結果

問題 No.2598 Kadomatsu on Tree
ユーザー 👑 p-adicp-adic
提出日時 2024-01-23 15:39:10
言語 C++17
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 1,857 ms / 2,000 ms
コード長 61,961 bytes
コンパイル時間 5,684 ms
コンパイル使用メモリ 278,924 KB
実行使用メモリ 248,960 KB
最終ジャッジ日時 2024-09-28 06:46:42
合計ジャッジ時間 66,232 ms
ジャッジサーバーID
(参考情報)
judge2 / judge3
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 2 ms
5,248 KB
testcase_01 AC 2 ms
5,248 KB
testcase_02 AC 3 ms
5,376 KB
testcase_03 AC 2 ms
5,376 KB
testcase_04 AC 3 ms
5,376 KB
testcase_05 AC 2 ms
5,376 KB
testcase_06 AC 3 ms
5,376 KB
testcase_07 AC 3 ms
5,376 KB
testcase_08 AC 2 ms
5,376 KB
testcase_09 AC 3 ms
5,376 KB
testcase_10 AC 3 ms
5,376 KB
testcase_11 AC 3 ms
5,376 KB
testcase_12 AC 3 ms
5,376 KB
testcase_13 AC 35 ms
8,192 KB
testcase_14 AC 32 ms
8,832 KB
testcase_15 AC 21 ms
7,168 KB
testcase_16 AC 31 ms
9,216 KB
testcase_17 AC 14 ms
5,760 KB
testcase_18 AC 24 ms
7,680 KB
testcase_19 AC 29 ms
8,960 KB
testcase_20 AC 15 ms
6,144 KB
testcase_21 AC 19 ms
6,912 KB
testcase_22 AC 26 ms
8,064 KB
testcase_23 AC 1,607 ms
236,416 KB
testcase_24 AC 1,606 ms
240,896 KB
testcase_25 AC 1,624 ms
241,280 KB
testcase_26 AC 1,673 ms
237,312 KB
testcase_27 AC 1,571 ms
237,952 KB
testcase_28 AC 1,602 ms
240,760 KB
testcase_29 AC 1,626 ms
242,176 KB
testcase_30 AC 1,594 ms
236,672 KB
testcase_31 AC 1,599 ms
241,664 KB
testcase_32 AC 1,730 ms
239,488 KB
testcase_33 AC 1,680 ms
244,096 KB
testcase_34 AC 1,571 ms
234,624 KB
testcase_35 AC 1,645 ms
245,632 KB
testcase_36 AC 1,777 ms
244,772 KB
testcase_37 AC 1,857 ms
244,096 KB
testcase_38 AC 1,657 ms
245,888 KB
testcase_39 AC 1,611 ms
245,880 KB
testcase_40 AC 1,612 ms
245,888 KB
testcase_41 AC 1,628 ms
246,016 KB
testcase_42 AC 1,597 ms
245,888 KB
testcase_43 AC 1,606 ms
246,016 KB
testcase_44 AC 1,657 ms
245,888 KB
testcase_45 AC 1,633 ms
245,888 KB
testcase_46 AC 1,609 ms
245,888 KB
testcase_47 AC 1,640 ms
245,888 KB
testcase_48 AC 1,522 ms
235,520 KB
testcase_49 AC 128 ms
25,344 KB
testcase_50 AC 958 ms
156,416 KB
testcase_51 AC 239 ms
47,360 KB
testcase_52 AC 431 ms
77,184 KB
testcase_53 AC 685 ms
102,016 KB
testcase_54 AC 546 ms
85,344 KB
testcase_55 AC 183 ms
35,584 KB
testcase_56 AC 1,103 ms
151,168 KB
testcase_57 AC 352 ms
60,032 KB
testcase_58 AC 1,551 ms
238,592 KB
testcase_59 AC 1,593 ms
241,792 KB
testcase_60 AC 1,543 ms
240,000 KB
testcase_61 AC 1,610 ms
248,960 KB
testcase_62 AC 1,678 ms
242,688 KB
権限があれば一括ダウンロードができます

ソースコード

diff #

#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  // #define USE_GETLINE
#endif

#ifdef INCLUDE_MAIN

IN VO Solve()
{
  CIN( int , N ); int M = N - 1;
  gE<int>.resize( N );
  FOR( j , 0 , M ){
    CIN_ASSERT( uj , 1 , N ); CIN_ASSERT( vj , 1 , N );
    uj--; vj--;
    gE<int>[uj].push_back( vj ); gE<int>[vj].push_back( uj );
  }
  gA<int>.resize( N );
  FOR( i , 0 , N ){
    CIN( ll , Ai );
    gA<int>[i] = Ai;
  }
  Graph tree{ N , Get( gE<int> ) };
  DepthFirstSearchOnTree dfst{ tree };
  // a[0]:左端が増大する{列の個数,門松度の合計}
  // a[1]:左端が減少する{列の個数,門松度の合計}
  // a[2]:左端が変化なし{列の個数,門松度の合計}
  using T = vector<vector<MP>>;
  auto ADD = [&]( const T& t0 , const T& t1 )
  {
    return T{ { t0[0][0] + t1[0][0] , t0[0][1] + t1[0][1] } , { t0[1][0] + t1[1][0] , t0[1][1] + t1[1][1] }, { t0[2][0] + t1[2][0] , t0[2][1] + t1[2][1] } };
  };
  auto END = [&]( const T& t , const int& i ){ return ADD( t , {{0,0},{0,0},{1,0}}); };
  auto UPDATE = [&](const T& t,const bool&,const int& i,const int& j)
  {
    return gA<int>[j] < gA<int>[i] ? T{ { 0 , 0 } , { t[0][0] + t[1][0] + t[2][0] , t[0][0] + t[0][1] + t[1][1] + t[2][1] } , { 0 , 0 } } :
    gA<int>[j] > gA<int>[i] ? T{ { t[0][0] + t[1][0] + t[2][0] , t[0][1] + t[1][0] + t[1][1] + t[2][1] } , { 0 , 0 } , { 0 , 0 } } :
    T{ { 0 , 0 } , { 0 , 0 } , { t[0][0] + t[1][0] + t[2][0] , t[0][1] + t[1][1] + t[2][1] } };
  };
  AbstractMonoid monoid{ ADD , T{3,{0,0}} };
  vector<T> d;
  dfst.RerootingDP( monoid , END , UPDATE , d );
  MP answer = 0;
  FOR( i , 0 , N ){
    answer += d[i][0][1] + d[i][1][1] + d[i][2][1];
  }
  answer /= 2;
  RETURN( answer );
}
REPEAT_MAIN(1);

#else // INCLUDE_MAIN

#ifdef INCLUDE_SUB

// グラフ用
TE <TY T> Map<T,T> gF;
TE <TY T> VE<T> gA;
TE <TY PATH> VE<LI<PATH>> gE;
TE <TY T , TE <TY...> TY V> IN auto Get( CO V<T>& a ) { return [&]( CRI i = 0 ){ RE a[i]; }; }

// COMPAREに使用。圧縮時は削除する。
ll Naive( int N , int M , int K )
{
  ll answer = N + M + K;
  return answer;
}

// COMPAREに使用。圧縮時は削除する。
ll Answer( ll N , ll M , ll K )
{
  // START_WATCH;
  ll answer = N + M + K;

  // // TLに準じる乱択や全探索。デフォルトの猶予は100.0[ms]。
  // CEXPR( double , TL , 2000.0 );
  // while( CHECK_WATCH( TL ) ){

  // }
  return answer;
}

// 圧縮時は中身だけ削除する。
IN VO Experiment()
{
  // CEXPR( int , bound , 10 );
  // FOREQ( N , 0 , bound ){
  //   FOREQ( M , 0 , bound ){
  //     FOREQ( K , 0 , bound ){
  //   	COUT( N , M , K , ":" , Naive( N , M , K ) );
  //     }
  //   }
  //   // cout << Naive( N ) << ",\n"[N==bound];
  // }
}

// 圧縮時は中身だけ削除する。
IN VO SmallTest()
{
  // CEXPR( int , bound , 10 );
  // FOREQ( N , 0 , bound ){
  //   FOREQ( M , 0 , bound ){
  //     FOREQ( K , 0 , bound ){
  //   	COMPARE( N , M , K );
  //     }
  //   }
  //   // COMPARE( N );
  // }
}

#define INCLUDE_MAIN
#include __FILE__

#else // INCLUDE_SUB

#ifdef INCLUDE_LIBRARY

/*

C-x 3 C-x o C-x C-fによるファイル操作用

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

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

DFSOnTree
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/DepthFirstSearch/Tree/a.hpp

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

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

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

UnionFind
c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/UnionFindForest/compress.txt

*/

// VVV 常設でないライブラリは以下に挿入する。

// 構築 O(1)/O(|V_G|)(未初期化/初期化)
// Next()の反復でinitから到達可能な頂点を全探索 O(initの連結成分における辺の本数)
// Next()の反復とShift()で全探索 O(|V_G|+|E_G|)
// initからの到達可能性判定と深さ計算 O(initの連結成分における辺の本数)
// 連結成分の色分けと数え上げ O(|V_G|+|E_G|) 
template <typename GRAPH>
class VirtualBreadthFirstSearch :
  public PointedSet<int>
{

protected:
  GRAPH& m_G;
  bool m_initialised;
  // 次の探索点たちを格納。
  list<int> m_next;
  // 到達済みか否かを格納。
  vector<bool> m_found;
  // 到達済みかつ到達済みの点から辺を辿って到達した場合、その点を格納。
  vector<int> m_prev;

public:
  inline VirtualBreadthFirstSearch( GRAPH& G );
  inline VirtualBreadthFirstSearch( GRAPH& G , const int& init );

  // m_nextとm_foundとm_prevを初期化する。
  inline void Initialise();
  // m_nextとm_foundとm_prevを初期化した上でinitを最初の探索点に設定する。
  inline void Initialise( const int& init );
  // m_nextを初期化した上でm_foundとm_prevを非初期化せずinitを次の探索点に設定する。
  inline void Shift( const int& init );

  inline const int& size() const noexcept;
  inline vector<bool>::reference found( const int& i );
  inline const int& prev( const int& i );

  inline int Next();

  // 最初の未到達点(初期化時点ではinit)から到達できる未到達点の深さを格納し、
  // 到達できない未到達点や既到達点は深さの代わりに-1を格納。
  void SetDepth( vector<int>& depth );
  // 無向グラフである場合にのみサポート。
  // 未到達点の全体のなす部分グラフにおける連結成分の色分けと連結成分数を格納。
  void SetConnectedComponent( vector<int>& cc_num , int& count );

private:
  virtual void Push( list<int>& next , const int& i ) = 0;

