// 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_N , 1e3 );
  GETLINE_COUNT_ASSERT( N_str , ' ' , 1 );
  STOI( N_str , N , 1 , bound_N );
  vector<int> XP( N + 1 ) , inP( N + 1 ) , ord( N + 1 );
  FOREQ( i , 2 , N ){
    COUT( "?" , 1 , i );
    GETLINE_COUNT_ASSERT( X_str , ' ' , 1 );
    STOI( X_str , X , 0 , N * ( N - 1 ) / 2 );
    XP[i] = X;
    inP[i] = XP[i] - XP[i-1];
    ord[i] = i - inP[i] - 1;
  }
  NonNegativeLineSubset<int> S( N );
  FOREQ( i , 1 , N ){
    S.insert( i );
  }
  vector<int> P( N + 1 );
  FOREQINV( i , N , 1 ){
    S.erase( P[i] = *S.MinimumGeq( 0 , ord[i] ) );
  }
  COUTNS( "! " );
  FOREQ( i , 1 , N ){
    COUTNS( P[i] , " \n"[i==N] );
  }
}
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。

/* 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
// CEXPRがCEに依存しているので削除しない。
// redefinitionを避けるため圧縮元はincludeしない。

// Module
// Graph
// が必要な場合はここに追加する。

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

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

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

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

#define SFINAE_FOR_BIT_BS enable_if_t<is_invocable_r_v<bool,F,U,int>>*
TE <TY U,TY ABELIAN_GROUP>CL AbstractBIT{PU:ABELIAN_GROUP m_M;int m_SZ;VE<U> m_fenwick;int m_pw;IN AbstractBIT(ABELIAN_GROUP M,CRI SZ = 0);IN AbstractBIT(ABELIAN_GROUP M,CO VE<U>& a);TE <TY...Args> IN VO Initialise(CO Args&... args);IN VO Set(CRI i,CO U& u);VO Add(CRI i,CO U& u);IN CRI SZ()CO NE;IN U OP[](CRI i);IN U Get(CRI i);IN CO U& LSBSegmentSum(CRI j)CO;U InitialSegmentSum(CRI i_final);IN U IntervalSum(int i_start,int i_final);TE <TY F,SFINAE_FOR_BIT_BS = nullptr> int Search(CO F& f);TE <TY F,SFINAE_FOR_BIT_BS = nullptr> IN int Search(CRI i_start,CO F& f);IN int Search(CO U& u);IN int Search(CRI i_start,CO U& u);IN VO COruct();};TE <TY ABELIAN_GROUP,TY...Args> AbstractBIT(ABELIAN_GROUP M,CO Args&... args)-> AbstractBIT<inner_t<ABELIAN_GROUP>,ABELIAN_GROUP>;TE <TY U = ll>CL BIT:PU AbstractBIT<U,AdditiveGroup<U>>{PU:TE <TY...Args> IN BIT(CO Args&... args);};TE <TY U> BIT(CO VE<U>& a)-> BIT<U>;
TE <TY U,TY ABELIAN_GROUP> IN AbstractBIT<U,ABELIAN_GROUP>::AbstractBIT(ABELIAN_GROUP M,CRI SZ):m_M(MO(M)),m_SZ(SZ),m_fenwick(m_SZ + 1,m_M.Zero()),m_pw(1){COruct();}TE <TY U,TY ABELIAN_GROUP> IN AbstractBIT<U,ABELIAN_GROUP>::AbstractBIT(ABELIAN_GROUP M,CO VE<U>& a):m_M(MO(M)),m_SZ(a.SZ()),m_fenwick(m_SZ + 1,m_M.Zero()),m_pw(1){COruct();for(int j = 1;j <= m_SZ;j++){int i = j - 1,i_lim = j -(j & -j);U& fenwick_j = m_fenwick[j]= a[i];WH(i > i_lim){fenwick_j = m_M.Sum(MO(fenwick_j),m_fenwick[i]);i -=(i & -i);}}}TE <TY U,TY ABELIAN_GROUP> IN VO AbstractBIT<U,ABELIAN_GROUP>::COruct(){ST_AS(is_same_v<U,inner_t<ABELIAN_GROUP>>);WH(m_pw < m_SZ){m_pw <<= 1;}}TE <TY U> TE <TY...Args> IN BIT<U>::BIT(CO Args&... args):AbstractBIT<U,AdditiveGroup<U>>(AdditiveGroup<U>(),args...){}TE <TY U,TY ABELIAN_GROUP> TE <TY...Args> IN VO AbstractBIT<U,ABELIAN_GROUP>::Initialise(CO Args&... args){AbstractBIT<U,ABELIAN_GROUP> temp{m_M,args...};m_SZ = temp.m_SZ;m_fenwick = MO(temp.m_fenwick);m_pw = temp.m_pw;}TE <TY U,TY ABELIAN_GROUP> IN VO AbstractBIT<U,ABELIAN_GROUP>::Set(CRI i,CO U& u){Add(i,m_M.Sum(m_M.Inverse(IntervalSum(i,i)),u));}TE <TY U,TY ABELIAN_GROUP>VO AbstractBIT<U,ABELIAN_GROUP>::Add(CRI i,CO U& u){if(i < 0){RE;}int j = i + 1;WH(j <= m_SZ){U& fenwick_j = m_fenwick[j];fenwick_j = m_M.Sum(MO(fenwick_j),u);j +=(j & -j);}RE;}TE <TY U,TY ABELIAN_GROUP> IN CRI AbstractBIT<U,ABELIAN_GROUP>::SZ()CO NE{RE m_SZ;}TE <TY U,TY ABELIAN_GROUP> IN U AbstractBIT<U,ABELIAN_GROUP>::OP[](CRI i){AS(0 <= i && i < m_SZ);RE IntervalSum(i,i);}TE <TY U,TY ABELIAN_GROUP> IN U AbstractBIT<U,ABELIAN_GROUP>::Get(CRI i){RE OP[](i);}TE <TY U,TY ABELIAN_GROUP> IN CO U& AbstractBIT<U,ABELIAN_GROUP>::LSBSegmentSum(CRI j)CO{AS(0 < j && j <= m_SZ);RE m_fenwick[j];}TE <TY U,TY ABELIAN_GROUP>U AbstractBIT<U,ABELIAN_GROUP>::InitialSegmentSum(CRI i_final){U sum = m_M.Zero();int j = min(i_final + 1,m_SZ);WH(j > 0){sum = m_M.Sum(MO(sum),m_fenwick[j]);j -= j & -j;}RE sum;}TE <TY U,TY ABELIAN_GROUP> IN U AbstractBIT<U,ABELIAN_GROUP>::IntervalSum(int i_start,int i_final){i_start=max(0,i_start);i_final=min(SZ()- 1,i_final);RE i_start <= i_final?m_M.Sum(m_M.Inverse(InitialSegmentSum(i_start - 1)),InitialSegmentSum(i_final)):m_M.Zero();}TE <TY U,TY ABELIAN_GROUP> TE <TY F,SFINAE_FOR_BIT_BS>int AbstractBIT<U,ABELIAN_GROUP>::Search(CO F& f){int j = 0;int pw = m_pw;U sum = m_M.Zero();U sum_next = sum;WH(pw > 0){int j_next = j | pw;if(j_next <= m_SZ){sum_next = m_M.Sum(MO(sum_next),m_fenwick[j_next]);if(f(sum_next,j_next - 1)){sum_next = sum;}else{sum = sum_next;j = j_next;}}pw >>= 1;}RE j;}TE <TY U,TY ABELIAN_GROUP> TE <TY F,SFINAE_FOR_BIT_BS> IN int AbstractBIT<U,ABELIAN_GROUP>::Search(CRI i_start,CO F& f){CO U u_inv = m_M.Inverse(InitialSegmentSum(i_start - 1));RE max(i_start,Search([&](CO U& sum,CRI i){RE i_start <= i && f(m_M.Sum(u_inv,sum),i);}));}TE <TY U,TY ABELIAN_GROUP> IN int AbstractBIT<U,ABELIAN_GROUP>::Search(CO U& u){RE Search([&](CO U& sum,CRI){RE !(sum < u);});}TE <TY U,TY ABELIAN_GROUP> IN int AbstractBIT<U,ABELIAN_GROUP>::Search(CRI i_start,CO U& u){RE max(i_start,Search(m_M.Sum(InitialSegmentSum(i_start - 1),u)));}TE <CL Traits,TY U,TY ABELIAN_GROUP> IN OS& OP<<(OS& os,AbstractBIT<U,ABELIAN_GROUP>& bit){auto&& SZ = bit.SZ();for(int i = 0;i < SZ;i++){(i == 0?os:os << " ")<< bit[i];}RE os;}

