// 入力制約／フォーマットチェック
#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  #define USE_GETLINE
#endif
#ifdef INCLUDE_MAIN

void Solve()
{
  CEXPR( int , bound_HW , 25 );
  CEXPR( ll , bound_K , 1e18 );
  CEXPR( ll , bound_M , ll(1e9) + 7 );
  GETLINE_COUNT( HWKM_str , 4 , ' ' );
  STOI( HWKM_str , H , 1 , bound_HW );
  STOI( HWKM_str , W , 1 , bound_HW );
  STOI( HWKM_str , K , 0 , bound_K );
  STOI( HWKM_str , M , ll(1e8) , bound_M );
  int HW = H * W; assert( HW <= bound_HW );
  FOREQ( d , 2 , 31623 ){
    assert( M % d != 0 );
  }
  vector count( 6 , vector( 26 , vector<int>() ) );
  count[ 1 ][ 1 ] = { 0, 1 };
  count[ 1 ][ 2 ] = { 0, 2, 1 };
  count[ 1 ][ 3 ] = { 0, 5, 2, 1 };
  count[ 1 ][ 4 ] = { 0, 12, 5, 2, 1 };
  count[ 1 ][ 5 ] = { 0, 28, 12, 5, 2, 1 };
  count[ 1 ][ 6 ] = { 0, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 7 ] = { 0, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 8 ] = { 0, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 9 ] = { 0, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 10 ] = { 0, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 11 ] = { 0, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 12 ] = { 0, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 13 ] = { 0, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 14 ] = { 0, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 15 ] = { 0, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 16 ] = { 0, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 17 ] = { 0, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 18 ] = { 0, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 19 ] = { 0, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 20 ] = { 0, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 21 ] = { 0, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 22 ] = { 0, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 23 ] = { 0, 26214400, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 24 ] = { 0, 54525952, 26214400, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 1 ][ 25 ] = { 0, 113246208, 54525952, 26214400, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 };
  count[ 2 ][ 2 ] = { 0, 8, 4, 4, 1 };
  count[ 2 ][ 3 ] = { 0, 40, 17, 14, 10, 6, 1 };
  count[ 2 ][ 4 ] = { 0, 192, 80, 64, 39, 28, 20, 8, 1 };
  count[ 2 ][ 5 ] = { 0, 896, 368, 296, 184, 120, 81, 58, 34, 10, 1 };
  count[ 2 ][ 6 ] = { 0, 4096, 1664, 1344, 856, 560, 368, 240, 167, 112, 52, 12, 1 };
  count[ 2 ][ 7 ] = { 0, 18432, 7424, 6016, 3904, 2568, 1712, 1112, 728, 488, 337, 198, 74, 14, 1 };
  count[ 2 ][ 8 ] = { 0, 81920, 32768, 26624, 17536, 11584, 7816, 5136, 3384, 2208, 1456, 992, 647, 324, 100, 16, 1 };
  count[ 2 ][ 9 ] = { 0, 360448, 143360, 116736, 77824, 51584, 35136, 23304, 15520, 10232, 6704, 4392, 2936, 1976, 1169, 498, 130, 18, 1 };
  count[ 2 ][ 10 ] = { 0, 1572864, 622592, 507904, 342016, 227328, 156032, 104256, 70024, 46640, 30888, 20320, 13304, 8784, 5904, 3792, 1991, 728, 164, 20, 1 };
  count[ 2 ][ 11 ] = { 0, 6815744, 2686976, 2195456, 1490944, 993280, 686080, 461184, 311872, 209416, 139968, 93048, 61440, 40328, 26480, 17624, 11672, 6952, 3217, 1022, 202, 22, 1 };
  count[ 2 ][ 12 ] = { 0, 29360128, 11534336, 9437184, 6455296, 4308992, 2992128, 2021376, 1374592, 929088, 625544, 419408, 279656, 185376, 122088, 80112, 52888, 35200, 22416, 12160, 4967, 1388, 244, 24, 1 };
  count[ 3 ][ 3 ] = { 0, 400, 144, 106, 84, 69, 52, 32, 9, 1 };
  count[ 3 ][ 4 ] = { 0, 3840, 1360, 952, 705, 550, 445, 360, 270, 164, 62, 12, 1 };
  count[ 3 ][ 5 ] = { 0, 35840, 12544, 8720, 6408, 4897, 3772, 2972, 2371, 1873, 1385, 854, 376, 101, 15, 1 };
  count[ 3 ][ 6 ] = { 0, 327680, 113664, 78592, 57936, 44456, 34161, 26334, 20329, 15846, 12510, 9758, 7131, 4478, 2164, 714, 149, 18, 1 };
  count[ 3 ][ 7 ] = { 0, 2949120, 1015808, 699392, 516864, 398928, 308296, 238385, 183352, 140476, 108244, 84130, 65926, 50898, 36946, 23558, 12123, 4602, 1205, 206, 21, 1 };
  count[ 3 ][ 8 ] = { 0, 26214400, 8978432, 6160384, 4561920, 3537664, 2748496, 2136040, 1652625, 1269794, 973997, 746362, 574842, 446028, 347449, 266040, 192322, 124150, 66909, 28072, 8654, 1876, 272, 24, 1 };
  count[ 4 ][ 4 ] = { 0, 73728, 25600, 17152, 11728, 9040, 7504, 6288, 5340, 4436, 3432, 2360, 1296, 492, 116, 16, 1 };
  count[ 4 ][ 5 ] = { 0, 1376256, 471040, 314368, 213248, 158976, 128272, 105936, 89480, 75998, 64363, 53486, 42473, 31078, 19942, 10358, 3972, 1054, 186, 20, 1 };
  count[ 4 ][ 6 ] = { 0, 25165824, 8519680, 5668864, 3856384, 2863104, 2288896, 1868032, 1553240, 1299184, 1090650, 919360, 774295, 643000, 517972, 394544, 274893, 167464, 83894, 32492, 9316, 1920, 272, 24, 1 };
  count[ 5 ][ 5 ] = { 0, 51380224, 17301504, 11501568, 7749632, 5599232, 4361216, 3564416, 3023760, 2603088, 2265120, 1971974, 1703398, 1448460, 1194986, 939036, 687581, 454452, 258314, 119182, 42437, 11295, 2192, 296, 25, 1 };
  if( H > W ){
    swap( H , W );
  }
  DynamicMod::SetModulo( M );
  vector<DynamicMod> power( HW + 1 );
  FOREQ( c , 0 , HW ){
    power[c] = Power( DynamicMod::Derepresent( c ) , K );
  }
  DynamicMod power_sum{} , coef{};
  FOREQ( c , 1 , HW ){
    power_sum += power[c] * count[H][W][c];
  }
  FOREQ( n , 1 , HW ){
    coef += DynamicMod::one() / ( HW - n + 1 );
  }
  RETURN( power_sum * coef * Power( DynamicMod::Derepresent( 2 ) , -HW ) );
}
REPEAT_MAIN(1);