};

template <typename GRAPH>
class BreadthFirstSearch :
  public VirtualBreadthFirstSearch<GRAPH>
{

public:
  template <typename...Args> inline BreadthFirstSearch( GRAPH& G , const Args&... args );

private:
  inline void Push( list<int>& next , const int& i );

};

template <typename GRAPH> inline VirtualBreadthFirstSearch<GRAPH>::VirtualBreadthFirstSearch( GRAPH& G ) : m_G( G ) , m_initialised( false ) , m_next() , m_found() , m_prev() { static_assert( is_same_v<inner_t<GRAPH>,int> ); }
template <typename GRAPH> inline VirtualBreadthFirstSearch<GRAPH>::VirtualBreadthFirstSearch( GRAPH& G , const int& init ) : VirtualBreadthFirstSearch<GRAPH>( G ) { Initialise( init ); }
template <typename GRAPH> template <typename...Args> inline BreadthFirstSearch<GRAPH>::BreadthFirstSearch( GRAPH& G , const Args&... args ) : VirtualBreadthFirstSearch<GRAPH>( G , args... ) {}

template <typename GRAPH> inline void VirtualBreadthFirstSearch<GRAPH>::Initialise() { m_initialised = true; const int& V = size(); m_next.clear(); m_found = vector<bool>( V ); m_prev = vector<int>( V , -1 ); }
template <typename GRAPH> inline void VirtualBreadthFirstSearch<GRAPH>::Initialise( const int& init ) { assert( ( this->init() = init ) < size() ); Initialise(); m_next.push_back( init ); m_found[init] = true; }
template <typename GRAPH> inline void VirtualBreadthFirstSearch<GRAPH>::Shift( const int& init ) { if( m_initialised ){ const int& V = size(); assert( ( this->init() = init ) < V ); m_next.clear(); if( ! m_found[init] ){ m_next.push_back( init ); m_found[init] = true; } } else { Initialise( init ); } }

template <typename GRAPH> inline const int& VirtualBreadthFirstSearch<GRAPH>::size() const noexcept { return m_G.size(); }
template <typename GRAPH> inline vector<bool>::reference VirtualBreadthFirstSearch<GRAPH>::found( const int& i ) { assert( i < size() ); if( !m_initialised ){ Initialise(); } return m_found[i]; }
template <typename GRAPH> inline const int& VirtualBreadthFirstSearch<GRAPH>::prev( const int& i ) { assert( i < size() ); if( !m_initialised ){ Initialise(); } return m_prev[i]; }

template <typename GRAPH> inline int VirtualBreadthFirstSearch<GRAPH>::Next()
{

  if( m_next.empty() ){

    return -1;

  }

  const int i_curr = m_next.front();
  m_next.pop_front();
  auto&& edge = m_G.Edge( i_curr );

  while( ! edge.empty() ){

    const int& i = edge.front();
    auto&& found_i = m_found[i];

    if( ! found_i ){

      Push( m_next , i );
      m_prev[i] = i_curr;
      found_i = true;

    }

    edge.pop_front();

  }

  return i_curr;

}

template <typename GRAPH>
void VirtualBreadthFirstSearch<GRAPH>::SetDepth( vector<int>& depth )
{

  depth = vector<int>( size() , -1 );
  int i = Next();
  depth[i] = 0;
  
  while( ( i = Next() ) != -1 ){

    depth[i] = depth[prev( i )] + 1;

  }

  return;  
}

template <typename GRAPH>
void VirtualBreadthFirstSearch<GRAPH>::SetConnectedComponent( vector<int>& cc_num , int& count )
{

  const int& V = size();
  cc_num = vector<int>( V , -1 );
  count = 0;

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

    if( cc_num[i] == -1 ){

      Shift( i );
      int j = Next();

      if( j != -1 ){

	while( j != -1 ){

	  // 無向グラフである場合は常にtrue。
	  assert( cc_num[j] == -1 );
	  cc_num[j] = count;
	  j = Next();

	}

	count++;

      }

    }

  }

  return;

}

template <typename GRAPH> inline void BreadthFirstSearch<GRAPH>::Push( list<int>& next , const int& i ) { next.push_back( i ); }


template <typename GRAPH>
class DepthFirstSearch :
  public VirtualBreadthFirstSearch<GRAPH>
{

public:
  template <typename...Args> inline DepthFirstSearch( GRAPH& G , const Args&... args );

private:
  inline void Push( list<int>& next , const int& i );

};

template <typename GRAPH> template <typename...Args> inline DepthFirstSearch<GRAPH>::DepthFirstSearch( GRAPH& G , const Args&... args ) : VirtualBreadthFirstSearch<GRAPH>( G , args... ) {}

template <typename GRAPH> inline void DepthFirstSearch<GRAPH>::Push( list<int>& next , const int& i ) { next.push_front( i ); }


// digitはAncestorとLCAにのみ使用。普段は0で良い。
// 2^16 = 65536
// 2^17 = 131072
// 2^18 = 262144

// Gが無向グラフとしての木である場合にのみサポート。
template <typename TREE>
class DepthFirstSearchOnTree :
  public DepthFirstSearch<TREE>
{

private:
  vector<int> m_reversed;

  vector<vector<int>> m_children;
  vector<int> m_children_num;
  bool m_set_children;

  vector<int> m_depth;
  bool m_set_depth;

  vector<int> m_height;
  bool m_set_height;

  vector<int> m_weight;
  bool m_set_weight;

  int m_digit;
  vector<vector<int>> m_doubling;
  bool m_set_doubling;

public:
  inline DepthFirstSearchOnTree( TREE& T , const int& root = 0 , const int& digit = 0 );
  inline void Initialise( ) = delete;
  inline void Initialise( const int& init ) = delete;
  inline void Shift( const int& init ) = delete;

  inline const int& Root() const;
  inline const int& Parent( const int& i );
  inline const vector<int>& Children( const int& i );
  inline const int& Depth( const int& i );
  inline const int& Height( const int& i );
  inline const int& Weight( const int& i );
  
  // 探索順にノードを番号づける。
  inline const int& NodeNumber( const int& i , const bool& reversed = false ) const;
  // 共通の親を持つノード間で昇順に番号づける。
  inline const int& ChildrenNumber( const int& i );  

  // 各ノードの高さ < 2^digitの場合のみサポート。
  // n階の親Parent^n( i )を返す。
  int Ancestor( int i , int n );
  int LCA( int i , int j );
  // LCAからi,j側に進める場合に進んだ先の頂点のラベルをi_prev,j_prevに格納する。
  int LCA( int i , int j , int& i_prev , int& j_prev );

  // Uを適当な型とし、
  // Fは写像f:U^{< \omega} \times N -> Uに相当する型。
  // 型推論のためにfはデフォルト引数で呼び出し可能とする。

  // 入力の範囲内で要件
  // (2) 任意の非負整数n,iとTの要素のみからなる任意の長さnの任意の列(u1,...,un)と
  //     その並び換え(v1,...,vn)に対しf((u1,...,un),i)=f((v1,...,vn),i)である。
  // を満たす場合のみサポート。
  // dp[j] = f(jの子ノードkを渡るdp[k]の列,j)
  // を満たす配列dpの根での値dp[m_init]をO(m_V)で求める。
  template <typename F> ret_t<F> RootingDP( F& f );

  // Uを適当な型とし、
  // Fは写像f:U \times N -> Uに相当する型。
  // Eは写像g:U \times \{0,1\} \times N^2 -> Uに相当する型。

  // 入力の範囲内で要件
  // (1) MがUのモノイド構造である。(以下演算を*と置く)
  // (2) 任意の非負整数n,iとTの要素のみからなる任意の長さnの任意の列(u1,...,un)と
  //     その並び換え(v1,...,vn)に対し
  //     f(u1*u2*...*un,j)=f(v1*v2*...*vn,j)である。
  // を満たす場合のみサポート。
  // dp[i][j] =
  // f(iを根とみなした時のjの子ノードkを渡るg(dp[i][k],jはiの子孫,k,j)のMに関する積,j)
  // を満たす二重配列dpの対角成分dp[i][i]をO(|V_T|)で求めてdに格納する。
  template <typename MONOID , typename F , typename G> void RerootingDP( MONOID M , F& f , G& g , vector<inner_t<MONOID>>& d );
  // fはノードjごとのデータ(グラフ構造に依存しない)、gは有向辺b?(j,k):(k,j)ごとのデータに対応。
  // 例えば「パスの数」を求める時はm_Tが和、fが+1(葉かどうかに関係ない)、gがidでよい。

private:
  void SetChildren();
  void SetDepth();
  void SetHeight();
  void SetWeight();

  // 各ノードの高さ < 2^digitの場合のみサポート。
  // LCA()を呼ぶ前にAncestor()経由で完全にダブリングを設定するため、
  // 遅延評価してしまう
  // ../../../../Mathematics/Function/Iteration/Doubling/
  // のダブリングでは代用しない。
  void SetDoubling();

};

template <typename TREE> inline DepthFirstSearchOnTree<TREE>::DepthFirstSearchOnTree( TREE& T , const int& root , const int& digit ) :
  DepthFirstSearch<TREE>( T , root ) , m_reversed( this->size() ) , m_children() , m_set_children() , m_depth() , m_set_depth() , m_height() , m_set_height() , m_weight() , m_set_weight() , m_digit( digit ) , m_doubling( m_digit ) , m_set_doubling()
{

  int n = this->size();
  
  while( --n >= 0 ){
    
    m_reversed[n] = this->Next();

  }

}

template <typename TREE> inline const int& DepthFirstSearchOnTree<TREE>::Root() const { return this->init(); }

template <typename TREE> inline const int& DepthFirstSearchOnTree<TREE>::Parent( const int& i ) { return this->prev( i ); }
template <typename TREE> inline const vector<int>& DepthFirstSearchOnTree<TREE>::Children( const int& i ) { if( ! m_set_children ){ SetChildren(); } return m_children[i]; }
template <typename TREE> inline const int& DepthFirstSearchOnTree<TREE>::Depth( const int& i ) { if( ! m_set_depth ){ SetDepth(); } return m_depth[i]; }
template <typename TREE> inline const int& DepthFirstSearchOnTree<TREE>::Height( const int& i ) { if( ! m_set_height ){ SetHeight(); } return m_height[i]; }
template <typename TREE> inline const int& DepthFirstSearchOnTree<TREE>::Weight( const int& i ) { if( ! m_set_weight ){ SetWeight(); } return m_weight[i]; }

template <typename TREE> inline const int& DepthFirstSearchOnTree<TREE>::NodeNumber( const int& i , const bool& reversed ) const { return m_reversed[reversed ? i : this->size() - 1 - i]; }
template <typename TREE> inline const int& DepthFirstSearchOnTree<TREE>::ChildrenNumber( const int& i ) { if( ! m_set_children ){ SetChildren(); } return m_children_num[i]; }

template <typename TREE>
int DepthFirstSearchOnTree<TREE>::Ancestor( int i , int n )
{

  if( ! m_set_doubling ){

    SetDoubling();