TE <TY BLS,TY INT>CL IteratorOfBoundedLineSubset{PU:BLS* m_p;INT m_i;IN IteratorOfBoundedLineSubset(BLS& S,INT i);IN bool OP==(CO IteratorOfBoundedLineSubset<BLS,INT>& IT)CO NE;IN bool OP!=(CO IteratorOfBoundedLineSubset<BLS,INT>& IT)CO NE;IN INT OP*()CO;IN IteratorOfBoundedLineSubset<BLS,INT>& OP++();IN IteratorOfBoundedLineSubset<BLS,INT> OP++(int);IN IteratorOfBoundedLineSubset<BLS,INT>& OP--();IN IteratorOfBoundedLineSubset<BLS,INT> OP--(int);IN VO Next();IN VO Prev();IN pair<INT,INT> ConnectedComponent()CO;IN IteratorOfBoundedLineSubset<BLS,INT>& erase_from(BLS& S);};
TE <TY BLS,TY INT> IN IteratorOfBoundedLineSubset<BLS,INT>::IteratorOfBoundedLineSubset(BLS& S,INT i):m_p(&S),m_i(MO(i)){}TE <TY BLS,TY INT> IN bool IteratorOfBoundedLineSubset<BLS,INT>::OP==(CO IteratorOfBoundedLineSubset<BLS,INT>& IT)CO NE{RE m_p == IT.m_p && m_i == IT.m_i;}TE <TY BLS,TY INT> IN bool IteratorOfBoundedLineSubset<BLS,INT>::OP!=(CO IteratorOfBoundedLineSubset<BLS,INT>& IT)CO NE{RE !(*TH == IT);}TE <TY BLS,TY INT> IN INT IteratorOfBoundedLineSubset<BLS,INT>::OP*()CO{RE m_i;}TE <TY BLS,TY INT> IN IteratorOfBoundedLineSubset<BLS,INT>& IteratorOfBoundedLineSubset<BLS,INT>::OP++(){AS(m_i <= m_p->ubound());RE *TH = m_p->MinimumGt(m_i);}TE <TY BLS,TY INT> IN IteratorOfBoundedLineSubset<BLS,INT> IteratorOfBoundedLineSubset<BLS,INT>::OP++(int){auto IT = *TH;++(*TH);RE IT;}TE <TY BLS,TY INT> IN IteratorOfBoundedLineSubset<BLS,INT>& IteratorOfBoundedLineSubset<BLS,INT>::OP--(){AS(m_p->BE().m_i <= m_i);RE *TH = m_p->MaximumLt(m_i);}TE <TY BLS,TY INT> IN IteratorOfBoundedLineSubset<BLS,INT> IteratorOfBoundedLineSubset<BLS,INT>::OP--(int){auto IT = *TH;--(*TH);RE IT;}TE <TY BLS,TY INT> IN VO IteratorOfBoundedLineSubset<BLS,INT>::Next(){AS(m_i < m_p->ubound());CO INT r = m_p->RightEndPointOf(m_i);*TH = m_i <= r?m_p->MinimumGt(r):m_p->EN();}TE <TY BLS,TY INT> IN VO IteratorOfBoundedLineSubset<BLS,INT>::Prev(){AS(m_p->lbound()< m_i);CO INT l = m_p->LeftEndPointOf(m_i);*TH = l <= m_i?m_p->MaximumLt(l):m_p->EN();}TE <TY BLS,TY INT> IN pair<INT,INT> IteratorOfBoundedLineSubset<BLS,INT>::ConnectedComponent()CO{RE m_p->ConnectedComponentOf(m_i);}TE <TY BLS,TY INT> IN IteratorOfBoundedLineSubset<BLS,INT>& IteratorOfBoundedLineSubset<BLS,INT>::erase_from(BLS& S){AS(&S == m_p);auto IT_copy =(*TH)++;S.erase(IT_copy);RE *TH;}

TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR>CL VirtualBoundedLineSubset{PU:INT m_lbound;INT m_ubound;DATA_STR<INT> m_ds;US iterator = IteratorOfBoundedLineSubset<VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>,INT>;IN VO insert(CO INT& i);IN VO erase(CO INT& i)NE;IN iterator& erase(iterator& IT);IN VO clear();IN INT count(CO INT& i)NE;bool find(CO INT& i)NE;IN INT InitialSegmentCount(CO INT& i_final);IN INT IntervalCount(CO INT& i_start,CO INT& i_final);IN bool empty()NE;IN iterator BE()NE;IN iterator EN()NE;IN iterator MaximumLeq(CO INT& i,CO INT& k = 0);IN iterator MaximumLt(CO INT& i,CO INT& k = 0);IN iterator MinimumGeq(CO INT& i,CO INT& k = 0);IN iterator MinimumGt(CO INT& i,CO INT& k = 0);IN INT Maximum(CO INT& k = 0);IN INT Minimum(CO INT& k = 0);INT RightEndPointOf(CO INT& i,int d = -1,int comp_minus = -1,CO bool& in = false);INT LeftEndPointOf(CO INT& i,int d = -1,int comp_minus = -1,CO bool& in = false);IN pair<INT,INT> ConnectedComponentOf(CO INT& i,bool in = false);VE<pair<INT,INT>> GetConnectedComponent()NE;IN CO INT& lbound()CO NE;IN CO INT& ubound()CO NE;VI bool InRange(CO INT& i)= 0;VI RET_NOR Normalise(CO INT& i)= 0;VI RET_DEN Denormalise(CO decay_t<RET_NOR>& d)= 0;};
TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN VO VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::insert(CO INT& i){AS(InRange(i));m_ds.Set(Normalise(i),1);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN VO VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::erase(CO INT& i)NE{if(InRange(i)){m_ds.Set(Normalise(i),0);}}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator& VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::erase(TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator& IT){RE IT.erase_from(*TH);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN VO VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::clear(){m_ds.Initialise(m_ds.SZ());}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN INT VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::count(CO INT& i)NE{RE InRange(i)?m_ds[Normalise(i)]:0;}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN bool VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::find(CO INT& i)NE{RE count(i)> 0;}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN INT VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::InitialSegmentCount(CO INT& i_final){RE i_final < m_lbound?0:m_ds.InitialSegmentSum(Normalise(i_final));}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN INT VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::IntervalCount(CO INT& i_start,CO INT& i_final){auto&& l = Normalise(i_start);RE m_ds.IntervalSum((l < 0 || Denormalise(l)< i_start)?l + 1:l,Normalise(i_final));}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN bool VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::empty()NE{RE InitialSegmentCount(m_ubound)== 0;}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::BE()NE{RE MinimumGeq(m_lbound);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::EN()NE{RE TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator(*TH,m_ubound + 1);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::MaximumLeq(CO INT& i,CO INT& k){CO INT num = InitialSegmentCount(i)- k;if(num >= 0){CO int d = m_ds.Search([&](CO INT& sum,CRI j){RE num <= sum;});if(d < m_ds.SZ()){auto&& l = Denormalise(d);if(find(l)){RE TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator{*TH,l};}}}RE EN();}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::MaximumLt(CO INT& i,CO INT& k){CO int d = Normalise(i);if(d == 0){RE EN();}RE MaximumLeq(Denormalise(d - 1),k);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::MinimumGeq(CO INT& i,CO INT& k){CO int c = count(i);RE c > k?TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator{*TH,i}:MinimumGt(i,k - c);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::MinimumGt(CO INT& i,CO INT& k){CO INT num = InitialSegmentCount(i)+ k;CO int d = m_ds.Search([&](CO INT& sum,CRI j){RE num < sum;});if(d < m_ds.SZ()){auto&& r = Denormalise(d);if(find(r)){RE TY VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::iterator{*TH,r};}}RE EN();}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN INT VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::Maximum(CO INT& k){RE MaximumLeq(m_ubound,k);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN INT VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::Minimum(CO INT& k){RE MinimumGeq(m_lbound,k);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR>INT VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::RightEndPointOf(CO INT& i,int d,int comp_minus,CO bool& in){if(!in && !find(i)){RE i - 1;}if(d == -1){d = Normalise(i);comp_minus = d - InitialSegmentCount(i);}RE Denormalise(m_ds.Search([&](CO INT& sum,CRI j){RE d <= j && sum + comp_minus < j;})- 1);}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR>INT VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::LeftEndPointOf(CO INT& i,int d,int comp_minus,CO bool& in){if(!in && !find(i)){RE i + 1;}if(d == -1){d = Normalise(i);comp_minus = d - InitialSegmentCount(i);}RE Denormalise(m_ds.Search([&](CO INT& sum,CRI j){RE d <= j ||(find(j)&& sum + comp_minus == j);}));}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN pair<INT,INT> VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::ConnectedComponentOf(CO INT& i,bool in){if(!in){in = find(i);}CO int d = Normalise(i),comp_minus = d - InitialSegmentCount(i);RE{LeftEndPointOf(i,d,comp_minus,in),RightEndPointOf(i,d,comp_minus,in)};}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR>VE<pair<INT,INT>> VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::GetConnectedComponent()NE{VE<pair<INT,INT>> AN{};INT r;for(auto IT = BE();*IT <= m_ubound;IT = MinimumGt(r)){AN.push_back({*IT,r = RightEndPointOf(*IT)});}RE AN;}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN CO INT& VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::lbound()CO NE{RE m_lbound;}TE <TY INT,TY RET_NOR,TY RET_DEN,TE <TY...> TY DATA_STR> IN CO INT& VirtualBoundedLineSubset<INT,RET_NOR,RET_DEN,DATA_STR>::ubound()CO NE{RE m_ubound;}