#else // INCLUDE_MAIN
#ifdef INCLUDE_LIBRARY

// https://github.com/p-adic/cpp
// VVV ライブラリは以下に挿入する。redefinitionを避けるため圧縮元はincludeしない。

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

#define 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 <int NUM> CL DynamicMods;TE <int NUM>CL COantsForDynamicMods{PU:COantsForDynamicMods()= delete;ST uint g_M;ST uint g_M_minus;ST int g_order;ST int g_order_minus_1;ST int g_order_minus_1_neg;ST bool g_M_is_prime;};
TE <int NUM> uint COantsForDynamicMods<NUM>::g_M = 0;TE <int NUM> uint COantsForDynamicMods<NUM>::g_M_minus = -1;TE <int NUM> int COantsForDynamicMods<NUM>::g_order = 1;TE <int NUM> int COantsForDynamicMods<NUM>::g_order_minus_1 = 0;TE <int NUM> int COantsForDynamicMods<NUM>::g_order_minus_1_neg = 0;TE <int NUM> bool COantsForDynamicMods<NUM>::g_M_is_prime = false;

#define DC_OF_CM_FOR_DYNAMIC_MOD(OPR)IN bool OP OPR(CO DynamicMods<NUM>& n)CO NE
#define DC_OF_AR_FOR_DYNAMIC_MOD(OPR,EX)IN DynamicMods<NUM> OP OPR(DynamicMods<NUM> n)CO EX;
#define DF_OF_CM_FOR_DYNAMIC_MOD(OPR)TE <int NUM> IN bool DynamicMods<NUM>::OP OPR(CO DynamicMods<NUM>& n)CO NE{RE m_n OPR n.m_n;}
#define DF_OF_AR_FOR_DYNAMIC_MOD(OPR,EX,LEFT,OPR2)TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP OPR(DynamicMods<NUM> n)CO EX{RE MO(LEFT OPR2 ## = *TH);}TE <int NUM,TY T> IN DynamicMods<NUM> OP OPR(T n0,CO DynamicMods<NUM>& n1)EX{RE MO(DynamicMods<NUM>(MO(n0))OPR ## = n1);}
TE <int NUM>CL DynamicMods{PU:uint m_n;IN DynamicMods()NE;IN DynamicMods(CO DynamicMods<NUM>& n)NE;IN DynamicMods(DynamicMods<NUM>&& n)NE;TE <TY T> IN DynamicMods(T n)NE;IN DynamicMods<NUM>& OP=(DynamicMods<NUM> n)NE;IN DynamicMods<NUM>& OP+=(CO DynamicMods<NUM>& n)NE;IN DynamicMods<NUM>& OP-=(CO DynamicMods<NUM>& n)NE;IN DynamicMods<NUM>& OP*=(CO DynamicMods<NUM>& n)NE;IN DynamicMods<NUM>& OP/=(DynamicMods<NUM> n);TE <TY INT> IN DynamicMods<NUM>& OP<<=(INT n);TE <TY INT> IN DynamicMods<NUM>& OP>>=(INT n);IN DynamicMods<NUM>& OP++()NE;IN DynamicMods<NUM> OP++(int)NE;IN DynamicMods<NUM>& OP--()NE;IN DynamicMods<NUM> OP--(int)NE;DC_OF_CM_FOR_DYNAMIC_MOD(==);DC_OF_CM_FOR_DYNAMIC_MOD(!=);DC_OF_CM_FOR_DYNAMIC_MOD(<);DC_OF_CM_FOR_DYNAMIC_MOD(<=);DC_OF_CM_FOR_DYNAMIC_MOD(>);DC_OF_CM_FOR_DYNAMIC_MOD(>=);DC_OF_AR_FOR_DYNAMIC_MOD(+,NE);DC_OF_AR_FOR_DYNAMIC_MOD(-,NE);DC_OF_AR_FOR_DYNAMIC_MOD(*,NE);DC_OF_AR_FOR_DYNAMIC_MOD(/,);TE <TY INT> IN DynamicMods<NUM> OP^(INT EX)CO;TE <TY INT> IN DynamicMods<NUM> OP<<(INT n)CO;TE <TY INT> IN DynamicMods<NUM> OP>>(INT n)CO;IN DynamicMods<NUM> OP-()CO NE;IN DynamicMods<NUM>& SignInvert()NE;IN DynamicMods<NUM>& Invert();TE <TY INT> IN DynamicMods<NUM>& PW(INT EX);IN VO swap(DynamicMods<NUM>& n)NE;IN CRUI RP()CO NE;ST IN DynamicMods<NUM> DeRP(uint n)NE;ST IN CO DynamicMods<NUM>& Inverse(CRUI n);ST IN CO DynamicMods<NUM>& Factorial(CRUI n);ST IN CO DynamicMods<NUM>& FactorialInverse(CRUI n);ST IN DynamicMods<NUM> Combination(CRUI n,CRUI i);ST IN CO DynamicMods<NUM>& zero()NE;ST IN CO DynamicMods<NUM>& one()NE;ST IN CRUI GetModulo()NE;ST IN VO SetModulo(CRUI M,CRI order_minus_1 = -1)NE;TE <TY INT> IN DynamicMods<NUM>& PositivePW(INT EX)NE;TE <TY INT> IN DynamicMods<NUM>& NonNegativePW(INT EX)NE;US COants = COantsForDynamicMods<NUM>;};
US DynamicMod = DynamicMods<0>;
TE <int NUM> IN DynamicMods<NUM>::DynamicMods()NE:m_n(){}TE <int NUM> IN DynamicMods<NUM>::DynamicMods(CO DynamicMods<NUM>& n)NE:m_n(n.m_n){}TE <int NUM> IN DynamicMods<NUM>::DynamicMods(DynamicMods<NUM>&& n)NE:m_n(MO(n.m_n)){}TE <int NUM> TE <TY T> IN DynamicMods<NUM>::DynamicMods(T n)NE:m_n(Residue(uint(MO(n)),COants::g_M)){ST_AS(is_COructible_v<uint,decay_t<T> >);}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP=(DynamicMods<NUM> n)NE{m_n = MO(n.m_n);RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP+=(CO DynamicMods<NUM>& n)NE{(m_n += n.m_n)< COants::g_M?m_n:m_n -= COants::g_M;RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP-=(CO DynamicMods<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 DynamicMods<NUM>& DynamicMods<NUM>::OP*=(CO DynamicMods<NUM>& n)NE{m_n = Residue(MO(ull(m_n)* n.m_n),COants::g_M);RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP/=(DynamicMods<NUM> n){RE OP*=(n.Invert());}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::OP<<=(INT n){AS(n >= 0);RE *TH *= DynamicMods<NUM>(2).NonNegativePW(MO(n));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::OP>>=(INT n){AS(n >= 0);WH(n-- > 0){((m_n & 1)== 0?m_n:m_n += COants::g_M)>>= 1;}RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP++()NE{m_n < COants::g_M_minus?