  }

  assert( ( n >> m_digit ) == 0 );
    
  int d = 0;
  
  while( n != 0 ){

    if( ( n & 1 ) == 1 ){

      assert( ( i = m_doubling[d][i] ) != -1 );

    }
    
    d++;
    n >>= 1;

  }

  return i;

}

template <typename TREE>
int DepthFirstSearchOnTree<TREE>::LCA( int i , int j )
{

  int diff = Depth( i ) - Depth( j );

  if( diff < 0 ){

    swap( i , j );
    diff *= -1;

  }
    
  i = Ancestor( i , diff );

  if( i == j ){

    return i;
      
  }
    
  int d = m_digit;

  while( --d >= 0 ){

    const vector<int>& doubling_d = m_doubling[d];
    const int& doubling_d_i = doubling_d[i];
    const int& doubling_d_j = doubling_d[j];

    if( doubling_d_i != doubling_d_j ){

      i = doubling_d_i;
      j = doubling_d_j;
      assert( i != -1 );
      assert( j != -1 );

    }

  }

  return Parent( i );

}

template <typename TREE>
int DepthFirstSearchOnTree<TREE>::LCA( int i , int j , int& i_prev , int& j_prev )
{

  if( i == j ){

    i_prev = j_prev = -1;
    return i;

  }
  
  int diff = Depth( i ) - Depth( j );

  if( diff < 0 ){

    return LCA( j , i , j_prev , i_prev );

  }
  
  if( diff > 0 ){

    i_prev = Ancestor( i , diff - 1 );
    i = Parent( i_prev );
    assert( i != -1 );
    
    if( i == j ){

      j_prev = -1;
      return i;
      
    }
    
  } else if( ! m_set_doubling ){

    SetDoubling();

  }
    
  int d = m_digit;

  while( --d >= 0 ){

    const vector<int>& doubling_d = m_doubling[d];
    const int& doubling_d_i = doubling_d[i];
    const int& doubling_d_j = doubling_d[j];

    if( doubling_d_i != doubling_d_j ){

      i = doubling_d_i;
      j = doubling_d_j;
      assert( i != -1 );
      assert( j != -1 );

    }

  }

  i_prev = i;
  j_prev = j;
  return Parent( i_prev );

}

template <typename TREE>
void DepthFirstSearchOnTree<TREE>::SetChildren()
{

  assert( !m_set_children );
  m_set_children = true;
  const int& V = this->size();
  m_children.resize( V );
  m_children_num.resize( V );
  
  for( int i = 0 ; i < V ; i++ ){

    const int& j = Parent( i );

    if( j == -1 ){

      m_children_num[i] = -1;

    } else {
      
      vector<int>& m_children_j = m_children[j];
      m_children_num[i] = m_children_j.size();
      m_children_j.push_back( i );

    }

  }

  return;
  
}

template <typename TREE>
void DepthFirstSearchOnTree<TREE>::SetDepth()
{

  assert( !m_set_depth );
  m_set_depth = true;
  const int& V = this->size();
  m_depth.resize( V );
  
  for( int i = 0 ; i < V ; i++ ){

    const int& reversed_i = m_reversed[i];
    const int& parent_i = Parent( reversed_i );

    if( parent_i != -1 ){

      m_depth[i] += m_depth[parent_i] + 1;

    }

  }

  return;

}

template <typename TREE>
void DepthFirstSearchOnTree<TREE>::SetHeight()
{

  assert( !m_set_height );
  m_set_height = true;
  const int& V = this->size();
  m_height.resize( V );
  
  for( int i = 0 ; i < V ; i++ ){

    const int& reversed_i = m_reversed[i];
    const int& parent_i = Parent( reversed_i );

    if( parent_i != -1 ){

      int& height_parent_i = m_height[parent_i];
      const int& height_i = m_height[reversed_i];
      height_parent_i > height_i ? height_parent_i : height_parent_i = height_i + 1;

    }

  }

  return;

}

template <typename TREE>
void DepthFirstSearchOnTree<TREE>::SetWeight()
{

  assert( !m_set_weight );
  m_set_weight = true;
  const int& V = this->size();
  m_weight.resize( V );
  
  for( int i = 0 ; i < V ; i++ ){

    const int& reversed_i = m_reversed[i];
    const int& parent_i = Parent( reversed_i );

    if( parent_i != -1 ){

      m_weight[parent_i] += m_weight[reversed_i] + 1;

    }

  }

  return;

}

template <typename TREE>
void DepthFirstSearchOnTree<TREE>::SetDoubling()
{

  assert( !m_set_doubling );
  m_set_doubling = true;
  const int& V = this->size();
  
  {
    
    vector<int>& doubling_0 = m_doubling[0];
    doubling_0.reserve( V );
    const int& r = Root();

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

      doubling_0.push_back( Parent( i ) );

    }

  }
  
  for( int d = 1 ; d < m_digit ; d++ ){

    vector<int>& doubling_d = m_doubling[d];
    vector<int>& doubling_d_minus = m_doubling[d-1];
    doubling_d.reserve( V );

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

      const int& doubling_d_minus_i = doubling_d_minus[i];
      doubling_d.push_back( doubling_d_minus_i == -1 ? -1 : doubling_d_minus[doubling_d_minus_i] );

    }

  }

  return;

}

template <typename TREE> template <typename F>
ret_t<F> DepthFirstSearchOnTree<TREE>::RootingDP( F& f )
{

  using U = ret_t<F>;
  static_assert( is_invocable_r_v<U,F,list<U>,int> );

  if( ! m_set_children ){

    SetChildren();

  }
  
  const int& V = this->size();
  vector<list<U>> children_value( V );
  U temp;
  
  for( int n = 0 ; n < V ; n++ ){
    
    const int& i = NodeNumber( n , true );
    const int& j = Parent( i );
    temp = f( children_value[i] , i );

    if( j != -1 ){

      children_value[j].push_back( temp );

    }

  }

  return temp;

}
  
template <typename TREE> template <typename MONOID , typename F , typename G>
void DepthFirstSearchOnTree<TREE>::RerootingDP( MONOID M , F& f , G& g , vector<inner_t<MONOID>>& d )
{

  using U = inner_t<MONOID>;
  static_assert( is_invocable_r_v<U,F,U,int> && is_invocable_r_v<U,G,U,bool,int,int> );
  
  if( ! m_set_children ){

    SetChildren();

  }
  
  const int& V = this->size();
  const U& e = M.Unit();
  d.resize( V );

  // children_value[i][m]にiのm番目の子ノードjまでの計算値のfでの像を格納。
  vector<vector<U>> children_value( V );
  // left_sum[i][m]にchildren_value[i][0],...,children_value[i][m-1]の
  // gでの像のMに関する積を格納。
  vector<vector<U>> left_sum( V );
  // right_sum[i][m]にchildren_value[i][m+1],...,children_value[i][size_i-1]の
  // gでの像のM関する積を格納。
  vector<vector<U>> right_sum( V );
  
  for( int i = 0 ; i < V ; i++ ){

    children_value[i].resize( m_children[i].size() );

  }
  
  for( int n = 0 ; n < V ; n++ ){
    
    const int& i = NodeNumber( n , true );
    const vector<U>& children_value_i = children_value[i];
    const int size_i = children_value_i.size();

    U temp = e;
    vector<U>& left_sum_i = left_sum[i];
    left_sum_i.reserve( size_i + 1 );
    left_sum_i.push_back( temp );
    
    for( int m = 0 ; m < size_i ; m++ ){

      left_sum_i.push_back( temp = M.Product( temp , g( children_value_i[m] , true , i , m_children[i][m] ) ) );

    }
    
    const int& j = Parent( i );

    if( j != -1 ){
      
      children_value[j][m_children_num[i]] = f( temp , i );

    }

    temp = e;
    vector<U>& right_sum_i = right_sum[i];
    right_sum_i.resize( size_i );

    for( int m = 1 ; m <= size_i ; m++ ){

      right_sum_i[ size_i - m ] = temp;
      temp = M.Product( g( children_value_i[size_i - m] , true , i , m_children[i][size_i - m] ) , temp );

    }

  }

  // left_sum[i][m]にchildren_value[i][0],...,children_value[i][m-1]の
  // gでの像のMに関する積を格納していたが、さらにこれに親ノードの寄与も追加する。
  for( int n = 1 ; n < V ; n++ ){
    
    const int& i = NodeNumber( n );
    const int& j = Parent( i );
    const int& k = ChildrenNumber( i );
    vector<U>& left_sum_i = left_sum[i];
    vector<U>& right_sum_i = right_sum[i];
    const int size_i = right_sum_i.size();
    // children_value[j][0],...,children_value[j][k-1]のgでの像と
    // rest_j(既に計算済み)と
    // children_value[j][k+1],...,children_value[j][size_i-1]のgでの像と
    // のMに関する積のfでの像のgでの像。
    const U rest_i = g( f( M.Product( left_sum[j][k] , right_sum[j][k] ) , j ) , false , i , j );
    
    for( int m = 0 ; m <= size_i ; m++ ){

      // left_sum_imにrest_iと
      // children_value[i][0],...,children_value[i][m-1]のgでの像
      // のMに関する積を格納。
      U& left_sum_im = left_sum_i[m];
      left_sum_im = M.Product( rest_i , left_sum_im );

    }

  }

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

    // left_sum[i].back()はchildren_value_i[0],...,children_value_i[size_i-1]の
    // gでの像と親の寄与のMに関する積。
    d[i] = f( left_sum[i].back() , i );

  }

  return;