TE <TY INT,TE <TY...> TY DATA_STR>CL AbstractNonNegativeLineSubset:VI PU VirtualBoundedLineSubset<INT,CO INT&,CO INT&,DATA_STR>{PU:PU:IN AbstractNonNegativeLineSubset(CO INT& ubound);IN bool InRange(CO INT& i);CE CO INT& Normalise(CO INT& i);CE CO INT& Denormalise(CO INT& d);};
TE <TY INT>US NonNegativeLineSubset = AbstractNonNegativeLineSubset<INT,BIT>;
TE <TY INT,TE <TY...> TY DATA_STR> IN AbstractNonNegativeLineSubset<INT,DATA_STR>::AbstractNonNegativeLineSubset(CO INT& ubound){AS(-1 <= ubound);TH->m_lbound = 0;TH->m_ubound = ubound;TH->m_ds.Initialise(int(TH->m_ubound + 1));}TE <TY INT,TE <TY...> TY DATA_STR> IN bool AbstractNonNegativeLineSubset<INT,DATA_STR>::InRange(CO INT& i){RE 0 <= i && i <= TH->m_ubound;}TE <TY INT,TE <TY...> TY DATA_STR> CE CO INT& AbstractNonNegativeLineSubset<INT,DATA_STR>::Normalise(CO INT& i){RE i;}TE <TY INT,TE <TY...> TY DATA_STR> CE CO INT& AbstractNonNegativeLineSubset<INT,DATA_STR>::Denormalise(CO INT& d){RE d;}

