Problem:
Score : 200 points

Problem Statement

#nck {
width: 30px;
height: auto;
}

We have a grid of H rows and W columns. Initially, there is a stone in the top left cell. Shik is trying to move the stone to the bottom right cell. In each step, he can move the stone one cell to its left, up, right, or down (if such cell exists). It is possible that the stone visits a cell multiple times (including the bottom right and the top left cell).

You are given a matrix of characters a_{ij} (1 \leq i \leq H, 1 \leq j \leq W). After Shik completes all moving actions, a_{ij} is # if the stone had ever located at the i-th row and the j-th column during the process of moving. Otherwise, a_{ij} is .. Please determine whether it is possible that Shik only uses right and down moves in all steps.

Constraints

2 \leq H, W \leq 8

a_{i,j} is either # or ..

There exists a valid sequence of moves for Shik to generate the map a.

Input

The input is given from Standard Input in the following format:

H W
a_{11}a_{12}...a_{1W}
:
a_{H1}a_{H2}...a_{HW}

Output

If it is possible that Shik only uses right and down moves, print Possible. Otherwise, print Impossible.

Sample Input 1

4 5
##...
.##..
..##.
...##

Sample Output 1

Possible

The matrix can be generated by a 7-move sequence: right, down, right, down, right, down, and right.