++m_n:m_n = 0;RE *TH;}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP++(int)NE{DynamicMods<NUM> n{*TH};OP++();RE n;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP--()NE{m_n == 0?m_n = COants::g_M_minus:--m_n;RE *TH;}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP--(int)NE{DynamicMods<NUM> n{*TH};OP--();RE n;}DF_OF_CM_FOR_DYNAMIC_MOD(==);DF_OF_CM_FOR_DYNAMIC_MOD(!=);DF_OF_CM_FOR_DYNAMIC_MOD(>);DF_OF_CM_FOR_DYNAMIC_MOD(>=);DF_OF_CM_FOR_DYNAMIC_MOD(<);DF_OF_CM_FOR_DYNAMIC_MOD(<=);DF_OF_AR_FOR_DYNAMIC_MOD(+,NE,n,+);DF_OF_AR_FOR_DYNAMIC_MOD(-,NE,n.SignInvert(),+);DF_OF_AR_FOR_DYNAMIC_MOD(*,NE,n,*);DF_OF_AR_FOR_DYNAMIC_MOD(/,,n.Invert(),*);TE <int NUM> TE <TY INT> IN DynamicMods<NUM> DynamicMods<NUM>::OP^(INT EX)CO{RE MO(DynamicMods<NUM>(*TH).PW(MO(EX)));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM> DynamicMods<NUM>::OP<<(INT n)CO{RE MO(DynamicMods<NUM>(*TH)<<= MO(n));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM> DynamicMods<NUM>::OP>>(INT n)CO{RE MO(DynamicMods<NUM>(*TH)>>= MO(n));}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP-()CO NE{RE MO(DynamicMods<NUM>(*TH).SignInvert());}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::SignInvert()NE{m_n > 0?m_n = COants::g_M - m_n:m_n;RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::Invert(){RE m_n <(COants::g_M_is_prime?1e6:3e4)?*TH = Inverse(m_n):NonNegativePW(COants::g_order_minus_1);}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::PositivePW(INT EX)NE{DynamicMods<NUM> PW{*TH};EX--;WH(EX != 0){(EX & 1)== 1?*TH *= PW:*TH;EX >>= 1;PW *= PW;}RE *TH;}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::NonNegativePW(INT EX)NE{RE EX == 0?(m_n = 1,*TH):PositivePW(MO(EX));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::PW(INT EX){bool neg = EX < 0;RE neg?PositivePW(ll(MO(EX %= COants::g_order))* COants::g_order_minus_1_neg % COants::g_order):NonNegativePW(MO(EX));}TE <int NUM> IN VO DynamicMods<NUM>::swap(DynamicMods<NUM>& n)NE{std::swap(m_n,n.m_n);}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::Inverse(CRUI n){ST VE<DynamicMods<NUM>> memory ={zero(),one()};ST uint LE_curr = 2;AS(COants::g_M == 1||(0 < n && n < COants::g_M));WH(LE_curr <= n){memory.push_back(COants::g_M_is_prime?DeRP(COants::g_M - memory[COants::g_M % LE_curr].m_n * ull(COants::g_M / LE_curr)% COants::g_M):DeRP(n).NonNegativePW(COants::g_order_minus_1));LE_curr++;}RE memory[n];}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::Factorial(CRUI n){ST VE<DynamicMods<NUM>> memory ={one(),one()};ST uint LE_curr = 2;if(COants::g_M <= n){RE zero();}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 DynamicMods<NUM>& DynamicMods<NUM>::FactorialInverse(CRUI n){ST VE<DynamicMods<NUM>> memory ={one(),one()};ST uint 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 DynamicMods<NUM> DynamicMods<NUM>::Combination(CRUI n,CRUI i){RE i <= n?Factorial(n)* FactorialInverse(i)* FactorialInverse(n - i):zero();}TE <int NUM> IN CRUI DynamicMods<NUM>::RP()CO NE{RE m_n;}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::DeRP(uint n)NE{DynamicMods<NUM> n_copy{};n_copy.m_n = MO(n);RE n_copy;}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::zero()NE{ST CO DynamicMods<NUM> z{};RE z;}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::one()NE{ST CO DynamicMods<NUM> o{1};RE o;}TE <int NUM> IN CRUI DynamicMods<NUM>::GetModulo()NE{RE COants::g_M;}TE <int NUM> IN VO DynamicMods<NUM>::SetModulo(CRUI M,CRI order_minus_1)NE{COants::g_M = M;COants::g_M_minus = M - 1;COants::g_order = order_minus_1 == -1?M - 1:order_minus_1+1;COants::g_order_minus_1 = COants::g_order-1;COants::g_order_minus_1_neg = -COants::g_order_minus_1;COants::g_M_is_prime = order_minus_1 == -1;}TE <int NUM> IN DynamicMods<NUM> Inverse(CO DynamicMods<NUM>& n){RE MO(DynamicMods<NUM>(n).Invert());}TE <int NUM,TY INT> IN DynamicMods<NUM> PW(DynamicMods<NUM> n,INT EX){RE MO(n.PW(MO(EX)));}TE <int NUM> IN VO swap(DynamicMods<NUM>& n0,DynamicMods<NUM>& n1)NE{n0.swap(n1);}TE <int NUM> IN string to_string(CO DynamicMods<NUM>& n)NE{RE to_string(n.RP())+ " + " + to_string(DynamicMods<NUM>::GetModulo())+ "Z";}TE <int NUM,CL Traits> IN IS& OP>>(IS& is,DynamicMods<NUM>& n){ll m;is >> m;n = m;RE is;}TE <int NUM,CL Traits> IN OS& OP<<(OS& os,CO DynamicMods<NUM>& n){RE os << n.RP();}