}

// AAA 常設でないライブラリは以上に挿入する。

#define INCLUDE_SUB
#include __FILE__

#else // INCLUDE_LIBRARY

#ifdef DEBUG
  #define _GLIBCXX_DEBUG
  #define REPEAT_MAIN( BOUND ) START_MAIN; signal( SIGABRT , &AlertAbort ); AutoCheck( exec_mode , use_getline ); if( exec_mode == sample_debug_mode || exec_mode == submission_debug_mode || exec_mode == library_search_mode ){ RE 0; } else if( exec_mode == experiment_mode ){ Experiment(); RE 0; } else if( exec_mode == small_test_mode ){ SmallTest(); RE 0; }; DEXPR( int , bound_test_case_num , BOUND , min( BOUND , 100 ) ); int test_case_num = 1; if( exec_mode == solve_mode ){ if CE( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } } else if( exec_mode == random_test_mode ){ CERR( "ランダムテストを行う回数を指定してください。" ); SET_LL( test_case_num ); } FINISH_MAIN
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , DEBUG_VALUE )
  #define ASSERT( A , MIN , MAX ) CERR( "ASSERTチェック: " , ( MIN ) , ( ( MIN ) <= A ? "<=" : ">" ) , A , ( A <= ( MAX ) ? "<=" : ">" ) , ( MAX ) ); AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) if( exec_mode == solve_mode ){ SET_LL( A ); ASSERT( A , MIN , MAX ); } else if( exec_mode == random_test_mode ){ CERR( #A , " = " , ( A = GetRand( MIN , MAX ) ) ); } else { AS( false ); }
  #define SOLVE_ONLY ST_AS( __FUNCTION__[0] == 'S' )
  #define CERR( ... ) VariadicCout( cerr , __VA_ARGS__ ) << endl
  #define COUT( ... ) VariadicCout( cout << "出力: " , __VA_ARGS__ ) << endl
  #define CERR_A( A , N ) OUTPUT_ARRAY( cerr , A , N ) << endl
  #define COUT_A( A , N ) cout << "出力: "; OUTPUT_ARRAY( cout , A , N ) << endl
  #define CERR_ITR( A ) OUTPUT_ITR( cerr , A ) << endl
  #define COUT_ITR( A ) cout << "出力: "; OUTPUT_ITR( cout , A ) << endl
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if CE( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } FINISH_MAIN
  #define DEXPR( LL , BOUND , VALUE , DEBUG_VALUE ) CEXPR( LL , BOUND , VALUE )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #define SET_ASSERT( A , MIN , MAX ) SET_LL( A ); ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #define CERR( ... ) 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define CERR_A( A , N ) 
  #define COUT_A( A , N ) OUTPUT_ARRAY( cout , A , N ) << ENDL
  #define CERR_ITR( A ) 
  #define COUT_ITR( A ) OUTPUT_ITR( cout , A ) << ENDL
#endif
#ifdef REACTIVE
  #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 , ... ) SOLVE_ONLY; string __VA_ARGS__; VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
  #define GETLINE( ... ) SOLVE_ONLY; GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
#else
  #define SET_LL( A ) cin >> A
  #define CIN( LL , ... ) SOLVE_ONLY; LL __VA_ARGS__; VariadicCin( cin , __VA_ARGS__ )
  #define SET_A( A , N ) SOLVE_ONLY; FOR( VARIABLE_FOR_CIN_A , 0 , N ){ cin >> A[VARIABLE_FOR_CIN_A]; }
  #define CIN_A( LL , A , N ) VE<LL> A( N ); SET_A( A , N );
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define FINISH_MAIN REPEAT( test_case_num ){ if CE( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } Solve(); CERR( "" ); } }
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now()
#define CURRENT_TIME static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME < TL_MS - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( 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 AUTO_ITR( ARRAY ) auto itr_ ## ARRAY = ARRAY .BE() , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( AUTO_ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT_ ## HOW_MANY_TIMES , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS )
#define OUTPUT_ARRAY( OS , A , N ) FOR( VARIABLE_FOR_OUTPUT_ARRAY , 0 , N ){ OS << A[VARIABLE_FOR_OUTPUT_ARRAY] << (VARIABLE_FOR_OUTPUT_ARRAY==N-1?"":" "); } OS
#define OUTPUT_ITR( OS , A ) { auto ITERATOR_FOR_OUTPUT_ITR = A.BE() , EN_FOR_OUTPUT_ITR = A.EN(); bool VARIABLE_FOR_OUTPUT_ITR = ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR; WH( VARIABLE_FOR_OUTPUT_ITR ){ OS << *ITERATOR_FOR_COUT_ITR; ( VARIABLE_FOR_OUTPUT_ITR = ++ITERATOR_FOR_COUT_ITR != END_FOR_COUT_ITR ) ? OS : OS << " "; } } OS
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ ); auto answer = Answer( __VA_ARGS__ ); bool match = naive == answer; COUT( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive == " , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ RE; }

// 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 ST_AS static_assert
#define reMO_CO remove_const
#define is_COructible_v is_constructible_v
#define rBE rbegin
#define reSZ

// 型のエイリアス
#define decldecay_t( VAR ) decay_t<decltype( VAR )>
TE <TY F , TY...Args> US ret_t = decltype( declval<F>()( declval<Args>()... ) );
TE <TY T> US inner_t = TY T::type;
US uint = unsigned int;
US ll = long long;
US ull = unsigned long long;
US ld = long double;
US lld = __float128;
TE <TY INT> US T2 = pair<INT,INT>;
TE <TY INT> US T3 = tuple<INT,INT,INT>;
TE <TY INT> US T4 = tuple<INT,INT,INT,INT>;
US path = pair<int,ll>;

// 入出力用
TE <CL Traits> IN basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is ) { RE is; }
TE <CL Traits , TY Arg , TY... ARGS> IN basic_istream<char,Traits>& VariadicCin( basic_istream<char,Traits>& is , Arg& arg , ARGS&... args ) { RE VariadicCin( is >> arg , args... ); }
TE <CL Traits> IN basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , CO char& separator ) { RE is; }
TE <CL Traits , TY Arg , TY... ARGS> IN basic_istream<char,Traits>& VariadicGetline( basic_istream<char,Traits>& is , CO char& separator , Arg& arg , ARGS&... args ) { RE VariadicGetline( getline( is , arg , separator ) , separator , args... ); }
TE <CL Traits , TY Arg> IN basic_ostream<char,Traits>& operator<<( basic_ostream<char,Traits>& os , CO VE<Arg>& arg ) { auto BE = arg.BE() , EN = arg.EN(); auto itr = BE; WH( itr != EN ){ ( itr == BE ? os : os << " " ) << *itr; itr++; } RE os; }
TE <CL Traits , TY Arg> IN basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , CO Arg& arg ) { RE os << arg; }
TE <CL Traits , TY Arg1 , TY Arg2 , TY... ARGS> IN basic_ostream<char,Traits>& VariadicCout( basic_ostream<char,Traits>& os , CO Arg1& arg1 , CO Arg2& arg2 , CO ARGS&... args ) { RE VariadicCout( os << arg1 << " " , arg2 , args... ); }

// データ構造用
TE <TY T , TE <TY...> TY V> IN V<T> OP+( CO V<T>& a0 , CO V<T>& a1 ) { if( a0.empty() ){ RE a1; } if( a1.empty() ){ RE a0; } AS( a0.SZ() == a1.SZ() ); V<T> answer{}; for( auto itr0 = a0.BE() , itr1 = a1.BE() , EN0 = a0.EN(); itr0 != EN0 ; itr0++ , itr1++ ){ answer.push_back( *itr0 + *itr1 ); } RE answer; }
TE <TY T , TY U> IN pair<T,U> OP+( CO pair<T,U>& t0 , CO pair<T,U>& t1 ) { RE { t0.first + t1.first , t0.second + t1.second }; }
TE <TY T , TY U , TY V> IN tuple<T,U,V> OP+( CO tuple<T,U,V>& t0 , CO tuple<T,U,V>& t1 ) { RE { get<0>( t0 ) + get<0>( t1 ) , get<1>( t0 ) + get<1>( t1 ) , get<2>( t0 ) + get<2>( t1 ) }; }
TE <TY T , TY U , TY V , TY W> IN tuple<T,U,V,W> OP+( CO tuple<T,U,V,W>& t0 , CO tuple<T,U,V,W>& t1 ) { RE { get<0>( t0 ) + get<0>( t1 ) , get<1>( t0 ) + get<1>( t1 ) , get<2>( t0 ) + get<2>( t1 ) , get<3>( t0 ) + get<3>( t1 ) }; }

// デバッグ用
#ifdef DEBUG
  IN VO 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。" ); }
  VO AutoCheck( int& exec_mode , CO bool& use_getline );
  IN VO Solve();
  IN VO Experiment();
  IN VO SmallTest();
  IN VO RandomTest();
  ll GetRand( CRL Rand_min , CRL Rand_max );
  IN VO BreakPoint( CRI LINE ) {}
  int exec_mode;
  CEXPR( int , solve_mode , 0 );
  CEXPR( int , sample_debug_mode , 1 );
  CEXPR( int , submission_debug_mode , 2 );
  CEXPR( int , library_search_mode , 3 );
  CEXPR( int , experiment_mode , 4 );
  CEXPR( int , small_test_mode , 5 );
  CEXPR( int , random_test_mode , 6 );
  #ifdef USE_GETLINE
    CEXPR( bool , use_getline , true );
  #else
    CEXPR( bool , use_getline , false );
  #endif
#else
  ll GetRand( CRL Rand_min , CRL Rand_max ) { ll answer = time( NULL ); RE answer * rand() % ( Rand_max + 1 - Rand_min ) + Rand_min; }
#endif

// VVV 常設ライブラリは以下に挿入する。
// Map
// c:/Users/user/Documents/Programming/Mathematics/Function/Map/compress.txt
CL is_ordered{PU:is_ordered()= delete;TE <TY T> ST CE auto Check(CO T& t)-> decltype(t < t,true_type());ST CE false_type Check(...);TE <TY T> ST CE CO bool value = is_same_v< decltype(Check(declval<T>())),true_type >;};
TE <TY T , TY U>US Map = conditional_t<is_COructible_v<unordered_map<T,int>>,unordered_map<T,U>,conditional_t<is_ordered::value<T>,map<T,U>,VO>>;