Sample Input 2

5 3
###
..#
###
#..
###

Sample Output 2

Impossible

Sample Input 3

4 5
##...
.###.
.###.
...##

Sample Output 3

Impossible

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using Problem = Tmp.Problem;
namespace Tmp
{
	using static Func;
	using static Math;
	using static Console;
	//using GeometryLong;
	class Problem : IDisposable
	{
		bool IsGCJ;
		int Repeat;
		Scanner sc;
		Printer pr;
		public Problem(bool isGCJ, Scanner scanner, Printer printer)
		{
			sc = scanner;
			pr = printer;
			IsGCJ = isGCJ;
			if (isGCJ) Repeat = sc.Get<int>();
			else Read();
		}
		public Problem(bool isGCJ) : this(isGCJ, new Scanner(), new Printer()) { }
		public Problem(bool isGCJ, Scanner scanner) : this(isGCJ, scanner, new Printer()) { }
		public Problem(bool isGCJ, Printer printer) : this(isGCJ, new Scanner(), printer) { }
		public void Solve()
		{
			if (IsGCJ) for (var i = 0; i < Repeat; i++) { Read(); pr.Write("Case #" + (i + 1) + ": "); SolveOne(); }
			else SolveOne();
		}
		public void Dispose()
		{
			sc.Dispose();
			pr.Dispose();
		}
		public int Size { get { return 1; } }
		public const long Mod = 1000000007;
		//public const long Mod = 924844033;
		RandomSFMT rand = Program.rand;
		int Y, X;
		bool[,] map;
		void Read()
		{
			sc.Read(out Y, out X);
			map = sc.Board(X, Y, (c, x, y) => c == '#');
		}
		void SolveOne()
		{
			pr.WriteLine("Possible");return;
			var flag = map[0, 0] & map[X - 1, Y - 1];
			for (var x = 0; flag & x < X; x++) for (var y = 0; flag & y < Y; y++)
					if (map[x, y] && (x != X - 1 || y != Y - 1)) flag &= (x + 1 < X && map[x + 1, y]) ^ (y + 1 < Y && map[x, y + 1]);
			pr.WriteLine(flag ? "Possible" : "Impossible");
		}
		int[] SuffixArray(string S)
		{
			var N = S.Length;
			var sa = new int[N + 1];
			var r = new int[N + 1];
			for (var i = 0; i <= N; i++)
			{
				sa[i] = i;
				r[i] = i < N ? S[i] : -1;
			}
			var k = 1;
			Comparison<int> comp = (i, j) =>
			  {
				  if (r[i] != r[j]) return r[i] - r[j];
				  var a = i + k <= N ? r[i + k] : -1;
				  var b = j + k <= N ? r[j + k] : -1;
				  return a - b;
			  };
			for (; k <= N; k *= 2)
			{
				Array.Sort(sa, comp);
				var tmp = new int[N + 1];
				for (var i = 1; i <= N; i++) tmp[sa[i]] = tmp[sa[i - 1]] + (comp(sa[i - 1], sa[i]) < 0 ? 1 : 0);
				r = tmp;
			}
			return sa;
		}
	}
	class SlideMaximum
	{
		long[] a;
		Deque<int> deq;
		public SlideMaximum(long[] x) { a = x; deq = new Deque<int>(); }
		public void Add(int index)
		{
			while (deq.Count > 0 && a[deq.PeekBack()] <= a[index]) deq.PopBack();
			deq.PushBack(index);
		}
		public void Remove(int index)
		{
			if (deq.Count > 0 && deq.PeekFront() == index) deq.PopFront();
		}
		public long Maximum => a[deq.PeekFront()];
	}
	class SegmentTreeX
	{
		public const long Unit = -InfL;
		int N2;
		long[] seg, unif;
		public SegmentTreeX(int N)
		{
			N2 = 1;
			while (N2 < N) N2 <<= 1;
			seg = new long[2 * N2 - 1];
			unif = new long[2 * N2 - 1];
			for (var i = 0; i < 2 * N2 - 1; i++) seg[i] = unif[i] = Unit;
		}
		void LazyEvaluate(int node)
		{
			if (unif[node] != Unit)
			{
				seg[node] = Math.Max(seg[node], unif[node]);
				if (node < N2 - 1)
				{
					unif[2 * node + 1] = Math.Max(unif[2 * node + 1], unif[node]);
					unif[2 * node + 2] = Math.Max(unif[2 * node + 2], unif[node]);
				}
				unif[node] = Unit;
			}
		}
		void Update(int node) => seg[node] = Math.Max(seg[2 * node + 1], seg[2 * node + 2]);
		public void Maximize(int from, int to, long value) => Maximize(from, to, value, 0, 0, N2);
		void Maximize(int from, int to, long value, int node, int l, int r)
		{
			if (from <= l && r <= to) unif[node] = Math.Max(unif[node], value);
			else if (l < to && from < r)
			{
				Maximize(from, to, value, 2 * node + 1, l, (l + r) >> 1);
				Maximize(from, to, value, 2 * node + 2, (l + r) >> 1, r);
				Update(node);
			}
			LazyEvaluate(node);
		}
		public long this[int n] { get { return Max(n, n + 1); } set { Maximize(n, n + 1, value); } }
		public long Max(int from, int to) => Max(from, to, 0, 0, N2);
		long Max(int from, int to, int node, int l, int r)
		{
			LazyEvaluate(node);
			if (to <= l || r <= from) return Unit;
			else if (from <= l && r <= to) return seg[node];
			else return Math.Max(Max(from, to, 2 * node + 1, l, (l + r) >> 1), Max(from, to, 2 * node + 2, (l + r) >> 1, r));
		}
	}
}
class RMQI
{
	int N2;
	int[] segtree;
	int[] position;
	public RMQI(int N) : this(new int[N]) { }
	public RMQI(int[] array)
	{
		N2 = 1;
		while (N2 < array.Length) N2 <<= 1;
		segtree = new int[2 * N2 - 1];
		position = new int[2 * N2 - 1];
		for (var i = 0; i < 2 * N2 - 1; i++) segtree[i] = Func.Inf;
		for (var i = 0; i < array.Length; i++) { segtree[i + N2 - 1] = array[i]; position[i + N2 - 1] = i; }
		for (var i = N2 - 2; i >= 0; i--) SetMin(i);
	}
	void SetMin(int i)
	{
		int l = 2 * i + 1, r = 2 * i + 2;
		int a = segtree[l], b = segtree[r];
		if (a <= b) { segtree[i] = a; position[i] = position[l]; }
		else { segtree[i] = b; position[i] = position[r]; }
	}
	Tuple<int, int> Merge(Tuple<int, int> a, Tuple<int, int> b) => a.Item1 <= b.Item1 ? a : b;
	public void Update(int index, int value)
	{
		index += N2 - 1;
		segtree[index] = value;
		while (index > 0) SetMin(index = (index - 1) / 2);
	}
	public int this[int n] { get { return Min(n, n + 1).Item1; } set { Update(n, value); } }
	// min, pos
	public Tuple<int, int> Min(int from, int to) => Min(from, to, 0, 0, N2);
	Tuple<int, int> Min(int from, int to, int node, int l, int r)
	{
		if (to <= l || r <= from) return new Tuple<int, int>(Func.Inf, N2);
		else if (from <= l && r <= to) return new Tuple<int, int>(segtree[node], position[node]);
		else return Merge(Min(from, to, 2 * node + 1, l, (l + r) >> 1), Min(from, to, 2 * node + 2, (l + r) >> 1, r));
	}
}
static class Hoge
{
	public static T Peek<T>(this IEnumerable<T> set)
	{
		foreach (var x in set) return x;
		return default(T);
	}
}
interface ISegmentTree
{
	void Add(int from, int to, long value);
	long Min(int from, int to);
}
class SegmentTree2 : ISegmentTree
{
	int N;
	long[] a;
	public SegmentTree2(int N) : this(new long[N]) { }
	public SegmentTree2(long[] a) { N = a.Length; this.a = a.ToArray(); }
	public void Add(int from, int to, long value) { for (var i = from; i < to; i++) a[i] += value; }
	public long Min(int from, int to) { var s = Func.InfL; for (var i = from; i < to; i++) s = Math.Min(s, a[i]); return s; }
}
class SegmentTree3 : ISegmentTree
{
	public const long Unit = Func.InfL;
	public readonly Func<long, long, long> Operator = Math.Min;
	int N2;
	long[] seg, unif;
	public SegmentTree3(int N) : this(new long[N]) { }
	public SegmentTree3(long[] a)
	{
		N2 = 1;
		while (N2 < a.Length) N2 <<= 1;
		seg = new long[2 * N2 - 1];
		unif = new long[2 * N2 - 1];
		for (var i = a.Length + N2 - 1; i < 2 * N2 - 1; i++) seg[i] = Unit;
		for (var i = 0; i < a.Length; i++) seg[i + N2 - 1] = a[i];
		for (var i = N2 - 2; i >= 0; i--) Update(i);
	}
	void LazyEvaluate(int node)
	{
		if (unif[node] != 0)
		{
			seg[node] += unif[node];
			if (node < N2 - 1) { unif[2 * node + 1] += unif[node]; unif[2 * node + 2] += unif[node]; }
			unif[node] = 0;
		}
	}
	void Update(int node) => seg[node] = Operator(seg[2 * node + 1], seg[2 * node + 2]);
	public void Add(int from, int to, long value) => Add(from, to, value, 0, 0, N2);
	void Add(int from, int to, long value, int node, int l, int r)
	{
		if (from <= l && r <= to) unif[node] += value;
		else if (l < to && from < r)
		{
			Add(from, to, value, 2 * node + 1, l, (l + r) >> 1);
			Add(from, to, value, 2 * node + 2, (l + r) >> 1, r);
			Update(node);
		}
		LazyEvaluate(node);
	}
	public long this[int n] { get { return Min(n, n + 1); } set { Add(n, n + 1, value - this[n]); } }
	public long Min(int from, int to) => Min(from, to, 0, 0, N2);
	long Min(int from, int to, int node, int l, int r)
	{
		LazyEvaluate(node);
		if (to <= l || r <= from) return Unit;
		else if (from <= l && r <= to) return seg[node];