TE <TY INT1,TY INT2>INT1 GCD(CO INT1& b_0,CO INT2& b_1){INT1 a_0 = b_0 < 0?-b_0:b_0;INT1 a_1 = b_1 < 0?-b_1:b_1;WH(a_1 != 0){swap(a_0 %= a_1,a_1);}RE a_0;}TE <TY INT1,TY INT2> IN INT1 LCM(CO INT1& b_0,CO INT2& b_1){RE(b_0 == 0 && b_1 == 0)?0:(b_0 / GCD(b_0,b_1))* b_1;}

// AAA ライブラリは以上に挿入する。

#define INCLUDE_MAIN
#include __FILE__
#else // INCLUDE_LIBRARY
#ifdef DEBUG
  #define _GLIBCXX_DEBUG
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE2 )
  #define SIGNAL signal( SIGABRT , &AlertAbort );
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); assert( ( MIN ) <= A && A <= ( MAX ) )
  #define COUT( ... ) VariadicCout( cout << "出力：" , __VA_ARGS__ ) << endl
  #define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ ) << flush
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define CERRNS( ... ) VariadicCout( cerr , __VA_ARGS__ ) << flush
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout << "出力："  , A , N ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << endl
  int exec_mode = 0;
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define SIGNAL 
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) assert( ( MIN ) <= A && A <= ( MAX ) )
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ )
  #define CERR( ... ) 
  #define CERRNS( ... ) 
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout , A , N ) << ENDL
  #define CERR_A( A , N ) 
#endif
#ifdef REACTIVE
  #ifdef DEBUG
    #define RSET( A , ... ) A = __VA_ARGS__
  #else
    #define RSET( A , ... ) cin >> A
  #endif
  #define RCIN( LL , A , ... ) LL A; RSET( A , __VA_ARGS__ )
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#ifdef USE_GETLINE
  #define SET_LL( A ) { GETLINE( A ## _str ); A = stoll( A ## _str ); }
  #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
#else
  #define SET_LL( A ) cin >> A
  #define CIN( LL , ... ) LL __VA_ARGS__; VariadicCin( cin , __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 , ... ) vector<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
  #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) vector<vector<LL>> VAR( N0 + I0 ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , VAR[VARIABLE_FOR_CIN_AA + I0] ); }
#endif
#include <bits/stdc++.h>
using namespace std;
#define REPEAT_MAIN( BOUND ) int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr ); SIGNAL; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if constexpr( bound_test_case_num > 1 ){ CERR( "テストケースの個数を入力してください。" ); SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } REPEAT( test_case_num ){ if constexpr( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } Solve(); CERR( "" ); } CHECK_REDUNDANT_INPUT; }
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now()
#define CURRENT_TIME static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME < TL_MS - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) constexpr LL BOUND = VALUE
#define SET_ASSERT( A , MIN , MAX ) SET_LL( A ); ASSERT( A , MIN , MAX )
#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 FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( decldecay_t( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( decldecay_t( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( decldecay_t( INITIAL ) VAR = INITIAL ; VAR + 1 > FINAL ; VAR -- )
#define 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_ ## HOW_MANY_TIMES , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS )
#define RETURN( ... ) COUT( __VA_ARGS__ ); return

// 型のエイリアス
#define decldecay_t( VAR ) decay_t<decltype( VAR )>
template <typename F , typename...Args> using ret_t = decltype( declval<F>()( declval<Args>()... ) );
template <typename T> using inner_t = typename T::type;
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>;