// Algebra
// c:/Users/user/Documents/Programming/Mathematics/Algebra/compress.txt
#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:virtual PU UnderlyingSet<U>{PU:virtual CO U& Point()CO NE = 0;virtual U& Point() NE = 0;DC_OF_CPOINT(Unit);DC_OF_CPOINT(Zero);DC_OF_CPOINT(One);DC_OF_CPOINT(Infty);DC_OF_CPOINT(size);DC_OF_POINT(init);DC_OF_POINT(root);};TE <TY U>CL PointedSet:virtual PU VirtualPointedSet<U>{PU:U m_b_U;IN PointedSet(CO U& b_u = U());IN CO U& Point()CO NE;IN U& Point() NE;};TE <TY U>CL VirtualNSet:virtual PU UnderlyingSet<U>{PU:virtual U Transfer(CO U& u)= 0;IN U Inverse(CO U& u);};TE <TY U,TY F_U>CL AbstractNSet:virtual PU VirtualNSet<U>{PU:F_U& m_f_U;IN AbstractNSet(F_U& f_U);IN U Transfer(CO U& u);};TE <TY U>CL VirtualMagma:virtual PU UnderlyingSet<U>{PU:virtual U Product(CO U& u0,CO U& u1)= 0;IN U Sum(CO U& u0,CO U& u1);};TE <TY U,TY M_U>CL AbstractMagma:virtual PU VirtualMagma<U>{PU:M_U& m_m_U;IN AbstractMagma(M_U& m_U);IN U Product(CO U& u0,CO U& u1);};
TE <TY U> IN PointedSet<U>::PointedSet(CO U& b_U):m_b_U(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_CPOINT(size);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(f_U){ST_AS(is_invocable_r_v<U,F_U,U>);}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(m_U){ST_AS(is_invocable_r_v<U,M_U,U,U>);}TE <TY U,TY M_U> IN U AbstractMagma<U,M_U>::Product(CO U& u0,CO U& u1){RE m_m_U(u0,u1);}TE <TY U> IN U VirtualMagma<U>::Sum(CO U& u0,CO U& u1){RE Product(u0,u1);}

TE <TY U>CL VirtualMonoid:virtual PU VirtualMagma<U>,virtual PU VirtualPointedSet<U>{};TE <TY U = ll>CL AdditiveMonoid:virtual PU VirtualMonoid<U>,PU PointedSet<U>{PU:IN U Product(CO U& u0,CO U& u1);};TE <TY U = ll>CL MultiplicativeMonoid:virtual PU VirtualMonoid<U>,PU PointedSet<U>{PU:IN MultiplicativeMonoid(CO U& e_U);IN U Product(CO U& u0,CO U& u1);};TE <TY U,TY M_U>CL AbstractMonoid:virtual PU VirtualMonoid<U>,PU AbstractMagma<U,M_U>,PU PointedSet<U>{PU:IN AbstractMonoid(M_U& m_U,CO U& e_U);};
TE <TY U> IN MultiplicativeMonoid<U>::MultiplicativeMonoid(CO U& e_U):PointedSet<U>(e_U){}TE <TY U,TY M_U> IN AbstractMonoid<U,M_U>::AbstractMonoid(M_U& m_U,CO U& e_U):AbstractMagma<U,M_U>(m_U),PointedSet<U>(e_U){}TE <TY U> IN U AdditiveMonoid<U>::Product(CO U& u0,CO U& u1){RE u0 + u1;}TE <TY U> IN U MultiplicativeMonoid<U>::Product(CO U& u0,CO U& u1){RE u0 * u1;}

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

TE <TY U,TY GROUP,TY MONOID>CL VirtualRing{PU:GROUP m_R0;MONOID m_R1;IN VirtualRing(GROUP R0,MONOID R1);IN U Sum(CO U& u0,CO U& u1);IN CO U& Zero()CO NE;IN U Inverse(CO U& u);IN U Product(CO U& u0,CO U& u1);IN CO U& One()CO NE;IN GROUP& AdditiveGroup()NE;IN MONOID& MultiplicativeMonoid()NE;};TE <TY U = ll>CL Ring:virtual PU VirtualRing<U,AdditiveGroup<U>,MultiplicativeMonoid<U>>{PU:IN Ring(CO U& one_U);};TE <TY U,TY A_U,TY I_U,TY M_U>CL AbstractRing:virtual PU VirtualRing<U,AbstractGroup<U,A_U,I_U>,AbstractMonoid<U,M_U>>{PU:IN AbstractRing(A_U& a_U,CO U& z_U,I_U& i_U,M_U& m_U,CO U& e_U);};
TE <TY U,TY GROUP,TY MONOID> IN VirtualRing<U,GROUP,MONOID>::VirtualRing(GROUP R0,MONOID R1):m_R0(MO(R0)),m_R1(MO(R1)){}TE <TY U> IN Ring<U>::Ring(CO U& one_U):VirtualRing<U,AdditiveGroup<U>,MultiplicativeMonoid<U>>(AdditiveGroup<U>(),MultiplicativeMonoid<U>(one_U)){}TE <TY U,TY A_U,TY I_U,TY M_U> IN AbstractRing<U,A_U,I_U,M_U>::AbstractRing(A_U& a_U,CO U& z_U,I_U& i_U,M_U& m_U,CO U& e_U):VirtualRing<U,AbstractGroup<U,A_U,I_U>,AbstractMonoid<U,M_U>>(AbstractGroup<U,A_U,I_U>(a_U,z_U,i_U),AbstractMonoid<U,M_U>(m_U,e_U)){}TE <TY U,TY GROUP,TY MONOID> IN U VirtualRing<U,GROUP,MONOID>::Sum(CO U& u0,CO U& u1){RE m_R0.Sum(u0,u1);}TE <TY U,TY GROUP,TY MONOID> IN CO U& VirtualRing<U,GROUP,MONOID>::Zero()CO NE{RE m_R0.Zero();}TE <TY U,TY GROUP,TY MONOID> IN U VirtualRing<U,GROUP,MONOID>::Inverse(CO U& u){RE m_R0.Inverse(u);}TE <TY U,TY GROUP,TY MONOID> IN U VirtualRing<U,GROUP,MONOID>::Product(CO U& u0,CO U& u1){RE m_R1.Product(u0,u1);}TE <TY U,TY GROUP,TY MONOID> IN CO U& VirtualRing<U,GROUP,MONOID>::One()CO NE{RE m_R1.One();}TE <TY U,TY GROUP,TY MONOID> IN GROUP& VirtualRing<U,GROUP,MONOID>::AdditiveGroup()NE{RE m_R0;}TE <TY U,TY GROUP,TY MONOID> IN MONOID& VirtualRing<U,GROUP,MONOID>::MultiplicativeMonoid()NE{RE m_R1;}

// Graph
// c:/Users/user/Documents/Programming/Mathematics/Geometry/Graph/compress.txt
#define SFINAE_FOR_GRAPH TY T,TY E,enable_if_t<is_invocable_v<E,T>,void*> PTR
TE <TY T,TY R1,TY R2,TY E>CL VirtualGraph:PU PointedSet<int>{PU:E m_edge;IN VirtualGraph(CRI SZ,E edge);virtual R1 Enumeration(CRI i)= 0;virtual R2 Enumeration_inv(CO T& t)= 0;IN VO Reset();IN E& edge()NE;IN ret_t<E,T> Edge(CO T& t);US type = T;};TE <TY E>CL Graph:virtual PU VirtualGraph<int,CRI,CRI,E>{PU:IN Graph(CRI SZ,E edge);IN CRI Enumeration(CRI i);IN CRI Enumeration_inv(CRI t);TE <TY F> IN Graph<F> GetGraph(F edge)CO;};TE <TY T,TY Enum_T,TY Enum_T_inv,TY E>CL EnumerationGraph:virtual PU VirtualGraph<T,ret_t<Enum_T,int>,ret_t<Enum_T_inv,T>,E>{PU:Enum_T& m_enum_T;Enum_T_inv& m_enum_T_inv;IN EnumerationGraph(CRI SZ,Enum_T& enum_T,Enum_T_inv& enum_T_inv,E edge);IN ret_t<Enum_T,int> Enumeration(CRI i);IN ret_t<Enum_T_inv,T> Enumeration_inv(CO T& t);TE <TY F> IN EnumerationGraph<T,Enum_T,Enum_T_inv,F> GetGraph(F edge)CO;};TE <TY Enum_T,TY Enum_T_inv,TY E> EnumerationGraph(CRI SZ,Enum_T& enum_T,Enum_T_inv& enum_T_inv,E edge)-> EnumerationGraph<decldecay_t(declval<Enum_T>()(0)),Enum_T,Enum_T_inv,E>;TE <SFINAE_FOR_GRAPH = nullptr>CL MemorisationGraph:virtual PU VirtualGraph<T,T,CRI,E>{PU:int m_LE;VE<T> m_memory;Map<T,int> m_memory_inv;IN MemorisationGraph(CRI SZ,E edge);IN T Enumeration(CRI i);IN CRI Enumeration_inv(CO T& t);IN VO Reset();TE <TY F> IN MemorisationGraph<T,F> GetGraph(F edge)CO;};TE <TY E> MemorisationGraph(CRI SZ,E edge)-> MemorisationGraph<decldecay_t(declval<E>()().back()),E>;TE <TY E> MemorisationGraph(CRI SZ,E edge)-> MemorisationGraph<decldecay_t(get<0>(declval<E>()().back())),E>;
TE <TY T,TY R1,TY R2,TY E> IN VirtualGraph<T,R1,R2,E>::VirtualGraph(CRI SZ,E edge):PointedSet<int>(SZ),m_edge(MO(edge)){ST_AS(is_COructible_v<T,R1> && is_COructible_v<int,R2> && is_invocable_v<E,T>);}TE <TY E> IN Graph<E>::Graph(CRI SZ,E edge):VirtualGraph<int,CRI,CRI,E>(SZ,MO(edge)){}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> IN EnumerationGraph<T,Enum_T,Enum_T_inv,E>::EnumerationGraph(CRI SZ,Enum_T& enum_T,Enum_T_inv& enum_T_inv,E edge):VirtualGraph<T,ret_t<Enum_T,int>,ret_t<Enum_T_inv,T>,E>(SZ,MO(edge)),m_enum_T(enum_T),m_enum_T_inv(enum_T_inv){}TE <SFINAE_FOR_GRAPH> IN MemorisationGraph<T,E,PTR>::MemorisationGraph(CRI SZ,E edge):VirtualGraph<T,T,CRI,E>(SZ,MO(edge)),m_LE(),m_memory(),m_memory_inv(){}TE <TY T,TY R1,TY R2,TY E> IN E& VirtualGraph<T,R1,R2,E>::edge()NE{RE m_edge;}TE <TY T,TY R1,TY R2,TY E> IN ret_t<E,T> VirtualGraph<T,R1,R2,E>::Edge(CO T& t){RE m_edge(t);}TE <TY E> IN CRI Graph<E>::Enumeration(CRI i){RE i;}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> IN ret_t<Enum_T,int> EnumerationGraph<T,Enum_T,Enum_T_inv,E>::Enumeration(CRI i){RE m_enum_T(i);}TE <SFINAE_FOR_GRAPH> IN T MemorisationGraph<T,E,PTR>::Enumeration(CRI i){AS(0 <= i && i < m_LE);RE m_memory[i];}TE <TY E> IN CRI Graph<E>::Enumeration_inv(CRI i){RE i;}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> IN ret_t<Enum_T_inv,T> EnumerationGraph<T,Enum_T,Enum_T_inv,E>::Enumeration_inv(CO T& t){RE m_enum_T_inv(t);}TE <SFINAE_FOR_GRAPH> IN CRI MemorisationGraph<T,E,PTR>::Enumeration_inv(CO T& t){if(m_memory_inv.count(t)== 0){AS(m_LE < TH->SZ());m_memory.push_back(t);RE m_memory_inv[t]= m_LE++;}RE m_memory_inv[t];}TE <TY T,TY R1,TY R2,TY E> VO VirtualGraph<T,R1,R2,E>::Reset(){}TE <SFINAE_FOR_GRAPH> IN VO MemorisationGraph<T,E,PTR>::Reset(){m_LE = 0;m_memory.clear();m_memory_inv.clear();}TE <TY E> TE <TY F> IN Graph<F> Graph<E>::GetGraph(F edge)CO{RE Graph<F>(TH->SZ(),MO(edge));}TE <TY T,TY Enum_T,TY Enum_T_inv,TY E> TE <TY F> IN EnumerationGraph<T,Enum_T,Enum_T_inv,F> EnumerationGraph<T,Enum_T,Enum_T_inv,E>::GetGraph(F edge)CO{RE EnumerationGraph(TH->SZ(),m_enum_T,m_enum_T_inv,MO(edge));}TE <SFINAE_FOR_GRAPH> TE <TY F> IN MemorisationGraph<T,F> MemorisationGraph<T,E,PTR>::GetGraph(F edge)CO{RE MemorisationGraph(TH->SZ(),MO(edge));}