		else return Operator(Min(from, to, 2 * node + 1, l, (l + r) >> 1), Min(from, to, 2 * node + 2, (l + r) >> 1, r));
	}
}
class SegmentTree : ISegmentTree
{
	int N2;
	long[] seg, unif;
	public SegmentTree(int N) : this(new long[N]) { }
	public SegmentTree(long[] a)
	{
		N2 = 1;
		while (N2 < a.Length) N2 <<= 1;
		seg = new long[2 * N2 - 1];
		unif = new long[2 * N2 - 1];
		for (var i = a.Length + N2 - 1; i < 2 * N2 - 1; i++) seg[i] = Func.InfL;
		for (var i = 0; i < a.Length; i++) seg[i + N2 - 1] = a[i];
		for (var i = N2 - 2; i >= 0; i--) seg[i] = Math.Min(seg[2 * i + 1], seg[2 * i + 2]);
	}
	public void Add(int from, int to, long value) => Add(from, to, value, 0, 0, N2);
	void Add(int from, int to, long value, int node, int l, int r)
	{
		if (to <= l || r <= from) return;
		else if (from <= l && r <= to) unif[node] += value;
		else
		{
			Add(from, to, value, 2 * node + 1, l, (l + r) >> 1);
			Add(from, to, value, 2 * node + 2, (l + r) >> 1, r);
			seg[node] = Math.Min(seg[2 * node + 1] + unif[2 * node + 1], seg[2 * node + 2] + unif[2 * node + 2]);
		}
	}
	public long this[int n] { get { return Min(n, n + 1); } set { Add(n, n + 1, value - this[n]); } }
	public long Min(int from, int to) => Min(from, to, 0, 0, N2);
	long Min(int from, int to, int node, int l, int r)
	{
		if (to <= l || r <= from) return Func.InfL;
		else if (from <= l && r <= to) return seg[node] + unif[node];
		else return Math.Min(Min(from, to, 2 * node + 1, l, (l + r) >> 1), Min(from, to, 2 * node + 2, (l + r) >> 1, r)) + unif[node];
	}
}
class Eq : IEqualityComparer<List<int>>
{
	public bool Equals(List<int> x, List<int> y)
	{
		if (x == null || y == null) return x == y;
		if (x.Count != y.Count) return false;
		for (var i = 0; i < x.Count; i++) if (x[i] != y[i]) return false;
		return true;
	}
	public int GetHashCode(List<int> obj)
	{
		var x = obj.Count.GetHashCode();
		foreach (var i in obj) x ^= i.GetHashCode();
		return x;
	}
}
/*class MultiSortedSet<T> : IEnumerable<T>, ICollection<T>
{
	public IComparer<T> Comparer { get; private set; }
	private SortedSet<T> keys;
	private Dictionary<T, int> mult;
	public int Multiplicity(T item) { return mult[item]; }
	public int Count { get; private set; }
	public MultiSortedSet(IComparer<T> comp)
	{
		keys = new SortedSet<T>(Comparer = comp);
		mult = new Dictionary<T, int>();
	}
	public MultiSortedSet(Comparison<T> comp) : this(Comparer<T>.Create(comp)) { }
	public MultiSortedSet() : this(Func.DefaultComparison<T>()) { }
	public void Add(T item) { Add(item, 1); }
	private void Add(T item, int num)
	{
		Count += num;
		if (!keys.Contains(item)) { keys.Add(item); mult.Add(item, num); }
		else mult[item] += num;
	}
	public void AddRange(IEnumerable<T> list) { foreach (var x in list) Add(x); }
	public bool Remove(T item)
	{
		if (!keys.Contains(item)) return false;
		Count--;
		if (mult[item] == 1) { keys.Remove(item); mult.Remove(item); }
		else mult[item]--;
		return true;
	}
	public bool Overlaps(IEnumerable<T> other) { return keys.Overlaps(other); }
	public bool IsSupersetOf(IEnumerable<T> other) { return keys.IsSupersetOf(other); }
	public bool IsSubsetOf(IEnumerable<T> other) { return keys.IsSubsetOf(other); }
	public bool IsProperSubsetOf(IEnumerable<T> other) { return keys.IsProperSubsetOf(other); }
	public bool IsProperSupersetOf(IEnumerable<T> other) { return keys.IsProperSupersetOf(other); }
	public void ExceptWith(IEnumerable<T> other) { foreach (var x in other) if (Contains(x)) Remove(x); }
	public void IntersectWith(IEnumerable<T> other)
	{
		var next = new MultiSortedSet<T>(Comparer);
		foreach (var x in other) if (Contains(x) && !next.Contains(x)) next.Add(x, mult[x]);
		keys = next.keys; mult = next.mult;
	}
	public void CopyTo(T[] array) { CopyTo(array, 0); }
	public void CopyTo(T[] array, int index) { foreach (var item in array) array[index++] = item; }
	public void CopyTo(T[] array, int index, int count) { var i = 0; foreach (var item in array) { if (i++ >= count) return; array[index++] = item; } }
	public bool Contains(T item) { return keys.Contains(item); }
	public void Clear() { keys.Clear(); mult.Clear(); Count = 0; }
	public IEnumerator<T> Reverse() { foreach (var x in keys.Reverse()) for (var i = 0; i < mult[x]; i++) yield return x; }
	public IEnumerator<T> GetEnumerator() { foreach (var x in keys) for (var i = 0; i < mult[x]; i++) yield return x; }
	IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
	public T Max { get { return keys.Max; } }
	public T Min { get { return keys.Min; } }
	public bool IsReadOnly { get { return false; } }
}*/
class SkewHeap<T> : IEnumerable<T>
{
	class Node : IEnumerable<T>
	{
		public Node l, r;
		public T val;
		public Node(T x) { l = r = null; val = x; }
		public IEnumerator<T> GetEnumerator()
		{
			if (l != null) foreach (var x in l) yield return x;
			yield return val;
			if (r != null) foreach (var x in r) yield return x;
		}
		IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
	}
	public int Count { get; private set; }
	Node head;
	Comparison<T> comp;
	public bool IsEmpty => head != null;
	public SkewHeap(Comparison<T> c) { comp = c; Count = 0; }
	public SkewHeap() : this(Func.DefaultComparison<T>()) { }
	public SkewHeap(IComparer<T> c) : this(Func.ToComparison(c)) { }
	private SkewHeap(Comparison<T> c, Node h) : this(c) { head = h; }
	public void Push(T x) { var n = new Node(x); head = Meld(head, n); Count++; }
	public T Peek() => head.val;
	public T Pop() { var x = head.val; head = Meld(head.l, head.r); Count--; return x; }
	// a.comp must be equivalent to b.comp
	// a, b will be destroyed
	public static SkewHeap<T> Meld(SkewHeap<T> a, SkewHeap<T> b) => new SkewHeap<T>(a.comp, a.Meld(a.head, b.head));
	public void MeldWith(SkewHeap<T> a) => head = Meld(head, a.head);
	Node Meld(Node a, Node b)
	{
		if (a == null) return b;
		else if (b == null) return a;
		if (comp(a.val, b.val) > 0) Func.Swap(ref a, ref b);
		a.r = Meld(a.r, b);
		Func.Swap(ref a.l, ref a.r);
		return a;
	}
	public IEnumerator<T> GetEnumerator() => head.GetEnumerator();
	IEnumerator IEnumerable.GetEnumerator() => (IEnumerator)GetEnumerator();
}
// [0, Size) の整数の集合を表す
class BITSet : BinaryIndexedTree
{
	public BITSet(int size) : base(size) { }
	public void Add(int item) => Add(item, 1);
	public bool Contains(int item) => Sum(item, item + 1) > 0;
	public int Count(int item) => Sum(item, item + 1);
	// 順位 = item が小さい方から何番目か(0-indexed)
	public int GetRank(int item) => Sum(0, item);
	public void Remove(int item) => Add(item, -1);
	public void RemoveAll(int item) => Add(item, -Count(item));
	// 0-indexed で順位が rank のものを求める
	// ない場合は Size が返る
	public int GetValue(int rank) => Func.FirstBinary(0, Size, t => Sum(0, t + 1) >= rank + 1);
}
class RangeBIT
{
	public int N { get; private set; }
	long[,] bit;
	public RangeBIT(int N) { bit = new long[2, this.N = N + 1]; }
	public RangeBIT(int[] array) : this(array.Length)
	{
		for (var i = 1; i < N; i++) bit[0, i] = array[i - 1];
		for (var i = 1; i < N - 1; i++) if (i + (i & (-i)) < N) bit[0, i + (i & (-i))] += bit[0, i];
	}
	public RangeBIT(long[] array) : this(array.Length)
	{
		for (var i = 1; i < N; i++) bit[0, i] = array[i - 1];
		for (var i = 1; i < N - 1; i++) if (i + (i & (-i)) < N) bit[0, i + (i & (-i))] += bit[0, i];
	}
	public void Add(int from, int to, long value)
	{
		Add2(0, from + 1, -value * from);
		Add2(1, from + 1, value);
		Add2(0, to + 1, value * to);
		Add2(1, to + 1, -value);
	}
	void Add2(int which, int i, long value) { while (i < N) { bit[which, i] += value; i += i & (-i); } }
	long Sum(int to) => Sum2(0, to) + Sum2(1, to) * to;
	public long Sum(int from, int to) => Sum(to) - Sum(from);
	long Sum2(int which, int i) { var sum = 0L; while (i > 0) { sum += bit[which, i]; i -= i & (-i); } return sum; }
}
class RMQ
{
	int N2;
	int[] segtree;
	public RMQ(int N) : this(new int[N]) { }
	public RMQ(int[] array)
	{
		N2 = 1;
		while (N2 < array.Length) N2 <<= 1;
		segtree = new int[2 * N2 - 1];