/* VVV 常設ライブラリの非圧縮版は以下に挿入する。*/
// Random
ll GetRand( const int& Rand_min , const int& Rand_max ) { assert( Rand_min <= Rand_max ); ll answer = time( NULL ); return answer * rand() % ( Rand_max + 1 - Rand_min ) + Rand_min; }

// Set
#define DECLARATION_OF_HASH( ... )				\
  struct hash<__VA_ARGS__>					\
  {								\
								\
    inline size_t operator()( const __VA_ARGS__& n ) const;	\
								\
  };								\

class is_ordered
{

private:
  is_ordered() = delete;
  template <typename T> static constexpr auto Check( const T& t ) -> decltype( t < t , true_type() );
  static constexpr false_type Check( ... );

public:
  template <typename T> static constexpr const bool value = is_same_v< decltype( Check( declval<T>() ) ) , true_type >;

};

template <typename T>
using Set = conditional_t<is_constructible_v<unordered_set<T>>,unordered_set<T>,conditional_t<is_ordered::value<T>,set<T>,void>>;

// Tuple
#define DECLARATION_OF_ARITHMETIC_FOR_TUPLE( OPR )			\
  template <typename T , typename U , template <typename...> typename V> inline auto operator OPR ## =( V<T,U>& t0 , const V<T,U>& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )&; \
  template <typename T , typename U , typename V> inline tuple<T,U,V>& operator OPR ## =( tuple<T,U,V>& t0 , const tuple<T,U,V>& t1 ); \
  template <typename T , typename U , typename V , typename W> inline tuple<T,U,V,W>& operator OPR ## =( tuple<T,U,V,W>& t0 , const tuple<T,U,V,W>& t1 ); \
  template <typename ARG , typename T , typename U , template <typename...> typename V> inline auto operator OPR ## =( V<T,U>& t0 , const ARG& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )&; \
  template <typename ARG , typename T , typename U , typename V> inline tuple<T,U,V>& operator OPR ## =( tuple<T,U,V>& t0 , const ARG& t1 ); \
  template <typename ARG , typename T , typename U , typename V , typename W> inline tuple<T,U,V,W>& operator OPR ## =( tuple<T,U,V,W>& t0 , const ARG& t1 ); \
  template <template <typename...> typename V , typename...ARGS , typename ARG> inline auto operator OPR( const V<ARGS...>& t0 , const ARG& t1 ) -> decldecay_t( ( get<0>( t0 ) , t0 ) ) \

#define DEFINITION_OF_ARITHMETIC_FOR_TUPLE( OPR )			\
  template <typename T , typename U , template <typename...> typename V> inline auto operator OPR ## =( V<T,U>& t0 , const V<T,U>& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = get<0>( t1 ); get<1>( t0 ) OPR ## = get<1>( t1 ); return t0; } \
  template <typename T , typename U , typename V> inline tuple<T,U,V>& operator OPR ## =( tuple<T,U,V>& t0 , const tuple<T,U,V>& t1 ) { get<0>( t0 ) OPR ## = get<0>( t1 ); get<1>( t0 ) OPR ## = get<1>( t1 ); get<2>( t0 ) OPR ## = get<2>( t1 ); return t0; } \
  template <typename T , typename U , typename V , typename W> inline tuple<T,U,V,W>& operator OPR ## =( tuple<T,U,V,W>& t0 , const tuple<T,U,V,W>& t1 ) { 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 ); return t0; } \
  template <typename ARG , typename T , typename U , template <typename...> typename V> inline auto operator OPR ## =( V<T,U>& t0 , const ARG& t1 ) -> decltype( ( get<0>( t0 ) , t0 ) )& { get<0>( t0 ) OPR ## = t1; get<1>( t0 ) OPR ## = t1; return t0; } \
  template <typename ARG , typename T , typename U , typename V> inline tuple<T,U,V>& operator OPR ## =( tuple<T,U,V>& t0 , const ARG& t1 ) { get<0>( t0 ) OPR ## = t1; get<1>( t0 ) OPR ## = t1; get<2>( t0 ) OPR ## = t1; return t0; } \
  template <typename ARG , typename T , typename U , typename V , typename W> inline tuple<T,U,V,W>& operator OPR ## =( tuple<T,U,V,W>& t0 , const ARG& t1 ) { get<0>( t0 ) OPR ## = t1; get<1>( t0 ) OPR ## = t1; get<2>( t0 ) OPR ## = t1; get<3>( t0 ) OPR ## = t1; return t0; } \
  template <template <typename...> typename V , typename...ARGS , typename ARG> inline auto operator OPR( const V<ARGS...>& t0 , const ARG& t1 ) -> decldecay_t( ( get<0>( t0 ) , t0 ) ) { auto t = t0; return move( t OPR ## = t1 ); } \

#define DECLARATION_OF_INCREMENT_FOR_TUPLE( INCR )			\
  template <typename T , typename U , template <typename...> typename V> inline auto operator INCR( V<T,U>& t ) -> decltype( ( get<0>( t ) , t ) )&; \
  template <typename T , typename U , typename V> inline tuple<T,U,V>& operator INCR ( tuple<T,U,V>& t ); \
  template <typename T , typename U , typename V , typename W> inline tuple<T,U,V,W>& operator INCR ( tuple<T,U,V,W>& t ); \

#define DEFINITION_OF_INCREMENT_FOR_TUPLE( INCR )			\
  template <typename T , typename U , template <typename...> typename V> inline auto operator INCR( V<T,U>& t ) -> decltype( ( get<0>( t ) , t ) )& { INCR get<0>( t ); INCR get<1>( t ); return t; } \
  template <typename T , typename U , typename V> inline tuple<T,U,V>& operator INCR ( tuple<T,U,V>& t ) { INCR get<0>( t ); INCR get<1>( t ); INCR get<2>( t ); return t; } \
  template <typename T , typename U , typename V , typename W> inline tuple<T,U,V,W>& operator INCR ( tuple<T,U,V,W>& t ) { INCR get<0>( t ); INCR get<1>( t ); INCR get<2>( t ); INCR get<3>( t ); return t; }	\

