinateCompress<T>::Compress() { m_a.resize( m_size ); m_size = 0; for( auto itr = m_enum.begin() , end = m_enum.end() ; itr != end ; itr++ ){ m_a[itr->second = m_size++] = itr->first; } m_compressed = true; }

TE<int N>CL PWInverse_CE{PU:int m_val;CE PWInverse_CE();};
TE<int N>CE PWInverse_CE<N>::PWInverse_CE():m_val(1){WH(m_val < N){m_val <<= 1;}}
TE <TY T,int N>CL BIT{PU:T m_fenwick[N + 1];IN BIT();BIT(CO T(&a)[N]);IN T Get(CRI i)CO;IN VO Set(CRI i,CO T& n);IN VO Set(CO T(&a)[N]);IN VO Initialise();IN BIT<T,N>& OP+=(CO T(&a)[N]);VO Add(CRI i,CO T& n);T InitialSegmentSum(CRI i_final)CO;IN T IntervalSum(CRI i_start,CRI i_final)CO;int BinarySearch(CO T& n)CO;IN int BinarySearch(CRI i_start,CO T& n)CO;};
TE <TY T,int N> IN BIT<T,N>::BIT():m_fenwick(){static_assert(! is_same<T,int>::value);}TE <TY T,int N>BIT<T,N>::BIT(CO T(&a)[N]):m_fenwick(){static_assert(! is_same<T,int>::value);for(int j = 1;j <= N;j++){T& fenwick_j = m_fenwick[j];int i = j - 1;fenwick_j = a[i];int i_lim = j -(j & -j);WH(i != i_lim){fenwick_j += m_fenwick[i];i -=(i & -i);}}}TE <TY T,int N> IN T BIT<T,N>::Get(CRI i)CO{RE IntervalSum(i,i);}TE <TY T,int N> IN VO BIT<T,N>::Set(CRI i,CO T& n){Add(i,n - IntervalSum(i,i));}TE <TY T,int N> IN VO BIT<T,N>::Set(CO T(&a)[N]){BIT<T,N> a_copy{a};swap(m_fenwick,a_copy.m_fenwick);}TE <TY T,int N> IN VO BIT<T,N>::Initialise(){for(int j = 1;j <= N;j++){m_fenwick[j] = 0;}}TE <TY T,int N> IN BIT<T,N>& BIT<T,N>::OP+=(CO T(&a)[N]){for(int i = 0;i < N;i++){Add(i,a[i]);}RE *TH;}TE <TY T,int N>VO BIT<T,N>::Add(CRI i,CO T& n){int j = i + 1;WH(j <= N){m_fenwick[j] += n;j +=(j & -j);}RE;}TE <TY T,int N>T BIT<T,N>::InitialSegmentSum(CRI i_final)CO{T sum = 0;int j =(i_final < N?i_final:N - 1)+ 1;WH(j > 0){sum += m_fenwick[j];j -= j & -j;}RE sum;}TE <TY T,int N> IN T BIT<T,N>::IntervalSum(CRI i_start,CRI i_final)CO{RE InitialSegmentSum(i_final)- InitialSegmentSum(i_start - 1);}TE <TY T,int N>int BIT<T,N>::BinarySearch(CO T& n)CO{int j = 0;int PW = PWInverse_CE<N>().m_val;T sum{};T sum_next{};WH(PW > 0){int j_next = j | PW;if(j_next < N){sum_next += m_fenwick[j_next];if(sum_next < n){sum = sum_next;j = j_next;}else{sum_next = sum;}}PW >>= 1;}RE j;}TE <TY T,int N> IN int BIT<T,N>::BinarySearch(CRI i_start,CO T& n)CO{RE max(i_start,BinarySearch(InitialSegmentSum(i_start)+ n));}

template <typename T>
class ExtendedRational
{

private:
  T m_n;
  T m_d;

public:
  inline constexpr ExtendedRational( const T& n = 0 , const T& d = 1 );
  inline constexpr ExtendedRational( const ExtendedRational<T>& r );

  inline constexpr ExtendedRational<T>& operator=( const ExtendedRational<T>& r ) noexcept;

  inline constexpr bool operator==( const ExtendedRational<T>& r ) const noexcept;
  inline constexpr bool operator!=( const ExtendedRational<T>& r ) const noexcept;
  inline constexpr bool operator<=( const ExtendedRational<T>& r ) const noexcept;
  inline constexpr bool operator>=( const ExtendedRational<T>& r ) const noexcept;
  inline constexpr bool operator<( const ExtendedRational<T>& r ) const noexcept;
  inline constexpr bool operator>( const ExtendedRational<T>& r ) const noexcept;

  inline constexpr const T& GetNumerator() const noexcept;
  inline constexpr const T& GetDenominator() const noexcept;