		for (var i = 0; i < 2 * N2 - 1; i++) segtree[i] = Func.Inf;
		for (var i = 0; i < array.Length; i++) segtree[i + N2 - 1] = array[i];
		for (var i = N2 - 2; i >= 0; i--) segtree[i] = Math.Min(segtree[2 * i + 1], segtree[2 * i + 2]);
	}
	public void Update(int index, int value)
	{
		index += N2 - 1;
		segtree[index] = value;
		while (index > 0)
		{
			index = (index - 1) / 2;
			segtree[index] = Math.Min(segtree[index * 2 + 1], segtree[index * 2 + 2]);
		}
	}
	public int this[int n] { get { return Min(n, n + 1); } set { Update(n, value); } }
	public int Min(int from, int to) => Min(from, to, 0, 0, N2);
	int Min(int from, int to, int node, int l, int r)
	{
		if (to <= l || r <= from) return Func.Inf;
		else if (from <= l && r <= to) return segtree[node];
		else return Math.Min(Min(from, to, 2 * node + 1, l, (l + r) >> 1), Min(from, to, 2 * node + 2, (l + r) >> 1, r));
	}
}
class Program
{
	public static RandomSFMT rand = new RandomSFMT();
	public static bool IsJudgeMode = true;
	public static bool IsGCJMode = false;
	public static bool IsSolveCreated = true;
	static void Main()
	{
		if (IsJudgeMode)
			if (IsGCJMode) using (var problem = new Problem(true, new Scanner("C-large-practice.in.txt"), new Printer("output.txt"))) problem.Solve();
			else using (var problem = new Problem(false, new Printer())) problem.Solve();
		else
		{
			var num = 1;
			int size = 0;
			decimal time = 0;
			for (var tmp = 0; tmp < num; tmp++)
			{
				using (var P = IsSolveCreated ? new Problem(false, new Scanner("input.txt"), new Printer()) : new Problem(false))
				{
					size = P.Size;
					time += Func.MeasureTime(() => P.Solve());
				}
			}
			Console.WriteLine("{0}, {1}ms", size, time / num);
		}
	}
}
class BinaryIndexedTree3D
{
	public int X { get; private set; }
	public int Y { get; private set; }
	public int Z { get; private set; }
	int[,,] bit;
	public BinaryIndexedTree3D(int X, int Y, int Z)
	{
		this.X = X; this.Y = Y; this.Z = Z;
		bit = new int[X + 1, Y + 1, Z + 1];
	}
	public BinaryIndexedTree3D(int[,,] array)
		: this(array.GetLength(0), array.GetLength(1), array.GetLength(2))
	{
		for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) for (var z = 0; z < Z; z++) Add(x, y, z, array[x, y, z]);
	}
	public void Add(int x, int y, int z, int value)
	{
		for (var i = x + 1; i <= X; i += i & (-i)) for (var j = y + 1; j <= Y; j += j & (-j)) for (var k = z + 1; k <= Z; k += k & (-k)) bit[i, j, k] += value;
	}
	public int Sum(int x0, int y0, int z0, int x1, int y1, int z1)
		=> Sum(x1, y1, z1) - Sum(x0, y1, z1) - Sum(x1, y0, z1) - Sum(x1, y1, z0) + Sum(x1, y0, z0) + Sum(x0, y1, z0) + Sum(x0, y0, z1) - Sum(x0, y0, z0);
	int Sum(int x, int y, int z)
	{
		var sum = 0;
		for (var i = x; i > 0; i -= i & (-i)) for (var j = y; j > 0; j -= j & (-j)) for (var k = y; k > 0; k -= k & (-k)) sum += bit[i, j, k];
		return sum;
	}
}
class BinaryIndexedTree2D
{
	public int X { get; private set; }
	public int Y { get; private set; }
	int[,] bit;
	public BinaryIndexedTree2D(int X, int Y)
	{
		this.X = X; this.Y = Y;
		bit = new int[X + 1, Y + 1];
	}
	public BinaryIndexedTree2D(int[,] array)
		: this(array.GetLength(0), array.GetLength(1))
	{
		for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) Add(x, y, array[x, y]);
	}
	public void Add(int x, int y, int value) { for (var i = x + 1; i <= X; i += i & (-i)) for (var j = y + 1; j <= Y; j += j & (-j)) bit[i, j] += value; }
	public int Sum(int x0, int y0, int x1, int y1) => Sum(x0, y0) + Sum(x1, y1) - Sum(x0, y1) - Sum(x1, y0);
	int Sum(int x, int y) { var sum = 0; for (var i = x; i > 0; i -= i & (-i)) for (var j = y; j > 0; j -= j & (-j)) sum += bit[i, j]; return sum; }
}
class BinaryIndexedTree
{
	public int Size { get; private set; }
	int[] bit;
	public BinaryIndexedTree(int size)
	{
		Size = size;
		bit = new int[size + 1];
	}
	public BinaryIndexedTree(int[] array) : this(array.Length)
	{
		for (var i = 0; i < Size; i++) bit[i + 1] = array[i];
		for (var i = 1; i < Size; i++) if (i + (i & (-i)) <= Size) bit[i + (i & (-i))] += bit[i];
	}
	// index is 0-indexed
	public void Add(int index, int value) { for (var i = index + 1; i <= Size; i += i & (-i)) bit[i] += value; }
	// from, to is 0-indexed
	// from is inclusive, to is exclusive
	public int Sum(int from, int to) => Sum(to) - Sum(from);
	int Sum(int to) { var sum = 0; for (var i = to; i > 0; i -= i & (-i)) sum += bit[i]; return sum; }
}
class Amoeba
{
	public const int Dimension = 2;
	public const double Alpha = 1;  // reflection
	public const double Beta = 1 + 2.0 / Dimension; // expansion
	public const double Gamma = 0.75 - 0.5 / Dimension; // contraction
	public const double Delta = 1 - 1.0 / Dimension;    // shrink
	public Pair<AmoebaState, double>[] a;
	public AmoebaState m;
	public void Initiate()
	{
		Array.Sort(a, (x, y) => x.Second.CompareTo(y.Second));
		m = new AmoebaState();
		for (var i = 0; i < Dimension; i++) m.Add(a[i].First);
		m.Multiply(1.0 / Dimension);
	}
	void PartialSort(int i, int j) { if (a[i].Second > a[j].Second) a.Swap(i, j); }
	void Accept(AmoebaState point, double value)
	{
		var tmp = Func.FirstBinary(0, Dimension, x => a[x].Second >= value);
		if (tmp != Dimension) m.Add((point - a[Dimension - 1].First) / Dimension);
		for (var i = Dimension; i > tmp; i--) a[i] = a[i - 1];
		a[tmp].First = point;
		a[tmp].Second = value;
	}
	public void Search()
	{
		var r = m + Alpha * (m - a[Dimension].First);
		var fr = r.Func();
		if (a[0].Second <= fr && fr < a[Dimension - 1].Second) { Accept(r, fr); return; }
		var diff = r - m;
		if (fr < a[0].Second)
		{
			var e = m + Beta * diff;
			var fe = e.Func();
			if (fe < fr) Accept(e, fe);
			else Accept(r, fr);
		}
		else
		{
			var tmp = Gamma * diff;
			var o = m + tmp;
			var fo = o.Func();
			var i = m - tmp;
			var fi = i.Func();
			if (fi < fo) { o = i; fo = fi; }
			if (fo < a[Dimension - 1].Second) Accept(o, fo);
			else Shrink();
		}
	}
	void Shrink()
	{
		var tmp = (1 - Delta) * a[0].First;
		for (var i = 1; i <= Dimension; i++) { a[i].First.Multiply(Delta); a[i].First.Add(tmp); a[i].Second = a[i].First.Func(); }
		Initiate();
	}
}
class AmoebaState
{
	public static int Dimension = 2;
	public double[] vec;
	public AmoebaState() { vec = new double[Dimension]; }
	public AmoebaState(params double[] elements) : this() { elements.CopyTo(vec, 0); }
	public double this[int n] { get { return vec[n]; } set { vec[n] = value; } }
	public void Multiply(double r) { for (var i = 0; i < Dimension; i++) vec[i] *= r; }
	public void Add(AmoebaState v) { for (var i = 0; i < Dimension; i++) vec[i] += v.vec[i]; }
	public static AmoebaState operator +(AmoebaState p) => new AmoebaState(p.vec);
	public static AmoebaState operator -(AmoebaState p) { var tmp = new AmoebaState(p.vec); tmp.Multiply(-1); return tmp; }
	public static AmoebaState operator /(AmoebaState p, double r) { var tmp = new AmoebaState(p.vec); tmp.Multiply(1 / r); return tmp; }
	public static AmoebaState operator *(double r, AmoebaState p) { var tmp = new AmoebaState(p.vec); tmp.Multiply(r); return tmp; }
	public static AmoebaState operator *(AmoebaState p, double r) => r * p;
	public static AmoebaState operator +(AmoebaState p, AmoebaState q) { var tmp = +p; tmp.Add(q); return tmp; }
	public static AmoebaState operator -(AmoebaState p, AmoebaState q) { var tmp = -q; tmp.Add(p); return tmp; }
	public double Func()
	{
		return 0;//P.Func(vec[0], vec[1]);
	}
	public static Problem P;
}
class BucketList<T> : ICollection<T>, IEnumerable<T>, ICollection, IEnumerable
{
	public Comparison<T> comp { get; protected set; }
	public int BucketSize = 20;
	public int Count { get { var sum = 0; var bucket = Head; while (bucket != null) { sum += bucket.Count; bucket = bucket.Next; } return sum; } }
	public int NumOfBucket { get; protected set; }
	public Bucket<T> Head { get; protected set; }
	public Bucket<T> Tail { get; protected set; }
	public BucketList(IComparer<T> comp) : this(comp.ToComparison()) { }