DEFINITION_OF_ARITHMETIC_FOR_TUPLE( + );
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( - );
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( * );
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( / );
DEFINITION_OF_ARITHMETIC_FOR_TUPLE( % );

DEFINITION_OF_INCREMENT_FOR_TUPLE( ++ );
DEFINITION_OF_INCREMENT_FOR_TUPLE( -- );

template <class Traits , typename T> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , tuple<T>& arg ){ return is >> get<0>( arg ); }
template <class Traits , typename T , typename U , template <typename...> typename V> inline auto operator>>( basic_istream<char,Traits>& is , V<T,U>& arg ) -> decltype((get<0>(arg),is))& { return is >> get<0>( arg ) >> get<1>( arg ); }
template <class Traits , typename T , typename U , typename V> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , tuple<T,U,V>& arg ) { return is >> get<0>( arg ) >> get<1>( arg ) >> get<2>( arg ); }
template <class Traits , typename T , typename U , typename V , typename W> inline basic_istream<char,Traits>& operator>>( basic_istream<char,Traits>& is , tuple<T,U,V,W>& arg ) { return is >> get<0>( arg ) >> get<1>( arg ) >> get<2>( arg ) >> get<3>( arg ); }

template <class Traits , typename T> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const tuple<T>& arg ) { return os << get<0>( arg ); }
template <class Traits , typename T , typename U , template <typename...> typename V> inline auto operator<<( basic_ostream<char,Traits>& os , const V<T,U>& arg ) -> decltype((get<0>(arg),os))& { return os << get<0>( arg ) << " " << get<1>( arg ); }
template <class Traits , typename T , typename U , typename V> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const tuple<T,U,V>& arg ) { return os << get<0>( arg ) << " " << get<1>( arg ) << " " << get<2>( arg ); }
template <class Traits , typename T , typename U , typename V , typename W> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const tuple<T,U,V,W>& arg ) { return os << get<0>( arg ) << " " << get<1>( arg ) << " " << get<2>( arg ) << " " << get<3>( arg ); }

#define DEFINITION_OF_HASH_FOR_TUPLE( PAIR )				\
  template <typename T , typename U> inline size_t hash<PAIR<T,U>>::operator()( const PAIR<T,U>& n ) const { static const size_t seed = ( GetRand( 1e3 , 1e8 ) << 1 ) | 1; static const hash<T> h0; static const hash<U> h1; return ( h0( get<0>( n ) ) * seed ) ^ h1( get<1>( n ) ); } \

template <typename T> DECLARATION_OF_HASH( tuple<T> );
template <typename T , typename U> DECLARATION_OF_HASH( pair<T,U> );
template <typename T , typename U> DECLARATION_OF_HASH( tuple<T,U> );
template <typename T , typename U , typename V> DECLARATION_OF_HASH( tuple<T,U,V> );
template <typename T , typename U , typename V , typename W> DECLARATION_OF_HASH( tuple<T,U,V,W> );

template <typename T> inline size_t hash<tuple<T>>::operator()( const tuple<T>& n ) const { static const hash<T> h; return h(get<0>( n ) ); }
DEFINITION_OF_HASH_FOR_TUPLE( pair );
DEFINITION_OF_HASH_FOR_TUPLE( tuple );
template <typename T , typename U , typename V> inline size_t hash<tuple<T,U,V>>::operator()( const tuple<T,U,V>& n ) const { static const size_t seed = ( GetRand( 1e3 , 1e8 ) << 1 ) | 1; static const hash<pair<T,U>> h01; static const hash<V> h2; return ( h01( { get<0>( n ) , get<1>( n ) } ) * seed ) ^ h2( get<2>( n ) ); }
template <typename T , typename U , typename V , typename W> inline size_t hash<tuple<T,U,V,W>>::operator()( const tuple<T,U,V,W>& n ) const { static const size_t seed = ( GetRand( 1e3 , 1e8 ) << 1 ) | 1; static const hash<pair<T,U>> h01; static const hash<pair<V,W>> h23; return ( h01( { get<0>( n ) , get<1>( n ) } ) * seed ) ^ h23( { get<2>( n ) , get<3>( n ) } ); }

// Vector
#define DECLARATION_OF_ARITHMETIC_FOR_VECTOR( V , OPR )			\
  template <typename T> inline V<T>& operator OPR ## = ( V<T>& a , const T& t ); \
  template <typename T> inline V<T>& operator OPR ## = ( V<T>& a0 , const V<T>& a1 ); \
  template <typename T , typename U> inline V<T> operator OPR( V<T> a , const U& u ); \

#define DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , OPR )			\
  template <typename T> inline V<T>& operator OPR ## = ( V<T>& a , const T& t ) { for( auto& s : a ){ s OPR ## = t; } return a; } \
  template <typename T> inline V<T>& operator OPR ## = ( V<T>& a0 , const V<T>& a1 ) { assert( a0.size() <= a1.size() ); auto itr0 = a0.begin() , end0 = a0.end(); auto itr1 = a1.begin(); while( itr0 != end0 ){ *( itr0++ ) OPR ## = *( itr1++ ); } return a0; } \
  template <typename T , typename U> inline V<T> operator OPR( V<T> a , const U& u ) { return move( a OPR ## = u ); } \

#define DECLARATION_OF_INCREMENT_FOR_VECTOR( V , INCR )		\
  template <typename T> inline V<T>& operator INCR( V<T>& a );	\

#define DEFINITION_OF_INCREMENT_FOR_VECTOR( V , INCR )			\
  template <typename T> inline V<T>& operator INCR( V<T>& a ) { for( auto& i : a ){ INCR i; } return a; } \