};

template <typename T> inline constexpr bool operator==( const T& n , const ExtendedRational<T>& r ) noexcept;
template <typename T> inline constexpr bool operator!=( const T& n , const ExtendedRational<T>& r ) noexcept;
template <typename T> inline constexpr bool operator<=( const T& n , const ExtendedRational<T>& r ) noexcept;
template <typename T> inline constexpr bool operator>=( const T& n , const ExtendedRational<T>& r ) noexcept;
template <typename T> inline constexpr bool operator<( const T& n , const ExtendedRational<T>& r ) noexcept;
template <typename T> inline constexpr bool operator>( const T& n , const ExtendedRational<T>& r ) noexcept;

template <typename T , class Traits> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const ExtendedRational<T>& r );




template <typename T> inline constexpr ExtendedRational<T>::ExtendedRational( const T& n , const T& d ) : m_n( n ) , m_d( d ) { assert( n != 0 || d != 0 ); }
template <typename T> inline constexpr ExtendedRational<T>::ExtendedRational( const ExtendedRational<T>& r ) : m_n( r.m_n ) , m_d( r.m_d ) {}

template <typename T> inline constexpr ExtendedRational<T>& ExtendedRational<T>::operator=( const ExtendedRational<T>& r ) noexcept { m_n = r.m_n; m_d = r.m_d; return *this; }

template <typename T> inline constexpr bool ExtendedRational<T>::operator==( const ExtendedRational<T>& r ) const noexcept { return m_d == 0 ? r.m_d == 0 && ( m_n > 0 ) == ( r.m_n > 0 ) : m_n * r.m_d == r.m_n * m_d; }
template <typename T> inline constexpr bool ExtendedRational<T>::operator!=( const ExtendedRational<T>& r ) const noexcept { return !( *this == r ); }
template <typename T> inline constexpr bool ExtendedRational<T>::operator<=( const ExtendedRational<T>& r ) const noexcept { return m_d == 0 ? m_n < 0 : ( m_d > 0 ) == ( r.m_d >= 0 ) ? m_n * r.m_d <= r.m_n * m_d : m_n * r.m_d >= r.m_n * m_d; }
template <typename T> inline constexpr bool ExtendedRational<T>::operator>=( const ExtendedRational<T>& r ) const noexcept { return r <= *this; }
template <typename T> inline constexpr bool ExtendedRational<T>::operator<( const ExtendedRational<T>& r ) const noexcept { return m_d == 0 ? m_n < 0 : ( m_d > 0 ) == ( r.m_d >= 0 ) ? m_n * r.m_d < r.m_n * m_d : m_n * r.m_d > r.m_n * m_d; }
template <typename T> inline constexpr bool ExtendedRational<T>::operator>( const ExtendedRational<T>& r ) const noexcept { return r < *this; }

template <typename T> inline constexpr const T& ExtendedRational<T>::GetNumerator() const noexcept { return m_n; }
template <typename T> inline constexpr const T& ExtendedRational<T>::GetDenominator() const noexcept { return m_d; }

template <typename T> inline constexpr bool operator==( const T& n , const ExtendedRational<T>& r ) noexcept { return n * r.GetDenominator() == r.GetNumerator(); }
template <typename T> inline constexpr bool operator!=( const T& n , const ExtendedRational<T>& r ) noexcept { return !( n == r ); }
template <typename T> inline constexpr bool operator<=( const T& n , const ExtendedRational<T>& r ) noexcept { return r.GetDenominator() >= 0 ? n * r.GetDenominator() <= r.GetNumerator() : n * r.GetDenominator() >= r.GetNumerator(); }
template <typename T> inline constexpr bool operator>=( const T& n , const ExtendedRational<T>& r ) noexcept { return r <= n; }
template <typename T> inline constexpr bool operator<( const T& n , const ExtendedRational<T>& r ) noexcept { return r.GetDenominator() >= 0 ? n * r.GetDenominator() < r.GetNumerator() : n * r.GetDenominator() > r.GetNumerator(); }
template <typename T> inline constexpr bool operator>( const T& n , const ExtendedRational<T>& r ) noexcept { return r < n; }

template <typename T , class Traits> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const ExtendedRational<T>& r ) { return os << r.GetNumerator() << "/" << r.GetDenominator(); }


// 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。

// H,W,e<PATH>は宣言済み。
template <typename PATH> list<PATH> E( const int& i )
{
  // list<PATH> answer{};
  list<PATH> answer = e<PATH>[i];
  // VVV 入力によらない処理は以下に挿入する。