// ConstexprModulo
// c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Mod/ConstexprModulo/compress.txt
CEXPR(uint,P,998244353);TE <uint M,TY INT> CE INT& RS(INT& n)NE{RE n < 0?((((++n)*= -1)%= M)*= -1)+= M - 1:n %= M;}TE <uint M> CE uint& RS(uint& n)NE{RE n %= M;}TE <uint M> CE ull& RS(ull& n)NE{RE n %= M;}TE <TY INT> CE INT& RSP(INT& n)NE{CE CO uint trunc =(1 << 23)- 1;INT n_u = n >> 23;n &= trunc;INT n_uq =(n_u / 7)/ 17;n_u -= n_uq * 119;n += n_u << 23;RE n < n_uq?n += P - n_uq:n -= n_uq;}TE <> CE ull& RS<P,ull>(ull& n)NE{CE CO ull Pull = P;CE CO ull Pull2 =(Pull - 1)*(Pull - 1);RE RSP(n > Pull2?n -= Pull2:n);}TE <uint M,TY INT> CE INT RS(INT&& n)NE{RE MO(RS<M>(n));}TE <uint M,TY INT> CE INT RS(CO INT& n)NE{RE RS<M>(INT(n));}

#define SFINAE_FOR_MOD(DEFAULT)TY T,enable_if_t<is_COructible_v<uint,decay_t<T>>>* DEFAULT
#define DC_OF_CM_FOR_MOD(FUNC)CE bool OP FUNC(CO Mod<M>& n)CO NE
#define DC_OF_AR_FOR_MOD(FUNC)CE Mod<M> OP FUNC(CO Mod<M>& n)CO NE;TE <SFINAE_FOR_MOD(= nullptr)> CE Mod<M> OP FUNC(T&& n)CO NE;
#define DF_OF_CM_FOR_MOD(FUNC)TE <uint M> CE bool Mod<M>::OP FUNC(CO Mod<M>& n)CO NE{RE m_n FUNC n.m_n;}
#define DF_OF_AR_FOR_MOD(FUNC,FORMULA)TE <uint M> CE Mod<M> Mod<M>::OP FUNC(CO Mod<M>& n)CO NE{RE MO(Mod<M>(*TH)FUNC ## = n);}TE <uint M> TE <SFINAE_FOR_MOD()> CE Mod<M> Mod<M>::OP FUNC(T&& n)CO NE{RE FORMULA;}TE <uint M,SFINAE_FOR_MOD(= nullptr)> CE Mod<M> OP FUNC(T&& n0,CO Mod<M>& n1)NE{RE MO(Mod<M>(forward<T>(n0))FUNC ## = n1);}

TE <uint M>CL Mod{PU:uint m_n;CE Mod()NE;CE Mod(CO Mod<M>& n)NE;CE Mod(Mod<M>& n)NE;CE Mod(Mod<M>&& n)NE;TE <SFINAE_FOR_MOD(= nullptr)> CE Mod(CO T& n)NE;TE <SFINAE_FOR_MOD(= nullptr)> CE Mod(T& n)NE;TE <SFINAE_FOR_MOD(= nullptr)> CE Mod(T&& n)NE;CE Mod<M>& OP=(CO Mod<M>& n)NE;CE Mod<M>& OP=(Mod<M>&& n)NE;CE Mod<M>& OP+=(CO Mod<M>& n)NE;CE Mod<M>& OP-=(CO Mod<M>& n)NE;CE Mod<M>& OP*=(CO Mod<M>& n)NE;IN Mod<M>& OP/=(CO Mod<M>& n);CE Mod<M>& OP<<=(int n)NE;CE Mod<M>& OP>>=(int n)NE;CE Mod<M>& OP++()NE;CE Mod<M> OP++(int)NE;CE Mod<M>& OP--()NE;CE Mod<M> OP--(int)NE;DC_OF_CM_FOR_MOD(==);DC_OF_CM_FOR_MOD(!=);DC_OF_CM_FOR_MOD(<);DC_OF_CM_FOR_MOD(<=);DC_OF_CM_FOR_MOD(>);DC_OF_CM_FOR_MOD(>=);DC_OF_AR_FOR_MOD(+);DC_OF_AR_FOR_MOD(-);DC_OF_AR_FOR_MOD(*);DC_OF_AR_FOR_MOD(/);CE Mod<M> OP<<(int n)CO NE;CE Mod<M> OP>>(int n)CO NE;CE Mod<M> OP-()CO NE;CE Mod<M>& SignInvert()NE;CE Mod<M>& Double()NE;CE Mod<M>& Halve()NE;IN Mod<M>& Invert();TE <TY T> CE Mod<M>& PositivePW(T&& EX)NE;TE <TY T> CE Mod<M>& NonNegativePW(T&& EX)NE;TE <TY T> CE Mod<M>& PW(T&& EX);CE VO swap(Mod<M>& n)NE;CE CRUI RP()CO NE;ST CE Mod<M> DeRP(CRUI n)NE;ST CE uint& Normalise(uint& n)NE;ST IN CO Mod<M>& Inverse(CRUI n)NE;ST IN CO Mod<M>& Factorial(CRUI n)NE;ST IN CO Mod<M>& FactorialInverse(CRUI n)NE;ST IN Mod<M> Combination(CRUI n,CRUI i)NE;ST IN CO Mod<M>& zero()NE;ST IN CO Mod<M>& one()NE;TE <TY T> CE Mod<M>& Ref(T&& n)NE;};

#define SFINAE_FOR_MN(DEFAULT)TY T,enable_if_t<is_COructible_v<Mod<M>,decay_t<T>>>* DEFAULT
#define DC_OF_AR_FOR_MN(FUNC)IN MN<M> OP FUNC(CO MN<M>& n)CO NE;TE <SFINAE_FOR_MOD(= nullptr)> IN MN<M> OP FUNC(T&& n)CO NE;
#define DF_OF_CM_FOR_MN(FUNC)TE <uint M> IN bool MN<M>::OP FUNC(CO MN<M>& n)CO NE{RE m_n FUNC n.m_n;}
#define DF_OF_AR_FOR_MN(FUNC,FORMULA)TE <uint M> IN MN<M> MN<M>::OP FUNC(CO MN<M>& n)CO NE{RE MO(MN<M>(*TH)FUNC ## = n);}TE <uint M> TE <SFINAE_FOR_MOD()> IN MN<M> MN<M>::OP FUNC(T&& n)CO NE{RE FORMULA;}TE <uint M,SFINAE_FOR_MOD(= nullptr)> IN MN<M> OP FUNC(T&& n0,CO MN<M>& n1)NE{RE MO(MN<M>(forward<T>(n0))FUNC ## = n1);}

TE <uint M>CL MN:PU Mod<M>{PU:CE MN()NE;CE MN(CO MN<M>& n)NE;CE MN(MN<M>& n)NE;CE MN(MN<M>&& n)NE;TE <SFINAE_FOR_MN(= nullptr)> CE MN(CO T& n)NE;TE <SFINAE_FOR_MN(= nullptr)> CE MN(T&& n)NE;CE MN<M>& OP=(CO MN<M>& n)NE;CE MN<M>& OP=(MN<M>&& n)NE;CE MN<M>& OP+=(CO MN<M>& n)NE;CE MN<M>& OP-=(CO MN<M>& n)NE;CE MN<M>& OP*=(CO MN<M>& n)NE;IN MN<M>& OP/=(CO MN<M>& n);CE MN<M>& OP<<=(int n)NE;CE MN<M>& OP>>=(int n)NE;CE MN<M>& OP++()NE;CE MN<M> OP++(int)NE;CE MN<M>& OP--()NE;CE MN<M> OP--(int)NE;DC_OF_AR_FOR_MN(+);DC_OF_AR_FOR_MN(-);DC_OF_AR_FOR_MN(*);DC_OF_AR_FOR_MN(/);CE MN<M> OP<<(int n)CO NE;CE MN<M> OP>>(int n)CO NE;CE MN<M> OP-()CO NE;CE MN<M>& SignInvert()NE;CE MN<M>& Double()NE;CE MN<M>& Halve()NE;CE MN<M>& Invert();TE <TY T> CE MN<M>& PositivePW(T&& EX)NE;TE <TY T> CE MN<M>& NonNegativePW(T&& EX)NE;TE <TY T> CE MN<M>& PW(T&& EX);CE uint RP()CO NE;CE Mod<M> Reduce()CO NE;ST CE MN<M> DeRP(CRUI n)NE;ST IN CO MN<M>& Formise(CRUI n)NE;ST IN CO MN<M>& Inverse(CRUI n)NE;ST IN CO MN<M>& Factorial(CRUI n)NE;ST IN CO MN<M>& FactorialInverse(CRUI n)NE;ST IN MN<M> Combination(CRUI n,CRUI i)NE;ST IN CO MN<M>& zero()NE;ST IN CO MN<M>& one()NE;ST CE uint Form(CRUI n)NE;ST CE ull& Reduction(ull& n)NE;ST CE ull& ReducedMU(ull& n,CRUI m)NE;ST CE uint MU(CRUI n0,CRUI n1)NE;ST CE uint BaseSquareTruncation(uint& n)NE;TE <TY T> CE MN<M>& Ref(T&& n)NE;};TE <uint M> CE MN<M> Twice(CO MN<M>& n)NE;TE <uint M> CE MN<M> Half(CO MN<M>& n)NE;TE <uint M> CE MN<M> Inverse(CO MN<M>& n);TE <uint M,TY T> CE MN<M> PW(MN<M> n,T EX);TE <TY T> CE MN<2> PW(CO MN<2>& n,CO T& p);TE <TY T> CE T Square(CO T& t);TE <> CE MN<2> Square<MN<2>>(CO MN<2>& t);TE <uint M> CE VO swap(MN<M>& n0,MN<M>& n1)NE;TE <uint M> IN string to_string(CO MN<M>& n)NE;TE<uint M,CL Traits> IN basic_istream<char,Traits>& OP>>(basic_istream<char,Traits>& is,MN<M>& n);TE<uint M,CL Traits> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO MN<M>& n);