#define DECLARATION_OF_ARITHMETICS_FOR_VECTOR( V )			\
  DECLARATION_OF_ARITHMETIC_FOR_VECTOR( V , + );			\
  DECLARATION_OF_ARITHMETIC_FOR_VECTOR( V , - );			\
  DECLARATION_OF_ARITHMETIC_FOR_VECTOR( V , * );			\
  DECLARATION_OF_ARITHMETIC_FOR_VECTOR( V , / );			\
  DECLARATION_OF_ARITHMETIC_FOR_VECTOR( V , % );			\
  DECLARATION_OF_INCREMENT_FOR_VECTOR( V , ++ );			\
  DECLARATION_OF_INCREMENT_FOR_VECTOR( V , -- );			\
  template <typename T> inline V<T> operator*( const T& scalar , V<T> v ) \

#define DEFINITION_OF_ARITHMETICS_FOR_VECTOR( V )			\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , + );				\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , - );				\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , * );				\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , / );				\
  DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , % );				\
  DEFINITION_OF_INCREMENT_FOR_VECTOR( V , ++ );				\
  DEFINITION_OF_INCREMENT_FOR_VECTOR( V , -- );				\
  template <typename T> inline V<T> operator*( const T& scalar , V<T> v ) { for( auto& t : v ){ t *= scalar; } return move( v ); } \

DEFINITION_OF_ARITHMETICS_FOR_VECTOR( vector );
DEFINITION_OF_ARITHMETICS_FOR_VECTOR( list );

template <typename V> inline auto Get( V& a ) { return [&]( const int& i = 0 ) -> const decldecay_t( a[0] )& { return a[i]; }; }
template <typename T = int> inline vector<T> id( const int& size ) { vector<T> answer( size ); for( int i = 0 ; i < size ; i++ ){ answer[i] = i; } return answer; }

template <typename T> inline void Sort( vector<T>& a , const bool& reversed ) { if( reversed ){ static auto comp = []( const T& t0 , const T& t1 ) { return t1 < t0; }; sort( a.begin() , a.end() , comp ); } else { sort( a.begin() , a.end() ); } }

template <typename T> inline vector<int> IndexSort( const vector<T>& a , const bool& reversed ) { auto index = id<int>( a.size() ); if( reversed ){ sort( index.begin() , index.end() , [&]( const int& i , const int& j ) { return a[j] < a[i]; } ); } else { sort( index.begin() , index.end() , [&]( const int& i , const int& j ) { return a[i] < a[j]; } ); } return index; }

template <typename T , typename U = T> inline U Sum( const vector<T>& a ) { U answer{}; for( auto& x : a ){ answer += x; } return answer; }
template <typename T , typename U = T> inline U Product( const vector<T>& a ) { U answer{}; for( auto& x : a ){ answer *= x; } return answer; }

#define DECLARATION_OF_COUT_FOR_VECTOR( V ) template <class Traits , typename Arg> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const V<Arg>& arg )
#define DEFINITION_OF_COUT_FOR_VECTOR( V ) template <class Traits , typename Arg> inline basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , const V<Arg>& arg ) { auto begin = arg.begin() , end = arg.end(); auto itr = begin; while( itr != end ){ ( itr == begin ? os : os << " " ) << *itr; itr++; } return os; }

DEFINITION_OF_COUT_FOR_VECTOR( vector );
DEFINITION_OF_COUT_FOR_VECTOR( list );
DEFINITION_OF_COUT_FOR_VECTOR( set );
DEFINITION_OF_COUT_FOR_VECTOR( unordered_set );

inline void VariadicResize( const int& size ) {}
template <typename Arg , typename... ARGS> inline void VariadicResize( const int& size , Arg& arg , ARGS&... args ) { arg.resize( size ); VariadicResize( size , args... ); }

// Map
#define DECLARATION_OF_ARITHMETIC_FOR_MAP( MAP , OPR )			\
  template <typename T , typename U> inline MAP<T,U>& operator OPR ## = ( MAP<T,U>& a , const pair<T,U>& v ); \
  template <typename T , typename U> inline MAP<T,U>& operator OPR ## = ( MAP<T,U>& a0 , const MAP<T,U>& a1 ); \
  template <typename T , typename U , typename ARG> inline MAP<T,U> operator OPR( MAP<T,U> a , const ARG& arg ); \

#define DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , OPR )			\
  template <typename T , typename U> inline MAP<T,U>& operator OPR ## = ( MAP<T,U>& a , const pair<T,U>& v ) { a[v.first] OPR ## = v.second; return a; } \
  template <typename T , typename U> inline MAP<T,U>& operator OPR ## = ( MAP<T,U>& a0 , const MAP<T,U>& a1 ) { for( auto& [t,u] : a1 ){ a0[t] OPR ## = u; } return a0; } \
  template <typename T , typename U , typename ARG> inline MAP<T,U> operator OPR( MAP<T,U> a , const ARG& arg ) { return move( a OPR ## = arg ); } \

#define DECLARATION_OF_ARITHMETICS_FOR_MAP( MAP )	\
  DECLARATION_OF_ARITHMETIC_FOR_MAP( MAP , + );		\
  DECLARATION_OF_ARITHMETIC_FOR_MAP( MAP , - );		\
  DECLARATION_OF_ARITHMETIC_FOR_MAP( MAP , * );		\
  DECLARATION_OF_ARITHMETIC_FOR_MAP( MAP , / );		\
  DECLARATION_OF_ARITHMETIC_FOR_MAP( MAP , % );		\

#define DEFINITION_OF_ARITHMETICS_FOR_MAP( MAP ) \
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , + );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , - );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , * );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , / );	\
  DEFINITION_OF_ARITHMETIC_FOR_MAP( MAP , % );	\

template <typename T , typename U>
using Map = conditional_t<is_constructible_v<unordered_map<T,int>>,unordered_map<T,U>,conditional_t<is_ordered::value<T>,map<T,U>,void>>;