  // AAA 入力によらない処理は以上に挿入する。
  return answer;
}













// 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()
{
  // // 大きな素数
  // CEXPR( ll , P , 998244353 );
  // // CEXPR( ll , P , 1000000007 ); // Mod<P>を使う時はP2に変更。

  // データ構造使用畤のNの上限
  DEXPR( int , bound_N , 5000 , 100 ); // 0が3個
  // DEXPR( int , bound_N , 1000000000 , 100 ); // 0が9個
  // DEXPR( ll , bound_N , 1000000000000000000 , 100 ); // 0が18個

  // // データ構造使用畤のMの上限
  // // CEXPR( TYPE_OF( bound_N ) , bound_M , bound_N );
  // DEXPR( int , bound_M , 100000 , 100 ); // 0が5個
  // // DEXPR( int , bound_M , 1000000000 , 100 ); // 0が9個
  // // DEXPR( ll , bound_M , 1000000000000000000 , 100 ); // 0が18個

  // // 数
  // CIN( ll , N );
  // CIN( ll , M );
  CIN( ll , N , L , R );
  // // CIN_ASSERT( N , 1 , bound_N ); // ランダムテスト用。上限のデフォルト値は10^5。
  // // CIN_ASSERT( M , 1 , bound_M ); // ランダムテスト用。上限のデフォルト値は10^5。

  // // 文字列
  // CIN( string , S );
  // CIN( string , T );

  // // 配列
  CIN_A( ll , A , N );
  // // CIN_A( ll , B , N );
  // // ll A[N];
  // // ll B[N];
  // // ll A[bound_N]; // 関数（コンストラクタ）の引数に使う。長さのデフォルト値は10^5。
  // // ll B[bound_N]; // 関数（コンストラクタ）の引数に使う。長さのデフォルト値は10^5。
  // // FOR( i , 0 , N ){
  // //   cin >> A[i] >> B[i];
  // // }
  sort( A , A + N );
  CERR_A( A , N );
  ll last_minus = -1;
  ll count_zero = 0;
  ll first_plus = N;
  FOR( i , 0 , N ){
    if( A[i] < 0 ){
      last_minus = i;
    } else if ( A[i] == 0 ){
      count_zero++;
    } else {
      first_plus = i;
      break;
    }
  }
  ll answer = 1;
  if( R < 0 ){
    FOR( i , first_plus , N ){
      FOREQ( j , 0 , last_minus ){
	if( L <= A[i] * A[j] && A[i] * A[j] <= R ){
	  RETURN( 2 );
	}
      }
    }
  } else if( R == 0 ){
    if( count_zero != 0 ){
      if( last_minus != -1 ){
	answer = max( answer , 1 + count_zero );
      }
    }
    FOR( i , first_plus , N ){
      ll temp = count_zero + 1;
      FOREQ( j , 0 , last_minus ){
	if( L <= A[i] * A[j] && A[i] * A[j] <= R ){
	  temp++;
	  break;
	}
      }
      answer = max( answer , temp );
    }
  } else {
    CoordinateCompress<ExtendedRational<ll> > cc{};
    FOR( i , 0 , N ){
      cc.insert( A[i] );
      if( A[i] != 0 ){
	cc.insert( ExtendedRational<ll>( L , A[i] ) );
	cc.insert( ExtendedRational<ll>( R , A[i] ) );
      }
    }
    BIT<ll,bound_N*3> bit{};
    FOR( i , 0 , N ){
      bit.Add( cc.GetOrder( A[i] ) , 1 );
    }
    int N_minus = N - 1;
    FOR( i , 0 , N_minus ){
      if( A[i] != 0 || ( L <= 0 && 0 <= R ) ){
	ExtendedRational<ll> mi = A[i];
	ExtendedRational<ll> Mi = A[N_minus];
	if( A[i] < 0 ){
	  mi = max( mi , ExtendedRational<ll>( R , A[i] ) );
	  Mi = min( Mi , ExtendedRational<ll>( L , A[i] ) );
	} else if( A[i] > 0 ){
	  mi = max( mi , ExtendedRational<ll>( L , A[i] ) );
	  Mi = min( Mi , ExtendedRational<ll>( R , A[i] ) );
	}
	if( mi <= Mi ){
	  FOR( j , i + 1 , N ){
	    if( mi <= A[j] ){
	      if( A[j] <= Mi ){
		ExtendedRational<ll> mj = mi;
		ExtendedRational<ll> Mj = A[j];
		if( A[j] < 0 ){
		  mj = max( mj , ExtendedRational<ll>( R , A[j] ) );
		  Mj = min( Mj , ExtendedRational<ll>( L , A[j] ) );
		} else if( A[j] > 0 ){
		  mj = max( mj , ExtendedRational<ll>( L , A[j] ) );
		  Mj = min( Mj , ExtendedRational<ll>( R , A[j] ) );
		}
		if( mj <= Mj ){
		  answer = max( answer , bit.IntervalSum( cc.GetOrder( mj ) , cc.GetOrder( Mj ) ) + ( mj == A[i] ? 0 : 1 ) + ( Mj == A[j] ? 0 : 1 ) );
		  CERR( i , j , A[i] , A[j] , mj , Mj ,  answer );
		}
	      } else {
		break;
	      }
	    }
	  }
	}
      }
    }
  }