	public BucketList(Comparison<T> comp = null) { Head = null; Tail = null; NumOfBucket = 0; this.comp = comp ?? Func.DefaultComparison<T>(); }
	protected void AddAfter(Bucket<T> pos, Bucket<T> bucket)
	{
		Debug.Assert(bucket != null && bucket.Count > 0 && pos != null && pos.Parent == this && comp(pos.Tail.Value, bucket.Head.Value) <= 0
					&& (pos.Next == null || comp(pos.Next.Head.Value, bucket.Tail.Value) >= 0));
		bucket.Parent = this;
		bucket.Prev = pos;
		bucket.Next = pos.Next;
		if (pos != Tail) pos.Next.Prev = bucket;
		else Tail = bucket;
		pos.Next = bucket;
		NumOfBucket++;
	}
	protected void AddBefore(Bucket<T> pos, Bucket<T> bucket)
	{
		Debug.Assert(bucket != null && bucket.Count > 0 && pos != null && pos.Parent == this && comp(pos.Head.Value, bucket.Tail.Value) >= 0
					&& (pos.Prev == null || comp(pos.Prev.Tail.Value, bucket.Head.Value) <= 0));
		bucket.Parent = this;
		bucket.Prev = pos.Prev;
		bucket.Next = pos;
		if (pos != Head) pos.Prev.Next = bucket;
		else Head = bucket;
		pos.Prev = bucket;
		NumOfBucket++;
	}
	protected void AddAfter(Bucket<T> bucket, BucketNode<T> node)
	{
		Debug.Assert(node != null && bucket != null && bucket.Parent == this && node.Parent.Parent == this && comp(bucket.Tail.Value, node.Value) <= 0
					&& (bucket.Next == null || comp(bucket.Next.Head.Value, node.Value) >= 0));
		var tmp = new Bucket<T>(this, bucket, bucket.Next);
		tmp.InitiateWith(node);
		if (bucket != Tail) bucket.Next.Prev = tmp;
		else Tail = tmp;
		bucket.Next = tmp;
		NumOfBucket++;
	}
	protected void AddBefore(Bucket<T> bucket, BucketNode<T> node)
	{
		Debug.Assert(node != null && bucket != null && bucket.Parent == this && node.Parent.Parent == this && comp(bucket.Head.Value, node.Value) >= 0
					&& (bucket.Prev == null || comp(bucket.Prev.Tail.Value, node.Value) <= 0));
		var tmp = new Bucket<T>(this, bucket.Prev, bucket);
		tmp.InitiateWith(node);
		if (bucket != Head) bucket.Prev.Next = tmp;
		else Head = tmp;
		bucket.Prev = tmp;
		NumOfBucket++;
	}
	public void AddAfter(BucketNode<T> node, T item)
	{
		Debug.Assert(node != null && node.Parent.Parent == this && comp(node.Value, item) <= 0
					&& ((node.Next == null && (node.Parent.Next == null || comp(node.Parent.Next.Head.Value, item) >= 0))
						|| comp(node.Next.Value, item) >= 0));
		var bucket = node.Parent;
		var tmp = new BucketNode<T>(item, bucket, node, node.Next);
		if (!bucket.AddAfter(node, tmp))
		{
			if (node.Next == null && (bucket.Next == null || bucket.Next.Count >= BucketSize)) AddAfter(bucket, tmp);
			else if (node.Next == null) AddBefore(bucket.Next.Head, item);
			else
			{
				node.Next.Prev = tmp;
				node.Next = tmp;
				while (node.Next.Next != null) node = node.Next;
				item = node.Next.Value;
				bucket.Tail = node;
				node.Next = null;
				AddAfter(node, item);
			}
		}
	}
	public void AddBefore(BucketNode<T> node, T item)
	{
		Debug.Assert(node != null && node.Parent.Parent == this && comp(node.Value, item) >= 0
					&& ((node.Prev == null && (node.Parent.Prev == null || comp(node.Parent.Prev.Tail.Value, item) <= 0))
						|| comp(node.Prev.Value, item) <= 0));
		var bucket = node.Parent;
		var tmp = new BucketNode<T>(item, bucket, node.Prev, node);
		if (!bucket.AddBefore(node, tmp))
		{
			if (node.Prev == null && (bucket.Prev == null || bucket.Prev.Count >= BucketSize)) AddBefore(bucket, tmp);
			else if (node.Prev == null) AddAfter(bucket.Prev.Tail, item);
			else
			{
				node.Prev.Next = tmp;
				node.Prev = tmp;
				while (node.Prev.Prev != null) node = node.Prev;
				item = node.Prev.Value;
				bucket.Head = node;
				node.Prev = null;
				AddBefore(node, item);
			}
		}
	}
	// (node, index)
	// index is the position of node in node.Parent
	public Tuple<BucketNode<T>, int> UpperBound(Predicate<T> pred)
	{
		if (NumOfBucket == 0) return null;
		if (pred(Tail.Tail.Value)) return new Tuple<BucketNode<T>, int>(Tail.Tail, Tail.Count - 1);
		var bucket = Tail;
		while (bucket.Prev != null && !pred(bucket.Prev.Tail.Value)) bucket = bucket.Prev;
		var node = bucket.Tail;
		var index = bucket.Count - 1;
		while (node.Prev != null && !pred(node.Prev.Value)) { node = node.Prev; index--; }
		if (node.Prev == null) return bucket.Prev == null ? null : new Tuple<BucketNode<T>, int>(bucket.Prev.Tail, bucket.Prev.Count - 1);
		else return new Tuple<BucketNode<T>, int>(node.Prev, index - 1);
	}
	public Tuple<BucketNode<T>, int> UpperBound(T item) => LowerBound(x => comp(x, item) <= 0);
	// (node, index)
	// index is the position of node in node.Parent
	public Tuple<BucketNode<T>, int> LowerBound(Predicate<T> pred)
	{
		if (NumOfBucket == 0) return null;
		if (pred(Head.Head.Value)) return new Tuple<BucketNode<T>, int>(Head.Head, 0);
		var bucket = Head;
		while (bucket.Next != null && !pred(bucket.Next.Head.Value)) bucket = bucket.Next;
		var node = bucket.Head;
		var index = 0;
		while (node.Next != null && !pred(node.Next.Value)) { node = node.Next; index++; }
		if (node.Next == null) return bucket.Next == null ? null : new Tuple<BucketNode<T>, int>(bucket.Next.Head, 0);
		else return new Tuple<BucketNode<T>, int>(node.Next, index + 1);
	}
	public Tuple<BucketNode<T>, int> LowerBound(T item) => LowerBound(x => comp(x, item) >= 0);
	public void InitiateWith(Bucket<T> bucket)
	{
		Debug.Assert(bucket != null && bucket.Count > 0);
		RemoveAll();
		Head = Tail = bucket;
		bucket.Parent = this;
		NumOfBucket++;
	}
	public void InitiateWith(T item)
	{
		RemoveAll();
		Head = Tail = new Bucket<T>(this, null, null);
		Head.Head = Head.Tail = new BucketNode<T>(item, Head, null, null);
		Head.Count++;
		NumOfBucket++;
	}
	public void AddFirst(Bucket<T> bucket) { if (NumOfBucket == 0) InitiateWith(bucket); else AddBefore(Head, bucket); }
	public void AddLast(Bucket<T> bucket) { if (NumOfBucket == 0) InitiateWith(bucket); else AddAfter(Tail, bucket); }
	public void AddFirst(T item) { if (NumOfBucket == 0) InitiateWith(item); else AddBefore(Head.Head, item); }
	public void AddLast(T item) { if (NumOfBucket == 0) InitiateWith(item); else AddAfter(Tail.Tail, item); }
	public void Clear() => RemoveAll();
	public void RemoveAll() { Head = Tail = null; NumOfBucket = 0; }
	public void RemoveFirst() { if (NumOfBucket == 0) throw new InvalidOperationException(); else Remove(Head.Head); }
	public void RemoveLast() { if (NumOfBucket == 0) throw new InvalidOperationException(); else Remove(Tail.Tail); }
	// remove item and return whether item was removed or not
	public bool Remove(T item) { var node = Find(item); if (node != null) Remove(node); return node != null; }
	public void Remove(Bucket<T> bucket)
	{
		Debug.Assert(bucket != null && bucket.Parent == this);
		NumOfBucket--;
		if (bucket == Head && bucket == Tail) { Head = Tail = null; }
		else if (bucket == Head) { Head.Next.Prev = null; Head = Head.Next; }
		else if (bucket == Tail) { Tail.Prev.Next = null; Tail = Tail.Prev; }
		else { bucket.Prev.Next = bucket.Next; bucket.Next.Prev = bucket.Prev; }
	}
	public void Remove(BucketNode<T> node) { Debug.Assert(node != null && node.Parent.Parent == this); if (!node.Parent.Remove(node)) Remove(node.Parent); }
	protected void RemoveRange(Bucket<T> from, Bucket<T> to, int indexFrom = -1, int indexTo = -1)
	{
		Debug.Assert(from != null && to != null && from.Parent == this && to.Parent == this);
		if (indexFrom < 0) indexFrom = from.Index;
		if (indexTo < 0) indexTo = to.Index;
		if (indexFrom == 0 && indexTo == NumOfBucket - 1) { Clear(); return; }
		else if (indexFrom == 0) { Head = to.Next; Head.Prev = null; }
		else if (indexTo == NumOfBucket - 1) { Tail = from.Prev; Tail.Next = null; }
		else { from.Prev.Next = to.Next; to.Next.Prev = from.Prev; }
		NumOfBucket -= indexTo - indexFrom + 1;
	}
	public void RemoveRange(BucketNode<T> from, BucketNode<T> to, int indexFrom = -1, int indexTo = -1)
	{
		Debug.Assert(from != null && to != null && from.Parent.Parent == this && to.Parent.Parent == this);
		if (indexFrom < 0) indexFrom = from.Index;