// 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 SIGNAL signal( SIGABRT , &AlertAbort );
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE2 )
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック： " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); assert( ( MIN ) <= A && A <= ( MAX ) )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define CERRNS( ... ) VariadicCoutNonSep( cerr , __VA_ARGS__ )
  #define CERR_A( I , N , A ) CoutArray( cerr , I , N , A ) << 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 ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define CERR( ... ) 
  #define CERRNS( ... ) 
  #define CERR_A( I , N , A ) 
#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_SEPARATE( SEPARATOR , ... ) VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define SET( ... ) SET_SEPARATE( '\n' , __VA_ARGS__ )
  #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; SET_SEPARATE( SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
  #define FINISH_MAIN GETLINE( test_case_num_str ); test_case_num = stoi( test_case_num_str ); ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } CHECK_REDUNDANT_INPUT; }
#else
  #define SET( ... ) VariadicCin( cin , __VA_ARGS__ )
  #define CIN( LL , ... ) LL __VA_ARGS__; SET( __VA_ARGS__ )
  #define SET_A( I , N , ... ) VariadicResize( N + I , __VA_ARGS__ ); FOR( VARIABLE_FOR_SET_A , 0 , N ){ VariadicSet( cin , VARIABLE_FOR_SET_A + I , __VA_ARGS__ ); }
  #define CIN_A( LL , I , N , ... ) VE<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
  #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) VE<VE<LL>> VAR( N0 + I0 ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , VAR[VARIABLE_FOR_CIN_AA + I0] ); }
  #define FINISH_MAIN SET_ASSERT( test_case_num , 1 , test_case_num_bound ); } REPEAT( test_case_num ){ Solve(); } CHECK_REDUNDANT_INPUT; }
#endif
#include <bits/stdc++.h>
using namespace std;
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr ); SIGNAL;
#define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , test_case_num_bound , BOUND ); int test_case_num = 1; if constexpr( test_case_num_bound > 1 ){ CERR( "テストケースの個数を入力してください。" ); FINISH_MAIN;
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now(); double loop_average_time = 0.0 , loop_start_time = loop_average_time , current_time = loop_start_time; int loop_count = current_time; assert( loop_count == 0 )
#define CURRENT_TIME ( current_time = static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 ) )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME , loop_count == 0 ? loop_start_time = current_time : loop_average_time = ( current_time - loop_start_time ) / loop_count , ++loop_count , current_time < TL_MS - loop_average_time * 2 - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_ASSERT( A , MIN , MAX ) SET( 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 PR1( A1 , ... ) A1
#define PR2( A1 , A2 , ... ) A2
#define PR3( A1 , A2 , A3 , ... ) A3
#define FOR_( VAR , INITIAL , FINAL , UPPER , COMP , INCR ) for( decldecay_t( UPPER ) VAR = INITIAL ; VAR COMP ( FINAL ) ; VAR INCR )
#define FOR( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , < , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQ( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , PR1( __VA_ARGS__ ) , <= , PR3( __VA_ARGS__ , += PR2( __VA_ARGS__ , ? ) , ++ ) )
#define FOREQINV( VAR , INITIAL , ... ) FOR_( VAR , INITIAL , PR1( __VA_ARGS__ ) , INITIAL , + 1 > , PR3( __VA_ARGS__ , -= PR2( __VA_ARGS__ , ? ) , -- ) )
#define ITR( ARRAY ) auto begin_ ## ARRAY = ARRAY .BE() , itr_ ## ARRAY = begin_ ## ARRAY , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define RUN( ARRAY , ... ) for( auto&& __VA_ARGS__ : ARRAY )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS ); cerr << fixed << setprecision( DECIMAL_DIGITS )
#define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
#define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ )
#define COUT_A( I , N , A ) CoutArray( cout , I , N , A ) << ENDL
#define DERR( ... ) 
#define DERRNS( ... ) 
#define DERR_A( I , N , A ) 
#define WHAT( ... ) 
#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;
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。*/
// BinarySearch
constexpr bool reactive =
#ifdef REACTIVE
true;
#else
false;
#endif
// EXPRESSIONがANSWERの広義単調関数の時、EXPRESSION >= CONST_TARGETの整数解を格納。
#define BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , DESIRED_INEQUALITY , CONST_TARGET , INEQUALITY_FOR_CHECK , UPDATE_U , UPDATE_L , UPDATE_ANSWER , EXTERNAL ) \
  static_assert( ! is_same<decldecay_t( CONST_TARGET ),uint>::value && ! is_same<decldecay_t( CONST_TARGET ),ull>::value ); \
  ll ANSWER = MINIMUM;							\
  {									\
    ll ANSWER ## _L = MINIMUM;                                          \
    ll ANSWER ## _R = MAXIMUM;                                          \
    ANSWER = UPDATE_ANSWER;						\
    ll EXPRESSION_BS;							\
    const ll CONST_TARGET_BS = ( CONST_TARGET );			\
    ll DIFFERENCE_BS;							\
    while( ANSWER ## _L < ANSWER ## _R ){                               \
      DIFFERENCE_BS = ( EXPRESSION_BS = ( EXPRESSION ) ) - CONST_TARGET_BS; \
      if( DIFFERENCE_BS INEQUALITY_FOR_CHECK 0 ){			\
	ANSWER ## _R = UPDATE_U;                                        \
      } else {								\
	ANSWER ## _L = UPDATE_L;                                        \
      }									\
      ANSWER = UPDATE_ANSWER;						\
    }									\
    if( ANSWER ## _L > ANSWER ## _R || !( reactive || ( EXPRESSION ) DESIRED_INEQUALITY CONST_TARGET_BS ) ){ \
      ANSWER = EXTERNAL;						\
    }									\
  }									\

// 単調増加の時にEXPRESSION >= CONST_TARGETの最小解を格納。
#define MIN_GEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CONST_TARGET , >= , ANSWER , ANSWER + 1 , Mid( ANSWER ## _L , ANSWER ## _R ) , ( MAXIMUM ) + 1 )
// 単調増加の時にEXPRESSION <= CONST_TARGETの最大解を格納。
#define MAX_LEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CONST_TARGET , > , ANSWER - 1 , ANSWER , Mid( ANSWER ## _L + 1 , ANSWER ## _R ) , ( MINIMUM ) - 1 )
// 単調減少の時にEXPRESSION >= CONST_TARGETの最大解を格納。
#define MAX_GEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , >= , CONST_TARGET , < , ANSWER - 1 , ANSWER , Mid( ANSWER ## _L + 1 , ANSWER ## _R ) , ( MINIMUM ) - 1 )
// 単調減少の時にEXPRESSION <= CONST_TARGETの最小解を格納。
#define MIN_LEQ( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , CONST_TARGET ) BS( ANSWER , MINIMUM , MAXIMUM , EXPRESSION , <= , CONST_TARGET , <= , ANSWER , ANSWER + 1 , Mid( ANSWER ## _L , ANSWER ## _R ) , ( MAXIMUM ) + 1 )
template <typename INT> inline constexpr INT Mid( const INT& l , const INT& r ) { return l + ( ( r - l ) >> 1 ); }

// Random
ll GetRand( const ll& Rand_min , const ll& Rand_max ) { assert( Rand_min <= Rand_max ); ll answer = time( NULL ); return answer * rand() % ( Rand_max + 1 - Rand_min ) + Rand_min; }

// Set
#define DC_OF_HASH( ... ) DECLARATION_OF_HASH( __VA_ARGS__ )