TE <uint M>CL COantsForMod{PU:COantsForMod()= delete;ST CE CO bool g_even =((M & 1)== 0);ST CE CO uint g_memory_bound = 1000000;ST CE CO uint g_memory_LE = M < g_memory_bound?M:g_memory_bound;ST CE ull MNBasePW(ull&& EX)NE;ST CE uint g_M_minus = M - 1;ST CE uint g_M_minus_2 = M - 2;ST CE uint g_M_minus_2_neg = 2 - M;ST CE CO int g_MN_digit = 32;ST CE CO ull g_MN_base = ull(1)<< g_MN_digit;ST CE CO uint g_MN_base_minus = uint(g_MN_base - 1);ST CE CO uint g_MN_digit_half =(g_MN_digit + 1)>> 1;ST CE CO uint g_MN_base_sqrt_minus =(1 << g_MN_digit_half)- 1;ST CE CO uint g_MN_M_neg_inverse = uint((g_MN_base - MNBasePW((ull(1)<<(g_MN_digit - 1))- 1))& g_MN_base_minus);ST CE CO uint g_MN_base_mod = uint(g_MN_base % M);ST CE CO uint g_MN_base_square_mod = uint(((g_MN_base % M)*(g_MN_base % M))% M);};TE <uint M> CE ull COantsForMod<M>::MNBasePW(ull&& EX)NE{ull prod = 1;ull PW = M;WH(EX != 0){(EX & 1)== 1?(prod *= PW)&= g_MN_base_minus:prod;EX >>= 1;(PW *= PW)&= g_MN_base_minus;}RE prod;}

US MP = Mod<P>;US MNP = MN<P>;TE <uint M> CE uint MN<M>::Form(CRUI n)NE{ull n_copy = n;RE uint(MO(Reduction(n_copy *= COantsForMod<M>::g_MN_base_square_mod)));}TE <uint M> CE ull& MN<M>::Reduction(ull& n)NE{ull n_sub = n & COantsForMod<M>::g_MN_base_minus;RE((n +=((n_sub *= COantsForMod<M>::g_MN_M_neg_inverse)&= COantsForMod<M>::g_MN_base_minus)*= M)>>= COantsForMod<M>::g_MN_digit)< M?n:n -= M;}TE <uint M> CE ull& MN<M>::ReducedMU(ull& n,CRUI m)NE{RE Reduction(n *= m);}TE <uint M> CE uint MN<M>::MU(CRUI n0,CRUI n1)NE{ull n0_copy = n0;RE uint(MO(ReducedMU(ReducedMU(n0_copy,n1),COantsForMod<M>::g_MN_base_square_mod)));}TE <uint M> CE uint MN<M>::BaseSquareTruncation(uint& n)NE{CO uint n_u = n >> COantsForMod<M>::g_MN_digit_half;n &= COantsForMod<M>::g_MN_base_sqrt_minus;RE n_u;}TE <uint M> CE MN<M>::MN()NE:Mod<M>(){ST_AS(! COantsForMod<M>::g_even);}TE <uint M> CE MN<M>::MN(CO MN<M>& n)NE:Mod<M>(n){}TE <uint M> CE MN<M>::MN(MN<M>& n)NE:Mod<M>(n){}TE <uint M> CE MN<M>::MN(MN<M>&& n)NE:Mod<M>(MO(n)){}TE <uint M> TE <SFINAE_FOR_MN()> CE MN<M>::MN(CO T& n)NE:Mod<M>(n){ST_AS(! COantsForMod<M>::g_even);Mod<M>::m_n = Form(Mod<M>::m_n);}TE <uint M> TE <SFINAE_FOR_MN()> CE MN<M>::MN(T&& n)NE:Mod<M>(forward<T>(n)){ST_AS(! COantsForMod<M>::g_even);Mod<M>::m_n = Form(Mod<M>::m_n);}TE <uint M> CE MN<M>& MN<M>::OP=(CO MN<M>& n)NE{RE Ref(Mod<M>::OP=(n));}TE <uint M> CE MN<M>& MN<M>::OP=(MN<M>&& n)NE{RE Ref(Mod<M>::OP=(MO(n)));}TE <uint M> CE MN<M>& MN<M>::OP+=(CO MN<M>& n)NE{RE Ref(Mod<M>::OP+=(n));}TE <uint M> CE MN<M>& MN<M>::OP-=(CO MN<M>& n)NE{RE Ref(Mod<M>::OP-=(n));}TE <uint M> CE MN<M>& MN<M>::OP*=(CO MN<M>& n)NE{ull m_n_copy = Mod<M>::m_n;RE Ref(Mod<M>::m_n = MO(ReducedMU(m_n_copy,n.m_n)));}TE <uint M> IN MN<M>& MN<M>::OP/=(CO MN<M>& n){RE OP*=(MN<M>(n).Invert());}TE <uint M> CE MN<M>& MN<M>::OP<<=(int n)NE{RE Ref(Mod<M>::OP<<=(n));}TE <uint M> CE MN<M>& MN<M>::OP>>=(int n)NE{RE Ref(Mod<M>::OP>>=(n));}TE <uint M> CE MN<M>& MN<M>::OP++()NE{RE Ref(Mod<M>::Normalise(Mod<M>::m_n += COantsForMod<M>::g_MN_base_mod));}TE <uint M> CE MN<M> MN<M>::OP++(int)NE{MN<M> n{*TH};OP++();RE n;}TE <uint M> CE MN<M>& MN<M>::OP--()NE{RE Ref(Mod<M>::m_n < COantsForMod<M>::g_MN_base_mod?((Mod<M>::m_n += M)-= COantsForMod<M>::g_MN_base_mod):Mod<M>::m_n -= COantsForMod<M>::g_MN_base_mod);}TE <uint M> CE MN<M> MN<M>::OP--(int)NE{MN<M> n{*TH};OP--();RE n;}DF_OF_AR_FOR_MN(+,MN<M>(forward<T>(n))+= *TH);DF_OF_AR_FOR_MN(-,MN<M>(forward<T>(n)).SignInvert()+= *TH);DF_OF_AR_FOR_MN(*,MN<M>(forward<T>(n))*= *TH);DF_OF_AR_FOR_MN(/,MN<M>(forward<T>(n)).Invert()*= *TH);TE <uint M> CE MN<M> MN<M>::OP<<(int n)CO NE{RE MO(MN<M>(*TH)<<= n);}TE <uint M> CE MN<M> MN<M>::OP>>(int n)CO NE{RE MO(MN<M>(*TH)>>= n);}TE <uint M> CE MN<M> MN<M>::OP-()CO NE{RE MO(MN<M>(*TH).SignInvert());}TE <uint M> CE MN<M>& MN<M>::SignInvert()NE{RE Ref(Mod<M>::m_n > 0?Mod<M>::m_n = M - Mod<M>::m_n:Mod<M>::m_n);}TE <uint M> CE MN<M>& MN<M>::Double()NE{RE Ref(Mod<M>::Double());}TE <uint M> CE MN<M>& MN<M>::Halve()NE{RE Ref(Mod<M>::Halve());}TE <uint M> CE MN<M>& MN<M>::Invert(){assert(Mod<M>::m_n > 0);RE PositivePW(uint(COantsForMod<M>::g_M_minus_2));}TE <uint M> TE <TY T> CE MN<M>& MN<M>::PositivePW(T&& EX)NE{MN<M> PW{*TH};(--EX)%= COantsForMod<M>::g_M_minus_2;WH(EX != 0){(EX & 1)== 1?OP*=(PW):*TH;EX >>= 1;PW *= PW;}RE *TH;}TE <uint M> TE <TY T> CE MN<M>& MN<M>::NonNegativePW(T&& EX)NE{RE EX == 0?Ref(Mod<M>::m_n = COantsForMod<M>::g_MN_base_mod):PositivePW(forward<T>(EX));}TE <uint M> TE <TY T> CE MN<M>& MN<M>::PW(T&& EX){bool neg = EX < 0;assert(!(neg && Mod<M>::m_n == 0));RE neg?PositivePW(forward<T>(EX *= COantsForMod<M>::g_M_minus_2_neg)):NonNegativePW(forward<T>(EX));}TE <uint M> CE uint MN<M>::RP()CO NE{ull m_n_copy = Mod<M>::m_n;RE MO(Reduction(m_n_copy));}TE <uint M> CE Mod<M> MN<M>::Reduce()CO NE{ull m_n_copy = Mod<M>::m_n;RE Mod<M>::DeRP(MO(Reduction(m_n_copy)));}TE <uint M> CE MN<M> MN<M>::DeRP(CRUI n)NE{RE MN<M>(Mod<M>::DeRP(n));}TE <uint M> IN CO MN<M>& MN<M>::Formise(CRUI n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={zero(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = DeRP(LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN CO MN<M>& MN<M>::Inverse(CRUI n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={zero(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = MN<M>(Mod<M>::Inverse(LE_curr));LE_curr++;}RE memory[n];}TE <uint M> IN CO MN<M>& MN<M>::Factorial(CRUI n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;ST MN<M> val_curr{one()};ST MN<M> val_last{one()};WH(LE_curr <= n){memory[LE_curr++] = val_curr *= ++val_last;}RE memory[n];}TE <uint M> IN CO MN<M>& MN<M>::FactorialInverse(CRUI n)NE{ST MN<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;ST MN<M> val_curr{one()};ST MN<M> val_last{one()};WH(LE_curr <= n){memory[LE_curr] = val_curr *= Inverse(LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN MN<M> MN<M>::Combination(CRUI n,CRUI i)NE{RE i <= n?Factorial(n)*FactorialInverse(i)*FactorialInverse(n - i):zero();}TE <uint M> IN CO MN<M>& MN<M>::zero()NE{ST CE CO MN<M> z{};RE z;}TE <uint M> IN CO MN<M>& MN<M>::one()NE{ST CE CO MN<M> o{DeRP(1)};RE o;}TE <uint M> TE <TY T> CE MN<M>& MN<M>::Ref(T&& n)NE{RE *TH;}TE <uint M> CE MN<M> Twice(CO MN<M>& n)NE{RE MO(MN<M>(n).Double());}TE <uint M> CE MN<M> Half(CO MN<M>& n)NE{RE MO(MN<M>(n).Halve());}TE <uint M> CE MN<M> Inverse(CO MN<M>& n){RE MO(MN<M>(n).Invert());}TE <uint M,TY T> CE MN<M> PW(MN<M> n,T EX){RE MO(n.PW(EX));}TE <uint M> CE VO swap(MN<M>& n0,MN<M>& n1)NE{n0.swap(n1);}TE <uint M> IN string to_string(CO MN<M>& n)NE{RE to_string(n.RP())+ " + MZ";}TE<uint M,CL Traits> IN basic_istream<char,Traits>& OP>>(basic_istream<char,Traits>& is,MN<M>& n){ll m;is >> m;n = m;RE is;}TE<uint M,CL Traits> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO MN<M>& n){RE os << n.RP();}