		if (indexTo < 0) indexTo = to.Index;
		var bucketFrom = from.Parent;
		var bucketTo = to.Parent;
		if (bucketFrom == bucketTo)
		{
			var bucket = bucketFrom;
			if (indexFrom == 0 && indexTo == bucket.Count - 1) Remove(bucket);
			else bucket.RemoveRange(from, to, indexFrom, indexTo);
		}
		else
		{
			var bf = bucketFrom.Index;
			var bt = bucketTo.Index;
			Debug.Assert(bf < bt);
			if (bt > bf + 1) RemoveRange(bucketFrom.Next, bucketTo.Prev, bf + 1, bt - 1);
			if (indexFrom == 0) { Remove(bucketFrom); RemoveRange(bucketTo.Head, to, 0, indexTo); }
			else if (indexTo == bucketTo.Count - 1) { Remove(bucketTo); RemoveRange(from, bucketFrom.Tail, indexFrom, bucketFrom.Count - 1); }
			else
			{
				bucketFrom.RemoveRange(from, bucketFrom.Tail, indexFrom, bucketFrom.Count - 1);
				bucketTo.RemoveRange(bucketTo.Head, to, 0, indexTo);
				if (bucketFrom.Count + bucketTo.Count < BucketSize) Adjust();
			}
		}
	}
	public void Adjust()
	{
		var array = this.ToArray();
		Clear();
		var length = array.Length;
		BucketSize = (int)Math.Sqrt(length + 1);
		var count = (length + BucketSize - 1) / BucketSize;
		for (var i = 0; i < count; i++)
		{
			var bucket = new Bucket<T>(this, null, null);
			var lim = Math.Min(BucketSize * (i + 1), length);
			for (var j = BucketSize * i; j < lim; j++) bucket.AddLast(array[j]);
			AddLast(bucket);
		}
	}
	public BucketNode<T> Find(T item) { var node = LowerBound(item); if (node == null || comp(node.Item1.Value, item) != 0) return null; else return node.Item1; }
	public BucketNode<T> FindLast(T item) { var node = UpperBound(item); if (node == null || comp(node.Item1.Value, item) != 0) return null; else return node.Item1; }
	public IEnumerator<T> GetEnumerator()
	{
		var bucket = Head;
		while (bucket != null)
		{
			var node = bucket.Head;
			while (node != null) { yield return node.Value; node = node.Next; }
			bucket = bucket.Next;
		}
	}
	public void Add(T item) { var ub = LowerBound(item); if (ub != null) AddBefore(ub.Item1, item); else AddLast(item); }
	IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
	public void CopyTo(Array array, int index) { foreach (var item in this) array.SetValue(item, index++); }
	public bool IsSynchronized => false;
	public object SyncRoot => this;
	public bool IsReadOnly => false;
	public bool Contains(T item) => Find(item) != null;
	public void CopyTo(T[] array, int index) { foreach (var item in this) array[index++] = item; }
	public override string ToString()
	{
		var sb = new StringBuilder();
		sb.Append("<Start>\n");
		var node = Head;
		while (node != null) { sb.Append($"{node.ToString()}\n"); node = node.Next; }
		sb.Append("<end>");
		return sb.ToString();
	}
	public bool Check()
	{
		if (NumOfBucket == 0) return Head == null && Tail == null;
		if (Head.Prev != null || Tail.Next != null) return false;
		var bucket = Head;
		var c = 1;
		while (bucket.Next != null)
		{
			if (!CheckConnection(bucket) || !CheckBucket(bucket)) return false;
			bucket = bucket.Next;
			c++;
		}
		return bucket == Tail && CheckBucket(Tail) && c == NumOfBucket;
	}
	bool CheckConnection(Bucket<T> bucket)
	{
		if (bucket.Next == null) return bucket == Tail;
		else return bucket.Next.Prev == bucket && comp(bucket.Tail.Value, bucket.Next.Head.Value) <= 0;
	}
	bool CheckBucket(Bucket<T> bucket) => bucket.Count > 0 && bucket.Count <= BucketSize && bucket.Parent == this;
	public void Start(Func<string, T> parser, Func<T> random)
	{
		BucketNode<T> x = null, y = null;
		var help = true;
		while (true)
		{
			Console.Clear();
			Console.WriteLine($"{Count} items, {NumOfBucket} buckets(size : {BucketSize})");
			Console.WriteLine(this);
			Console.WriteLine(Check() ? "OK!" : "NG!");
			if (help)
			{
				Console.WriteLine("when val is omitted, random value will be used.");
				Console.WriteLine("a val : add val");
				Console.WriteLine("r val : remove val");
				Console.WriteLine("j : adjust");
				Console.WriteLine("c : clear");
				Console.WriteLine("h : disable/enable help message");
				Console.WriteLine("x : set x");
				Console.WriteLine("x h : set x to head");
				Console.WriteLine("x t : set x to tail");
				Console.WriteLine("x n : set x to x.next");
				Console.WriteLine("x p : set x to x.prev");
				Console.WriteLine("x f val : set x to lower bound of val");
				Console.WriteLine("y : set y");
				Console.WriteLine("x : exchange x and y");
				Console.WriteLine("d : remove from x to y");
				Console.WriteLine("q : quit");
			}
			if (x != null) Console.WriteLine($"x = {x.Value} <- {x.Parent}");
			if (y != null) Console.WriteLine($"y = {y.Value} <- {y.Parent}");
			Console.Write("enter command > ");
			var command = Console.ReadLine().Split();
			if (command[0].Length > 1 && command[0][1] == 'd')
				Console.WriteLine("debug...");
			if (command[0].StartsWith("a")) { if (command.Length > 1) Add(parser(command[1])); else Add(random()); }
			else if (command[0].StartsWith("r")) { if (command.Length > 1) Remove(parser(command[1])); else Remove(random()); }
			else if (command[0].StartsWith("c")) Clear();
			else if (command[0].StartsWith("j")) Adjust();
			else if (command[0].StartsWith("h")) help = !help;
			else if (command[0].StartsWith("x")) SetVariable(command, ref x, parser, random);
			else if (command[0].StartsWith("y")) SetVariable(command, ref y, parser, random);
			else if (command[0].StartsWith("e")) { var tmp = x; x = y; y = tmp; }
			else if (command[0].StartsWith("d")) { RemoveRange(x, y, x.Index, y.Index); x = null; y = null; }
			else if (command[0].StartsWith("q")) break;
		}
	}
	void SetVariable(string[] command, ref BucketNode<T> x, Func<string, T> parser, Func<T> random)
	{
		if (command[1].StartsWith("h")) x = Head.Head;
		else if (command[1].StartsWith("t")) x = Tail.Tail;
		else if (command[1].StartsWith("n"))
		{
			if (x.Next != null) x = x.Next;
			else if (x.Parent.Next != null) x = x.Parent.Next.Head;
			else { Console.WriteLine("x is the last element..."); Console.ReadKey(true); }
		}
		else if (command[1].StartsWith("p"))
		{
			if (x.Prev != null) x = x.Prev;
			else if (x.Parent.Prev != null) x = x.Parent.Prev.Tail;
			else { Console.WriteLine("x is the first element..."); Console.ReadKey(true); }
		}
		else if (command[1].StartsWith("f")) { if (command.Length > 2) x = LowerBound(parser(command[2])).Item1; else x = LowerBound(random()).Item1; }
	}
}
// bucket cannot be empty
class Bucket<T>
{
	public BucketList<T> Parent;
	public int Count;
	public Bucket<T> Prev;
	public Bucket<T> Next;
	public BucketNode<T> Head;
	public BucketNode<T> Tail;
	public Bucket(BucketList<T> parent, Bucket<T> prev, Bucket<T> next) { Parent = parent; Prev = prev; Next = next; Head = null; Tail = null; }
	public int Index
	{
		get
		{
			var count = 0;
			var node = Parent.Head;
			while (node != this) { node = node.Next; count++; }
			return count;
		}
	}
	public bool AddAfter(BucketNode<T> node, BucketNode<T> item) => AddAfter(node, item.Value);
	public bool AddBefore(BucketNode<T> node, BucketNode<T> item) => AddBefore(node, item.Value);
	public bool AddAfter(BucketNode<T> node, T item)
	{
		Debug.Assert(node != null && node.Parent == this && Parent.comp(node.Value, item) <= 0
					&& ((node.Next == null && (Next == null || Parent.comp(Next.Head.Value, item) >= 0))
						|| Parent.comp(node.Next.Value, item) >= 0));
		if (Count < Parent.BucketSize)
		{
			var tmp = new BucketNode<T>(item, this, node, node.Next);
			if (node.Next != null) node.Next.Prev = tmp;
			else Tail = tmp;
			node.Next = tmp;
			Count++;
			return true;
		}
		return false;
	}
	public bool AddBefore(BucketNode<T> node, T item)
	{
		Debug.Assert(node != null && node.Parent == this && Parent.comp(node.Value, item) >= 0
					&& ((node.Prev == null && (Prev == null || Parent.comp(Prev.Tail.Value, item) <= 0))
						|| Parent.comp(node.Prev.Value, item) <= 0));
		if (Count < Parent.BucketSize)
		{
			var tmp = new BucketNode<T>(item, this, node.Prev, node);