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

#define DEFINITION_OF_POP_FOR_SET( SET )                                \
  template <typename T> inline T pop_max( SET& S ) { assert( !S.empty() ); auto itr = --S.end(); T answer = *itr; S.erase( itr ); return answer; } \
  template <typename T> inline T pop_min( SET& S ) { assert( !S.empty() ); auto itr = S.begin(); T answer = *itr; S.erase( itr ); return answer; } \
  template <typename T> inline SET& operator<<=( SET& S , T t ) { S.insert( move( t ) ); return S; } \
  template <typename T , typename U> inline SET& operator<<=( SET& S , U&& u ) { S.insert( T{ forward<U>( u ) } ); return S; } \
  template <typename T> inline SET& operator>>=( SET& S , const T& t ) { auto itr = S.lower_bound( t ); assert( itr != S.end() && *itr == t ); S.erase( itr ); return S; } \
  template <typename T , typename U> inline SET& operator>>=( SET& S , const U& u ) { return S >>= T{ u }; } \
  template <typename T> inline const T& Get( const SET& S , int i ) { auto begin = S.begin() , end = S.end(); auto& itr = i < 0 ? ( ++i , --end ) : begin; while( i > 0 && itr != end ){ --i; ++itr; } while( i < 0 && itr != begin ){ ++i; --itr; } assert( i == 0 ); return *itr; } \

#define DEFINITION_OF_UNION_FOR_SET( SET )                              \
  template <typename T> inline SET& operator|=( SET& S0 , SET S1 ) { S0.merge( move( S1 ) ); return S0; } \
  template <typename T> inline SET operator|( SET S0 , SET S1 ) { return move( S0.size() < S1.size() ? S1 |= move( S0 ) : S0 |= move( S1 ) ); } \

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>>;

template <typename SET , typename T> inline typename SET::const_iterator MaximumLeq( const SET& S , const T& t ) { auto itr = S.upper_bound( t ); return itr == S.begin() ? S.end() : --itr; }
template <typename SET , typename T> inline typename SET::const_iterator MaximumLt( const SET& S , const T& t ) { auto itr = S.lower_bound( t ); return itr == S.begin() ? S.end() : --itr; }
template <typename SET , typename T> inline typename SET::const_iterator MinimumGeq( const SET& S , const T& t ) { return S.lower_bound( t ); }
template <typename SET , typename T> inline typename SET::const_iterator MinimumGt( const SET& S , const T& t ) { return S.upper_bound( t ); }

template <typename SET , typename ITERATOR> inline void EraseBack( SET& S , ITERATOR& itr ) { itr = S.erase( itr ); }
template <typename SET , typename ITERATOR> inline void EraseFront( SET& S , ITERATOR& itr ) { itr = S.erase( itr ); itr == S.begin() ? itr = S.end() : --itr; }

template <template <typename...> typename SET , typename T , typename...Args> inline bool In( const T& t , const SET<T,Args...>& S ) { return S.count( t ) > 0; }

DEFINITION_OF_POP_FOR_SET( set<T> );
DEFINITION_OF_POP_FOR_SET( unordered_set<T> );
DEFINITION_OF_POP_FOR_SET( multiset<T> );
DEFINITION_OF_POP_FOR_SET( unordered_multiset<T> );

DEFINITION_OF_UNION_FOR_SET( set<T> );
DEFINITION_OF_UNION_FOR_SET( unordered_set<T> );
DEFINITION_OF_UNION_FOR_SET( multiset<T> );
DEFINITION_OF_UNION_FOR_SET( unordered_multiset<T> );
DEFINITION_OF_UNION_FOR_SET( vector<T> );
DEFINITION_OF_UNION_FOR_SET( list<T> );

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

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

DEFINITION_OF_ARITHMETIC_FOR_TUPLE( + );
template <typename T , typename U , template <typename...> typename V> inline auto operator-( const V<T,U>& t ) -> decltype( get<0>( t ) , t ) { return { -get<0>( t ) , -get<1>( t ) }; }
template <typename T , typename U , typename V> inline tuple<T,U,V> operator-( const tuple<T,U,V>& t ) { return { -get<0>( t ) , -get<1>( t ) , -get<2>( t ) }; }
template <typename T , typename U , typename V , typename W> inline tuple<T,U,V,W> operator-( const tuple<T,U,V,W>& t ) { return { -get<0>( t ) , -get<1>( t ) , -get<2>( t ) , -get<3>( t ) }; }
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 ); }

template <int n>
class TupleAccessIndex
{};

template <typename...Types>
class Tuple :
public tuple<Types...>
{

public:
  inline Tuple( Types&&... args );
  template <typename...Args> inline Tuple( Args&&... args );
  
  template <int n> inline auto& operator[]( const TupleAccessIndex<n>& i ) noexcept;
  template <int n> inline const auto& operator[]( const TupleAccessIndex<n>& i ) const noexcept;

};

// structural binding用
template <typename...Types>
class tuple_size<Tuple<Types...>> :
  public tuple_size<tuple<Types...>>
{};

template <size_t n , typename...Types>
class tuple_element<n,Tuple<Types...>> :
  public tuple_element<n,tuple<Types...>>
{};

template <typename T , typename U> using Pair = Tuple<T,U>;
template <typename INT> using T2 = Tuple<INT,INT>;
template <typename INT> using T3 = Tuple<INT,INT,INT>;
template <typename INT> using T4 = Tuple<INT,INT,INT,INT>;

constexpr TupleAccessIndex<0> O{};
constexpr TupleAccessIndex<1> I{};
constexpr TupleAccessIndex<2> II{};
constexpr TupleAccessIndex<3> III{};

template <typename...Types> inline Tuple<Types...>::Tuple( Types&&... args ) : tuple<Types...>( move( args )... ) {}
template <typename...Types> template <typename...Args> inline Tuple<Types...>::Tuple( Args&&... args ) : tuple<Types...>( forward<Args>( args )... ) {}

template <typename...Types> template <int n> inline auto& Tuple<Types...>::operator[]( const TupleAccessIndex<n>& i ) noexcept { return get<n>( *this ); }
template <typename...Types> template <int n> inline const auto& Tuple<Types...>::operator[]( const TupleAccessIndex<n>& i ) const noexcept { return get<n>( *this ); }

template <typename RET , template <typename...> typename PAIR , typename INT> T2<RET> cast( const PAIR<INT,INT>& t ) { return {get<0>(t),get<1>(t)}; }
template <typename RET , typename INT> T3<RET> cast( const tuple<INT,INT,INT>& t ) { return {get<0>(t),get<1>(t),get<2>(t)}; }
template <typename RET , typename INT> T4<RET> cast( const tuple<INT,INT,INT,INT>& t ) { return {get<0>(t),get<1>(t),get<2>(t),get<3>(t)}; }