TE <uint M> CE Mod<M>::Mod()NE:m_n(){}TE <uint M> CE Mod<M>::Mod(CO Mod<M>& n)NE:m_n(n.m_n){}TE <uint M> CE Mod<M>::Mod(Mod<M>& n)NE:m_n(n.m_n){}TE <uint M> CE Mod<M>::Mod(Mod<M>&& n)NE:m_n(MO(n.m_n)){}TE <uint M> TE <SFINAE_FOR_MOD()> CE Mod<M>::Mod(CO T& n)NE:m_n(RS<M>(n)){}TE <uint M> TE <SFINAE_FOR_MOD()> CE Mod<M>::Mod(T& n)NE:m_n(RS<M>(decay_t<T>(n))){}TE <uint M> TE <SFINAE_FOR_MOD()> CE Mod<M>::Mod(T&& n)NE:m_n(RS<M>(forward<T>(n))){}TE <uint M> CE Mod<M>& Mod<M>::OP=(CO Mod<M>& n)NE{RE Ref(m_n = n.m_n);}TE <uint M> CE Mod<M>& Mod<M>::OP=(Mod<M>&& n)NE{RE Ref(m_n = MO(n.m_n));}TE <uint M> CE Mod<M>& Mod<M>::OP+=(CO Mod<M>& n)NE{RE Ref(Normalise(m_n += n.m_n));}TE <uint M> CE Mod<M>& Mod<M>::OP-=(CO Mod<M>& n)NE{RE Ref(m_n < n.m_n?(m_n += M)-= n.m_n:m_n -= n.m_n);}TE <uint M> CE Mod<M>& Mod<M>::OP*=(CO Mod<M>& n)NE{RE Ref(m_n = COantsForMod<M>::g_even?RS<M>(ull(m_n)* n.m_n):MN<M>::MU(m_n,n.m_n));}TE <> CE MP& MP::OP*=(CO MP& n)NE{ull m_n_copy = m_n;RE Ref(m_n = MO((m_n_copy *= n.m_n)< P?m_n_copy:RSP(m_n_copy)));}TE <uint M> IN Mod<M>& Mod<M>::OP/=(CO Mod<M>& n){RE OP*=(Mod<M>(n).Invert());}TE <uint M> CE Mod<M>& Mod<M>::OP<<=(int n)NE{WH(n-- > 0){Normalise(m_n <<= 1);}RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP>>=(int n)NE{WH(n-- > 0){((m_n & 1)== 0?m_n:m_n += M)>>= 1;}RE *TH;}TE <uint M> CE Mod<M>& Mod<M>::OP++()NE{RE Ref(m_n < COantsForMod<M>::g_M_minus?++m_n:m_n = 0);}TE <uint M> CE Mod<M> Mod<M>::OP++(int)NE{Mod<M> n{*TH};OP++();RE n;}TE <uint M> CE Mod<M>& Mod<M>::OP--()NE{RE Ref(m_n == 0?m_n = COantsForMod<M>::g_M_minus:--m_n);}TE <uint M> CE Mod<M> Mod<M>::OP--(int)NE{Mod<M> n{*TH};OP--();RE n;}DF_OF_CM_FOR_MOD(==);DF_OF_CM_FOR_MOD(!=);DF_OF_CM_FOR_MOD(>);DF_OF_CM_FOR_MOD(>=);DF_OF_CM_FOR_MOD(<);DF_OF_CM_FOR_MOD(<=);DF_OF_AR_FOR_MOD(+,Mod<M>(forward<T>(n))+= *TH);DF_OF_AR_FOR_MOD(-,Mod<M>(forward<T>(n)).SignInvert()+= *TH);DF_OF_AR_FOR_MOD(*,Mod<M>(forward<T>(n))*= *TH);DF_OF_AR_FOR_MOD(/,Mod<M>(forward<T>(n)).Invert()*= *TH);TE <uint M> CE Mod<M> Mod<M>::OP<<(int n)CO NE{RE MO(Mod<M>(*TH)<<= n);}TE <uint M> CE Mod<M> Mod<M>::OP>>(int n)CO NE{RE MO(Mod<M>(*TH)>>= n);}TE <uint M> CE Mod<M> Mod<M>::OP-()CO NE{RE MO(Mod<M>(*TH).SignInvert());}TE <uint M> CE Mod<M>& Mod<M>::SignInvert()NE{RE Ref(m_n > 0?m_n = M - m_n:m_n);}TE <uint M> CE Mod<M>& Mod<M>::Double()NE{RE Ref(Normalise(m_n <<= 1));}TE <uint M> CE Mod<M>& Mod<M>::Halve()NE{RE Ref(((m_n & 1)== 0?m_n:m_n += M)>>= 1);}TE <uint M> IN Mod<M>& Mod<M>::Invert(){assert(m_n > 0);uint m_n_neg;RE m_n < COantsForMod<M>::g_memory_LE?Ref(m_n = Inverse(m_n).m_n):((m_n_neg = M - m_n)< COantsForMod<M>::g_memory_LE)?Ref(m_n = M - Inverse(m_n_neg).m_n):PositivePW(uint(COantsForMod<M>::g_M_minus_2));}TE <> IN Mod<2>& Mod<2>::Invert(){assert(m_n > 0);RE *TH;}TE <uint M> TE <TY T> CE Mod<M>& Mod<M>::PositivePW(T&& EX)NE{Mod<M> PW{*TH};EX--;WH(EX != 0){(EX & 1)== 1?OP*=(PW):*TH;EX >>= 1;PW *= PW;}RE *TH;}TE <> TE <TY T> CE Mod<2>& Mod<2>::PositivePW(T&& EX)NE{RE *TH;}TE <uint M> TE <TY T> CE Mod<M>& Mod<M>::NonNegativePW(T&& EX)NE{RE EX == 0?Ref(m_n = 1):Ref(PositivePW(forward<T>(EX)));}TE <uint M> TE <TY T> CE Mod<M>& Mod<M>::PW(T&& EX){bool neg = EX < 0;assert(!(neg && Mod<M>::m_n == 0));RE neg?PositivePW(forward<T>(EX *= COantsForMod<M>::g_M_minus_2_neg)):NonNegativePW(forward<T>(EX));}TE <uint M> IN CO Mod<M>& Mod<M>::Inverse(CRUI n)NE{ST Mod<M> memory[COantsForMod<M>::g_memory_LE] ={zero(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr].m_n = M - MN<M>::MU(memory[M % LE_curr].m_n,M / LE_curr);LE_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::Factorial(CRUI n)NE{ST Mod<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = MN<M>::Factorial(LE_curr).Reduce();LE_curr++;}RE memory[n];}TE <uint M> IN CO Mod<M>& Mod<M>::FactorialInverse(CRUI n)NE{ST Mod<M> memory[COantsForMod<M>::g_memory_LE] ={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory[LE_curr] = MN<M>::FactorialInverse(LE_curr).Reduce();LE_curr++;}RE memory[n];}TE <uint M> IN Mod<M> Mod<M>::Combination(CRUI n,CRUI i)NE{RE MN<M>::Combination(n,i).Reduce();}TE <uint M> CE VO Mod<M>::swap(Mod<M>& n)NE{std::swap(m_n,n.m_n);}TE <uint M> CE CRUI Mod<M>::RP()CO NE{RE m_n;}TE <uint M> CE Mod<M> Mod<M>::DeRP(CRUI n)NE{Mod<M> n_copy{};n_copy.m_n = n;RE n_copy;}TE <uint M> CE uint& Mod<M>::Normalise(uint& n)NE{RE n < M?n:n -= M;}TE <uint M> IN CO Mod<M>& Mod<M>::zero()NE{ST CE CO Mod<M> z{};RE z;}TE <uint M> IN CO Mod<M>& Mod<M>::one()NE{ST CE CO Mod<M> o{DeRP(1)};RE o;}TE <uint M> TE <TY T> CE Mod<M>& Mod<M>::Ref(T&& n)NE{RE *TH;}TE <uint M> CE Mod<M> Twice(CO Mod<M>& n)NE{RE MO(Mod<M>(n).Double());}TE <uint M> CE Mod<M> Half(CO Mod<M>& n)NE{RE MO(Mod<M>(n).Halve());}TE <uint M> IN Mod<M> Inverse(CO Mod<M>& n){RE MO(Mod<M>(n).Invert());}TE <uint M> CE Mod<M> Inverse_COrexpr(CRUI n)NE{RE MO(Mod<M>::DeRP(RS<M>(n)).NonNegativePW(M - 2));}TE <uint M,TY T> CE Mod<M> PW(Mod<M> n,T EX){RE MO(n.PW(EX));}TE <TY T>CE Mod<2> PW(Mod<2> n,CO T& p){RE p == 0?Mod<2>::one():MO(n);}TE <uint M> CE VO swap(Mod<M>& n0,Mod<M>& n1)NE{n0.swap(n1);}TE <uint M> IN string to_string(CO Mod<M>& n)NE{RE to_string(n.RP())+ " + MZ";}TE<uint M,CL Traits> IN basic_istream<char,Traits>& OP>>(basic_istream<char,Traits>& is,Mod<M>& n){ll m;is >> m;n = m;RE is;}TE<uint M,CL Traits> IN basic_ostream<char,Traits>& OP<<(basic_ostream<char,Traits>& os,CO Mod<M>& n){RE os << n.RP();}
// AAA 常設ライブラリは以上に挿入する。

#define INCLUDE_LIBRARY
#include __FILE__

#endif // INCLUDE_LIBRARY

#endif // INCLUDE_SUB

#endif // INCLUDE_MAIN
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