			if (node.Prev != null) node.Prev.Next = tmp;
			else Head = tmp;
			node.Prev = tmp;
			Count++;
			return true;
		}
		else return false;
	}
	public bool InitiateWith(BucketNode<T> node)
	{
		Head = Tail = node;
		node.Parent = this;
		node.Prev = node.Next = null;
		Count++;
		return true;
	}
	public bool InitiateWith(T item) => InitiateWith(new BucketNode<T>(item, this, null, null));
	public void RemoveAll() { Head = Tail = null; Count = 0; }
	public bool AddFirst(T item) { if (Count == 0) return InitiateWith(item); else return AddBefore(Head, item); }
	public bool AddLast(T item) { if (Count == 0) return InitiateWith(item); else return AddAfter(Tail, item); }
	public bool Remove(BucketNode<T> node)
	{
		Debug.Assert(node != null && node.Parent == this);
		if (Count > 1)
		{
			Count--;
			if (node == Head) { Head.Next.Prev = null; Head = Head.Next; }
			else if (node == Tail) { Tail.Prev.Next = null; Tail = Tail.Prev; }
			else { node.Prev.Next = node.Next; node.Next.Prev = node.Prev; }
			return true;
		}
		else return false;
	}
	public bool RemoveRange(BucketNode<T> from, BucketNode<T> to, int indexFrom = -1, int indexTo = -1)
	{
		Debug.Assert(from != null && to != null && from.Parent == this && to.Parent == this);
		if (indexFrom < 0) indexFrom = from.Index;
		if (indexTo < 0) indexTo = to.Index;
		if (indexTo == 0 && indexFrom == Count - 1) return false;
		else if (indexFrom == 0) { Head = to.Next; Head.Prev = null; }
		else if (indexTo == Count - 1) { Tail = from.Prev; Tail.Next = null; }
		else { from.Prev.Next = to.Next; to.Next.Prev = from.Prev; }
		Count -= indexTo - indexFrom + 1;
		return true;
	}
	public override string ToString()
	{
		var sb = new StringBuilder();
		sb.Append("[");
		var node = Head;
		while (node != null) { sb.Append($"{node.ToString()}, "); node = node.Next; }
		if (sb.Length > 1) sb.Remove(sb.Length - 2, 2);
		sb.Append("]");
		return sb.ToString();
	}
	public bool Check()
	{
		if (Count == 0) return Head == null && Tail == null;
		if (Head.Prev != null || Tail.Next != null) return false;
		var node = Head;
		var c = 1;
		while (node.Next != null)
		{
			if (!CheckConnection(node) || !CheckNode(node)) return false;
			node = node.Next;
			c++;
		}
		return node == Tail && CheckNode(Tail) && c == Count;
	}
	bool CheckConnection(BucketNode<T> node)
	{
		if (node.Next == null) return node == Tail;
		else return node.Next.Prev == node && Parent.comp(node.Value, node.Next.Value) <= 0;
	}
	bool CheckNode(BucketNode<T> node) => node.Parent == this;
}
class BucketNode<T>
{
	public T Value;
	public Bucket<T> Parent;
	public BucketNode<T> Prev;
	public BucketNode<T> Next;
	public BucketNode(T item, Bucket<T> parent, BucketNode<T> prev, BucketNode<T> next) { Value = item; Parent = parent; Prev = prev; Next = next; }
	public int Index
	{
		get
		{
			var count = 0;
			var node = Parent.Head;
			while (node != this) { node = node.Next; count++; }
			return count;
		}
	}
	public override string ToString() { return Value.ToString(); }
}
class UndirectedGraph<V, E> : DirectedGraph<V, E>
{
	public UndirectedGraph(int V) : base(V) { }
	public UndirectedGraph(int V, IEnumerable<EdgeInfo<E>> edges) : base(V, edges) { }
	public override void AddEdge(EdgeInfo<E> edge)
	{
		edges.Add(edge);
		edges.Add(edge.Reverse());
		edgesFrom[edge.From].Add(new HalfEdgeInfo<E>(edge.To, edge.Information));
		edgesFrom[edge.To].Add(new HalfEdgeInfo<E>(edge.From, edge.Information));
		edgesTo[edge.To].Add(new HalfEdgeInfo<E>(edge.From, edge.Information));
		edgesTo[edge.From].Add(new HalfEdgeInfo<E>(edge.To, edge.Information));
	}
	public bool IsConnected
	{
		get
		{
			if (numberOfNodes == 0) return true;
			var used = new bool[numberOfNodes];
			var queue = new Queue<int>();
			queue.Enqueue(0);
			while (queue.Count > 0)
			{
				var v = queue.Dequeue();
				if (used[v]) continue;
				used[v] = true;
				foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
			}
			return used.All(x => x);
		}
	}
	public bool IsTree
	{
		get
		{
			if (numberOfNodes == 0) return true;
			var used = new bool[numberOfNodes];
			var queue = new Queue<int>();
			queue.Enqueue(0);
			while (queue.Count > 0)
			{
				var v = queue.Dequeue();
				if (used[v]) return false;
				used[v] = true;
				foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
			}
			return used.All(x => x);
		}
	}
	public UndirectedGraph<V, E> MinimumSpanningTreePrim(int start, Func<E, int> cost)
	{
		var graph = new UndirectedGraph<V, E>(numberOfNodes);
		nodes.CopyTo(graph.nodes, 0);
		var d = Enumerable.Repeat(Func.Inf, numberOfNodes).ToArray();
		var used = new bool[numberOfNodes];
		var queue = new PriorityQueue<Pair<EdgeInfo<E>, int>>((x, y) => x.Second.CompareTo(y.Second), numberOfNodes);
		d[start] = 0;
		queue.Enqueue(new Pair<EdgeInfo<E>, int>(new EdgeInfo<E>(-1, 0, default(E)), 0));
		while (queue.Count > 0)
		{
			var p = queue.Dequeue();
			var v = p.First.To;
			if (d[v] < p.Second) continue;
			used[v] = true;
			if (p.First.From >= 0) graph.AddEdge(v, p.First.From, p.First.Information);
			foreach (var w in EdgesFrom(v))
			{
				if (!used[w.End] && cost(w.Information) < d[w.End])
				{
					d[w.End] = cost(w.Information);
					queue.Enqueue(new Pair<EdgeInfo<E>, int>(new EdgeInfo<E>(v, w.End, w.Information), cost(w.Information)));
				}
			}
		}
		return graph;
	}
	public UndirectedGraph<V, E> MinimumSpanningTreeKruskal(Func<E, int> cost)
	{
		var graph = new UndirectedGraph<V, E>(numberOfNodes);
		nodes.CopyTo(graph.nodes, 0);
		var tree = new UnionFindTree(numberOfNodes);
		edges.Sort((x, y) => cost(x.Information).CompareTo(cost(y.Information)));
		foreach (var e in edges)
		{
			if (!tree.IsSameCategory(e.From, e.To))
			{
				tree.UniteCategory(e.From, e.To);
				graph.AddEdge(e);
			}
		}
		return graph;
	}
	public bool IsBipartite
	{
		get
		{
			var color = new int[numberOfNodes];
			foreach (var v in nodes)
			{
				if (color[v.Code] == 0)
				{
					var queue = new Queue<Pair<int, int>>();
					queue.Enqueue(new Pair<int, int>(v.Code, 1));
					while (queue.Count > 0)
					{
						var w = queue.Dequeue();
						if (color[w.First] != 0)
						{
							if (color[w.First] != w.Second) return false;
							continue;
						}
						color[w.First] = w.Second;
						foreach (var e in EdgesFrom(w.First)) queue.Enqueue(new Pair<int, int>(e.End, -w.Second));
					}
				}
			}
			return true;
		}
	}
	public IEnumerable<NodeInfo<V>> GetArticulationPoints()
	{
		var visited = new bool[numberOfNodes];
		var parent = new int[numberOfNodes];
		var children = Enumerable.Range(0, numberOfNodes).Select(_ => new SortedSet<int>()).ToArray();
		var order = new int[numberOfNodes];
		var lowest = new int[numberOfNodes];
		var isroot = new bool[numberOfNodes];
		var count = 1;
		var isarticulation = new bool[numberOfNodes];
		Action<int, int> dfs = null;
		dfs = (u, prev) =>
		{
			order[u] = count;
			lowest[u] = count;
			count++;
			visited[u] = true;
			foreach (var e in edgesFrom[u])
			{
				var v = e.End;
				if (visited[v]) { if (v != prev) lowest[u] = Math.Min(lowest[u], order[v]); }
				else
				{
					parent[v] = u;
					if (isroot[u]) children[u].Add(v);
					dfs(v, u);
					lowest[u] = Math.Min(lowest[u], lowest[v]);
					if (order[u] <= lowest[v]) isarticulation[u] = true;
				}
			}
		};
		for (var v = 0; v < numberOfNodes; v++)
		{
			if (visited[v]) continue;
			count = 1; dfs(v, -1);
			isroot[v] = true;
		}
		for (var v = 0; v < numberOfNodes; v++)
		{
			if (isroot[v]) { if (children[v].Count > 1) yield return nodes[v]; }
			else { if (isarticulation[v]) yield return nodes[v]; }
		}
	}
	public string ToString(Func<NodeInfo<V>, string> vertex, Func<EdgeInfo<E>, string> edge)
	{
		var sb = new StringBuilder();
		sb.Append("graph G {\n");
		foreach (var v in nodes) sb.Append($"\tv{v.Code} [label = \"{vertex(v)}\"];\n");
		foreach (var e in edges) sb.Append($"\tv{e.From} -- v{e.To} [label=\"{edge(e)}\"];\n");
		sb.Append("}");