#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 DEFINITION_OF_ARITHMETIC_FOR_VECTOR( V , OPR )			\
  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> inline V<T>& operator OPR ## = ( V<T>& a , const T& t ) { for( auto& x : a ){ x OPR## = t; } return a; } \
  template <typename T , typename U> inline V<T> operator OPR( V<T> a , const U& u ) { return move( a OPR ## = u ); } \

#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 DEFINITION_OF_SHIFT_FOR_VECTOR( V )			\
  template <typename T> inline V<T>& operator<<=( V<T>& a , T t ) { a.push_back( move( t ) ); return a; } \
  template <typename T , typename U> inline V<T>& operator<<=( V<T>& a , U&& u ) { return a <<= T{ forward<U>( u ) }; } \
  template <typename T> inline T pop( V<T>& a ) { assert( !a.empty() ); T answer = move( a.back() ); a.pop_back(); return answer; } \

#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-( V<T> a ) { return move( a *= T( -1 ) );} \
  template <typename T> inline V<T> operator*( const T& t , V<T> v ) { return move( v *= t ); } \
  DEFINITION_OF_SHIFT_FOR_VECTOR( V );                                  \

DEFINITION_OF_ARITHMETICS_FOR_VECTOR( vector );
DEFINITION_OF_ARITHMETICS_FOR_VECTOR( list );
DEFINITION_OF_SHIFT_FOR_VECTOR( basic_string );

template <typename V> inline auto Get( V& a ) { return [&]( const int& i = 0 ) -> const decldecay_t( a[0] )& { return a[i]; }; }
template <typename T> 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 V> inline void Sort( V& a , const bool& reversed = false ) { using T = decldecay_t(a[0]); 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 V0 , typename V1> inline void Sort( V0& a , V1& b , const bool& reversed = false ) { const int size = a.size(); assert( size == int( b.size() ) ); vector<pair<decldecay_t(a[0]),decldecay_t(b[0])>> v( size ); for( int i = 0 ; i < size ; i++ ){ v[i] = { move( a[i] ) , move( b[i] ) }; } Sort( v , reversed ); for( int i = 0 ; i < size ; i++ ){ a[i] = move( v[i].first ); b[i] = move( v[i].second ); } }
template <typename V> inline pair<vector<int>,vector<int>> IndexSort( const V& a , const bool& reversed = false ) { const int size = a.size(); auto index = id<int>( size ) , ord = index; sort( index.begin() , index.end() , [&]( const int& i , const int& j ) { const pair ti{ a[i] , i } , tj{ a[j] , j }; return reversed ? tj < ti : ti < tj; } ); for( int i = 0 ; i < size ; i++ ){ ord[index[i]] = i; } return { move( index ) , move( ord ) }; }

template <typename V> inline int len( const V& a ) { return a.size(); }

template <typename V> inline void Reverse( V& a ) { const int size = len( a ) , half = size / 2; for( int i = 0 ; i < half ; i++ ){ swap( a[i] , a[size-1-i] ); } }
;
template <typename V> inline V Reversed( V a ) { Reverse( a ); return move( a ); }

template <typename RET , template <typename...> typename V , typename T> inline V<RET> cast( const V<T>& a ) { V<RET> answer{}; for( auto& x : a ){ answer <<= a; } }

#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 );
DEFINITION_OF_COUT_FOR_VECTOR( multiset );

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 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 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; }

// Arithmetic/Iteration
#define SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( TYPE ) \
  template <> inline TYPE ArithmeticProgressionSum( const TYPE& l , const TYPE& r , const TYPE& d ) { return SpecialisedArithmeticProgressionSum( l , r , d ); } \

template <typename T , typename U , template <typename...> typename V , typename OPR> T LeftConnectiveProd( T t , const V<U>& f , OPR opr ) { for( auto& u : f ){ t = opr( move( t ) , u ); } return move( t ); }
template <typename T , typename U , template <typename...> typename V> inline T Sum( const V<U>& f ) { return LeftConnectiveProd( T{ 0 } , f , []( T t0 , const U& u1 ){ return move( t0 += u1 ); } ); }
template <typename T , typename U , template <typename...> typename V> inline T Prod( const V<U>& f ) { return LeftConnectiveProd( T{ 1 } , f , []( T t0 , const U& u1 ){ return move( t0 *= u1 ); } ); }

template <typename T> inline T& SetMax( T& t ) { return t; }
template <typename T , typename U , typename... Args> inline T& SetMax( T& t0 , const U& u1 , const Args&... args ) { return SetMax( t0 < u1 ? t0 = u1 : t0 , args... ); }
template <typename T> inline T& SetMin( T& t ) { return t; }
template <typename T , typename U , typename... Args> inline T& SetMin( T& t0 , const U& u1 , const Args&... args ) { return SetMin( u1 < t0 ? t0 = u1 : t0 , args... ); }