  // // 順列
  // int P[N];
  // int P_inv[N];
  // FOR( i , 0 , N ){
  //   cin >> P[i];
  //   P_inv[--P[i]] = i;
  // }
  
  // // グラフ
  // e<int>.resize( N );
  // // e<path>.resize( N );
  // FOR( j , 0 , M ){
  //   CIN_ASSERT( uj , 1 , N );
  //   CIN_ASSERT( vj , 1 , N );
  //   uj--;
  //   vj--;
  //   e<int>[uj].push_back( vj );
  //   e<int>[vj].push_back( uj );
  //   // CIN( ll , wj );
  //   // e<path>[uj].push_back( { vj , wj } );
  //   // e<path>[vj].push_back( { uj , wj } );
  // }

  // // 座標圧縮や単一クエリタイプなどのための入力格納
  // T3<ll> data[M];
  // FOR( j , 0 , M ){
  //   CIN( ll , x , y , z );
  //   data[j] = { x , y , z };
  // }

  // // 一般のクエリ
  // CIN( int , Q );
  // // DEXPR( int , bound_Q , 100000 , 100 ); // 基本不要。
  // // CIN_ASSERT( Q , 1 , bound_Q ); // 基本不要。
  // // T3<int> query[Q];
  // // T2<int> query[Q];
  // FOR( q , 0 , Q ){
  //   CIN( int , type );
  //   if( type == 1 ){
  //     CIN( int , x , y );
  //     // query[q] = { type , x , y };
  //   } else if( type == 2 ){
  //     CIN( int , x , y );
  //     // query[q] = { type , x , y };
  //   } else {
  //     CIN( int , x , y );
  //     // query[q] = { type , x , y };
  //   }
  //   // CIN( int , x , y );
  //   // // query[q] = { x , y };
  // }
  // // sort( query , query + Q );
  // // FOR( q , 0 , Q ){
  // //   auto& [x,y] = query[q];
  // //   // auto& [type,x,y] = query[q];
  // // }
  
  // // データ構造や壁配列使用畤のH,Wの上限
  // DEXPR( int , bound_H , 2000 , 30 );
  // // DEXPR( int , bound_H , 100000 , 10 ); // 0が5個
  // // CEXPR( int , bound_H , 1000000000 ); // 0が9個
  // CEXPR( int , bound_W , bound_H );
  // static_assert( ll( bound_H ) * bound_W < ll( 1 ) << 31 );
  // CEXPR( int , bound_HW , bound_H * bound_W );
  // // CEXPR( int , bound_HW , 100000 ); // 0が5個
  // // CEXPR( int , bound_HW , 1000000 ); // 0が6個

  // // グリッド
  // cin >> H >> W;
  // // SET_ASSERT( H , 1 , bound_H ); // ランダムテスト用。上限のデフォルト値は2*10^3。
  // // SET_ASSERT( W , 1 , bound_W ); // ランダムテスト用。上限のデフォルト値は2*10^3。
  // H_minus = H - 1;
  // W_minus = W - 1;
  // HW = H * W;
  // // assert( HW <= bound_HW ); // 基本不要。上限のデフォルト値は4*10^6。
  // string S[H];
  // FOR( i , 0 , H ){
  //   cin >> S[i];
  //   // SetEdgeOnGrid( S[i] , i , e<int> );
  //   // SetWallOnGrid( S[i] , i , non_wall );
  // }
  // // {h,w}へデコード: EnumHW( v )
  // // {h,w}をコード: EnumHW_inv( h , w );
  // // (i,j)->(k,h)の方向番号を取得: DirectionNumberOnGrid( i , j , k , h );
  // // v->wの方向番号を取得: DirectionNumberOnGrid( v , w );
  // // 方向番号の反転U<->D、R<->L: ReverseDirectionNumberOnGrid( n );

  // auto answer = Answer( N , M , K );
  RETURN( answer );
  // // COUT( answer );
  // // COUT_A( A , N );
}

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);