DEFINITION_OF_ARITHMETICS_FOR_MAP( map );
DEFINITION_OF_ARITHMETICS_FOR_MAP( unordered_map );

// StdStream
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>& VariadicSet( basic_istream<char,Traits>& is , const int& i ) { return is; }
template <class Traits , typename Arg , typename... ARGS> inline basic_istream<char,Traits>& VariadicSet( basic_istream<char,Traits>& is , const int& i , Arg& arg , ARGS&... args ) { return VariadicSet( is >> arg[i] , i , 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 , Arg&& arg ) { return os << forward<Arg>( arg ); }
template <class Traits , typename Arg1 , typename Arg2 , typename... ARGS> inline basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , Arg1&& arg1 , Arg2&& arg2 , ARGS&&... args ) { return VariadicCout( os << forward<Arg1>( arg1 ) << " " , forward<Arg2>( arg2 ) , forward<ARGS>( args )... ); }

template <class Traits , typename Arg> inline basic_ostream<char,Traits>& VariadicCoutNonSep( basic_ostream<char,Traits>& os , Arg&& arg ) { return os << forward<Arg>( arg ); }
template <class Traits , typename Arg1 , typename Arg2 , typename... ARGS> inline basic_ostream<char,Traits>& VariadicCoutNonSep( basic_ostream<char,Traits>& os , Arg1&& arg1 , Arg2&& arg2 , ARGS&&... args ) { return VariadicCoutNonSep( os << forward<Arg1>( arg1 ) , forward<Arg2>( arg2 ) , forward<ARGS>( args )... ); }

template <class Traits , typename ARRAY> inline basic_ostream<char,Traits>& CoutArray( basic_ostream<char,Traits>& os , const int& i_start , const int& i_ulim , ARRAY&& a ) { for( int i = i_start ; i < i_ulim ; i++ ){ ( i == i_start ? os : ( os << " " ) ) << a[i]; } return os; }
/* AAA 常設ライブラリの非圧縮版は以上に挿入する。*/

// 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。" ); }
#endif

// 入力フォーマットチェック用
// 1行中の変数の個数をSEPARATOR区切りで確認
#define GETLINE_COUNT( S , VARIABLE_NUMBER , SEPARATOR ) GETLINE( S ); int VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S = 0; int VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S  = S.size(); { int size = S.size(); int count = 0; for( int i = 0 ; i < size ; i++ ){ if( S[i] == SEPARATOR ){ count++; } } assert( VARIABLE_NUMBER == 0 ? size == 0 : count + 1 == VARIABLE_NUMBER ); }
// 余計な入力の有無を確認
#if defined( DEBUG ) || defined( REACTIVE )
  #define CHECK_REDUNDANT_INPUT 
#else
  #ifdef USE_GETLINE
    #define CHECK_REDUNDANT_INPUT string VARIABLE_FOR_CHECK_REDUNDANT_INPUT = ""; getline( cin , VARIABLE_FOR_CHECK_REDUNDANT_INPUT ); assert( VARIABLE_FOR_CHECK_REDUNDANT_INPUT == "" ); assert( ! cin )
  #else
    #define CHECK_REDUNDANT_INPUT string VARIABLE_FOR_CHECK_REDUNDANT_INPUT = ""; cin >> VARIABLE_FOR_CHECK_REDUNDANT_INPUT; assert( VARIABLE_FOR_CHECK_REDUNDANT_INPUT == "" ); assert( ! cin )
  #endif
#endif
// MIN <= N <= MAXを満たすNをSから構築
#define STOI( S , N , MIN , MAX ) decldecay_t( MAX ) N = 0; decldecay_t( MAX ) BOUND ## N = max( decldecay_t( MAX )( abs( MIN ) ) , abs( MAX ) ); { bool VARIABLE_FOR_POSITIVITY_FOR_GETLINE = true; assert( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ); if( S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) == "-" ){ VARIABLE_FOR_POSITIVITY_FOR_GETLINE = false; VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; assert( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ); } assert( S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) != " " ); string VARIABLE_FOR_LETTER_FOR_GETLINE{}; int VARIABLE_FOR_DIGIT_FOR_GETLINE{}; while( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ? ( VARIABLE_FOR_LETTER_FOR_GETLINE = S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) ) != " " : false ){ VARIABLE_FOR_DIGIT_FOR_GETLINE = stoi( VARIABLE_FOR_LETTER_FOR_GETLINE ); assert( N < BOUND ## N / 10 ? true : N == BOUND ## N / 10 && VARIABLE_FOR_DIGIT_FOR_GETLINE <= BOUND ## N % 10 ); N = N * 10 + VARIABLE_FOR_DIGIT_FOR_GETLINE; VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } if( ! VARIABLE_FOR_POSITIVITY_FOR_GETLINE ){ N *= -1; } if( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ){ VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } ASSERT( N , MIN , MAX ); }
#define STOI_A( S , I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); FOR( VARIABLE_FOR_STOI_A , 0 , N ){ STOI( S , A ##_VARIABLE_FOR_STOI_A , MIN , MAX ); A[VARIABLE_FOR_STOI_A + I] = A ##_VARIABLE_FOR_STOI_A; }
// Sをstring SEPARATORで区切りTを構築
#define SEPARATE( S , T , SEPARATOR ) string T{}; { assert( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ); int VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S_prev = VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S; assert( S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) != SEPARATOR ); string VARIABLE_FOR_LETTER_FOR_GETLINE{}; while( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ? ( VARIABLE_FOR_LETTER_FOR_GETLINE = S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S , 1 ) ) != SEPARATOR : false ){ VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } T = S.substr( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S_prev , VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S - VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S_prev ); if( VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ){ VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S ++; } }

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