template <typename T> inline const T& Max( const vector<T>& f ) { return *max_element( f.begin() , f.end() ); }
template <typename T , template <typename...> typename SET> inline const T& Max( const SET<T>& f ) { return *--f.end(); }
template <typename T , typename U , typename...Args> inline T Max( T t0 , const U& t1 , const Args&... args ) { return move( SetMax( t0 , t1 , args... ) ); }
template <typename T> inline const T& Min( const vector<T>& f ) { return *min_element( f.begin() , f.end() ); }
template <typename T , template <typename...> typename SET> inline const T& Min( const SET<T>& f ) { return *f.begin(); }
template <typename T , typename U , typename...Args> inline T Min( T t0 , const U& t1 , const Args&... args ) { return move( SetMin( t0 , t1 , args... ) ); }

template <typename T , typename UINT>
T Power( const T& t , const UINT& exponent , T init )
{

  return exponent > 1 ? Power( t * t , exponent >> 1 , move( exponent & 1 ? init *= t : init ) ) : move( exponent > 0 ? init *= t : ( assert( exponent == 0 ) , init ) );

}

template <typename T> inline T PowerMemorisation( const T& t , const int& exponent ) { assert( exponent >= 0 ); static Map<T,vector<T>> memory{}; auto& answer = memory[t]; if( answer.empty() ){ answer.push_back( 1 ); } while( int( answer.size() ) <= exponent ){ answer.push_back( answer.back() * t ); } return answer[exponent]; }

template <typename INT> inline INT ArithmeticProgressionSum( const INT& l , const INT& r , const INT& d = 1 ) { return ( l + r ) * ( ( r - l ) / d + 1 ) / 2; }
template <typename INT> inline INT SpecialisedArithmeticProgressionSum( const INT& l , const INT& r , const INT& d = 1 ) { assert( l - 1 <= r ); const INT c = ( r - l ) / d; return l - 1 == r ? 0 : ( c & 1 ) == 0 ? ( c + 1 ) * ( l + d * ( c >> 1 ) ) : ( ( c + 1 ) >> 1 ) * ( ( l << 1 ) + d * c ); }
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( int );
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( uint );
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( ll );
SPECIALSATION_OF_ARITHMETIC_PROGRESSION_SUM( ull );
template <typename INT> inline INT ArithmeticProgressionSum( const INT& r ) { return ArithmeticProgressionSum( INT{} , r ); }

template <typename INT> inline INT SquareSum( const INT& r ) { return r * ( r + 1 ) * ( 2 * r + 1 ) / 6; }

template <typename T , typename UINT> inline T GeometricProgressionSum( T rate , UINT exponent_max , const T& init ) { T rate_minus = rate - 1; return rate_minus == 0 ? init * ++exponent_max : ( Power( move( rate ) , move( ++exponent_max ) ) - 1 ) / move( rate_minus ) * init; }

template <typename T , typename UINT>
T GeometricProgressionLinearCombinationSum( vector<T> rate , vector<UINT> exponent_max , const vector<T>& init )
{

  const int size = init.size();
  assert( int( rate.size() ) == size && int( exponent_max.size() ) == size );
  T answer{};

  for( int i = 0 ; i < size ; i++ ){

    answer += GeometricProgressionSum( move( rate[i] ) , move( exponent_max[i] ) , init[i] );

  }

  return answer;

}

// Arithmetic/Sqrt
template <typename INT>
INT RoundDownSqrt( const INT& n )
{

  static_assert( is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull> );
  assert( n >= 0 );
  
  if( n <= 1 ){

    return n;
    
  }

  constexpr INT r_max = is_same_v<INT,int> ? 46341 : is_same_v<INT,uint> ? 65536 : is_same_v<INT,ll> ? 3037000500 : 4294967296;
  INT l = 1 , r = min( r_max , n );

  while( l < r - 1 ){

    const INT m = ( l + r ) >> 1;
    // m * m <= nか否かを判定。
    ( m <= n / m ? l : r ) = m;

  }

  return l;

}

template <typename INT>
INT RoundUpSqrt( const INT& n )
{
  
  static_assert( is_same_v<INT,int> || is_same_v<INT,uint> || is_same_v<INT,ll> || is_same_v<INT,ull> );
  assert( n >= 0 );

  if( n <= 2 ){

    return n;
    
  }

  constexpr INT r_max = is_same_v<INT,int> ? 46341 : is_same_v<INT,uint> ? 65536 : is_same_v<INT,ll> ? 3037000500 : 4294967296;
  const INT n_minus = n - 1;
  INT l = 1 , r = min( r_max , n );

  while( l + 1 < r ){

    const INT m = ( l + r ) >> 1;
    // m * m < nか否かを判定。
    ( m <= n_minus / m ? l : r ) = m;

  }

  return r;

}

template <typename INT> bool IsSquare( const INT& n ) { const INT r = RoundDownSqrt( n ); return n == r * r; }

/* 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 , SEPARATOR ) GETLINE( S ); int S ## _count = 0; int VARIABLE_FOR_INDEX_FOR_GETLINE_FOR_ ## S = 0; int VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S = S.size(); for( int i = 0 ; i < VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S ; i++ ){ if( S[i] == SEPARATOR ){ S ## _count++; } } if( VARIABLE_FOR_SIZE_FOR_GETLINE_FOR_ ## S > 0 ){ S ## _count++; }
#define GETLINE_COUNT_ASSERT( S , SEPARATOR , COUNT ) GETLINE_COUNT( S , SEPARATOR ); assert( S ## _count == COUNT )
// 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