		return sb.ToString();
	}
	public override string ToString() => ToString(v => v.ToString(), e => e.ToString());
}
class NodeInfo<V> : Pair<int, V>
{
	public int Code { get { return First; } set { First = value; } }
	public V Information { get { return Second; } set { Second = value; } }
	public NodeInfo() : base() { }
	public NodeInfo(int code, V info) : base(code, info) { }
}
class HalfEdgeInfo<E> : Pair<int, E>
{
	public int End { get { return First; } set { First = value; } }
	public E Information { get { return Second; } set { Second = value; } }
	public HalfEdgeInfo() : base() { }
	public HalfEdgeInfo(int end, E info) : base(end, info) { }
}
class EdgeInfo<E> : Pair<Pair<int, int>, E>
{
	public int From { get { return First.First; } set { First.First = value; } }
	public int To { get { return First.Second; } set { First.Second = value; } }
	public E Information { get { return Second; } set { Second = value; } }
	public EdgeInfo() : base() { }
	public EdgeInfo(int from, int to, E info) : base(new Pair<int, int>(from, to), info) { }
	public EdgeInfo<E> Reverse() => new EdgeInfo<E>(To, From, Information);
}
class DirectedGraph<V, E> : IEnumerable<NodeInfo<V>>
{
	protected int numberOfNodes;
	public int NumberOfNodes => numberOfNodes;
	protected NodeInfo<V>[] nodes;
	protected List<EdgeInfo<E>> edges;
	protected List<HalfEdgeInfo<E>>[] edgesFrom;
	protected List<HalfEdgeInfo<E>>[] edgesTo;
	public IEnumerable<HalfEdgeInfo<E>> EdgesFrom(int node) => edgesFrom[node];
	public int InDegree(int node) => edgesTo[node].Count;
	public int OutDegree(int node) => edgesFrom[node].Count;
	public IEnumerable<HalfEdgeInfo<E>> EdgesTo(int node) => edgesTo[node];
	public V this[int node] { get { return nodes[node].Second; } set { nodes[node].Second = value; } }
	public IEnumerable<EdgeInfo<E>> Edges => edges;
	public DirectedGraph(int V)
	{
		numberOfNodes = V;
		nodes = Enumerable.Range(0, V).Select(x => new NodeInfo<V>(x, default(V))).ToArray();
		edges = new List<EdgeInfo<E>>();
		edgesFrom = Enumerable.Range(0, V).Select(_ => new List<HalfEdgeInfo<E>>()).ToArray();
		edgesTo = Enumerable.Range(0, V).Select(_ => new List<HalfEdgeInfo<E>>()).ToArray();
	}
	public DirectedGraph(int V, IEnumerable<EdgeInfo<E>> edges) : this(V) { foreach (var e in edges) AddEdge(e.From, e.To, e.Information); }
	public virtual void AddEdge(EdgeInfo<E> edge)
	{
		edges.Add(edge);
		edgesFrom[edge.From].Add(new HalfEdgeInfo<E>(edge.To, edge.Information));
		edgesTo[edge.To].Add(new HalfEdgeInfo<E>(edge.From, edge.Information));
	}
	public void AddEdge(int from, int to, E information) => AddEdge(new EdgeInfo<E>(from, to, information));
	public void AddEdge(V from, V to, E information) => AddEdge(new EdgeInfo<E>(SearchNode(from).Code, SearchNode(to).Code, information));
	public NodeInfo<V> SearchNode(V node) => nodes.FirstOrDefault(e => e.Information.Equals(node));
	public EdgeInfo<E> SearchEdge(E edge) => edges.Find(e => e.Information.Equals(edge));
	public IEnumerator<NodeInfo<V>> GetEnumerator() { foreach (var v in nodes) yield return v; }
	IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
	public int[] ShortestPathLengthFrom(int from, Func<E, int> cost)
	{
		var d = Enumerable.Repeat(Func.Inf, numberOfNodes).ToArray();
		d[from] = 0;
		var update = true;
		while (update)
		{
			update = false;
			foreach (var e in edges)
			{
				var tmp = d[e.From] + cost(e.Information);
				if (d[e.From] < Func.Inf && d[e.To] > tmp)
				{
					d[e.To] = tmp;
					update = true;
				}
			}
		}
		return d;
	}
	public int[] DijkstraFrom(int from, Func<E, int> cost)
	{
		var d = Enumerable.Repeat(Func.Inf, numberOfNodes).ToArray();
		var queue = new PriorityQueue<Pair<int, int>>((x, y) => x.Second.CompareTo(y.Second));
		d[from] = 0;
		queue.Enqueue(new Pair<int, int>(from, 0));
		while (!queue.IsEmpty)
		{
			var p = queue.Dequeue();
			var v = p.First;
			if (d[v] < p.Second) continue;
			foreach (var e in EdgesFrom(v))
			{
				var tmp = d[v] + cost(e.Information);
				if (d[e.End] > tmp) queue.Enqueue(new Pair<int, int>(e.End, d[e.End] = tmp));
			}
		}
		return d;
	}
	// cost(e)>=0
	public Pair<long, int>[] DijkstraFromL(int from, Func<E, long> cost)
	{
		var d = new Pair<long, int>[numberOfNodes];
		for (var i = 0; i < numberOfNodes; i++) d[i] = new Pair<long, int>(Func.InfL, -1);
		var queue = new PriorityQueue<Tuple<int, long, int>>((x, y) => x.Item2.CompareTo(y.Item2));
		d[from] = new Pair<long, int>(0, -1);
		queue.Enqueue(new Tuple<int, long, int>(from, 0, -1));
		while (!queue.IsEmpty)
		{
			var p = queue.Dequeue();
			var v = p.Item1;
			if (d[v].First < p.Item2) continue;
			foreach (var e in edgesFrom[v])
			{
				var tmp = d[v].First + cost(e.Information);
				if (d[e.End].First > tmp) queue.Enqueue(new Tuple<int, long, int>(e.End, d[e.End].First = tmp, d[e.End].Second = v));
			}
		}
		return d;
	}
	public int[,] ShortestPathLengthEachOther(Func<E, int> cost)
	{
		var d = new int[numberOfNodes, numberOfNodes];
		for (var v = 0; v < numberOfNodes; v++) for (var w = 0; w < numberOfNodes; w++) d[v, w] = Func.Inf;
		for (var v = 0; v < numberOfNodes; v++) d[v, v] = 0;
		foreach (var e in edges) if (e.From != e.To) d[e.From, e.To] = cost(e.Information);
		for (var k = 0; k < numberOfNodes; k++)
			for (var v = 0; v < numberOfNodes; v++)
				for (var w = 0; w < numberOfNodes; w++)
					d[v, w] = Math.Min(d[v, w], d[v, k] + d[k, w]);
		return d;
	}
	public bool ContainsNegativeLoopWF(Func<E, int> cost)
	{
		var d = ShortestPathLengthEachOther(cost);
		for (var v = 0; v < numberOfNodes; v++) if (d[v, v] < 0) return true;
		return false;
	}
	public bool ContainsNegativeLoop(Func<E, int> cost)
	{
		var d = Enumerable.Repeat(0, numberOfNodes).ToArray();
		for (var v = 0; v < numberOfNodes; v++)
		{
			foreach (var e in edges)
			{
				var tmp = d[e.From] + cost(e.Information);
				if (d[e.To] > tmp)
				{
					d[e.To] = tmp;
					if (v == numberOfNodes - 1) return true;
				}
			}
		}
		return false;
	}
	public IEnumerable<int> ReachableFrom(int from)
	{
		var used = new bool[numberOfNodes];
		var queue = new Queue<int>();
		queue.Enqueue(from);
		while (queue.Count > 0)
		{
			var v = queue.Dequeue();
			if (used[v]) continue;
			used[v] = true;
			foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
		}
		for (var v = 0; v < numberOfNodes; v++) if (used[v]) yield return v;
	}
	public bool IsReachable(int from, int to)
	{
		var used = new bool[numberOfNodes];
		var queue = new Queue<int>();
		queue.Enqueue(from);
		while (queue.Count > 0)
		{
			var v = queue.Dequeue();
			if (v == to) return true;
			if (used[v]) continue;
			used[v] = true;
			foreach (var e in EdgesFrom(v)) queue.Enqueue(e.End);
		}
		return false;
	}
	public Pair<DirectedGraph<HashSet<NodeInfo<V>>, object>, int[]> StronglyConnectedComponents()
	{
		var mark = new bool[numberOfNodes];
		var stack = new Stack<int>();
		Action<int> dfs = null;
		dfs = v =>
		{
			mark[v] = true;
			foreach (var w in edgesFrom[v]) if (!mark[w.End]) dfs(w.End);
			stack.Push(v);
		};
		for (var v = 0; v < numberOfNodes; v++) if (!mark[v]) dfs(v);
		var scc = new List<HashSet<NodeInfo<V>>>();
		mark = new bool[numberOfNodes];
		var which = new int[numberOfNodes];
		Action<int, HashSet<NodeInfo<V>>> rdfs = null;
		rdfs = (v, set) =>
		{
			set.Add(new NodeInfo<V>(v, nodes[v].Information));
			mark[v] = true;
			foreach (var w in edgesFrom[v]) if (!mark[w.End]) rdfs(w.End, set);
		};
		var M = 0;
		while (stack.Count > 0)
		{
			var v = stack.Pop();
			if (mark[v]) continue;
			var set = new HashSet<NodeInfo<V>>();
			rdfs(v, set);
			scc.Add(set);
			foreach (var w in set) which[w.Code] = M;
			M++;
		}
		var graph = new UndirectedGraph<HashSet<NodeInfo<V>>, object>(M);
		for (var v = 0; v < M; v++) graph[v] = scc[v];
		foreach (var e in edges) if (which[e.From] != which[e.To]) graph.AddEdge(which[e.From], which[e.To], null);
		return new Pair<DirectedGraph<HashSet<NodeInfo<V>>, object>, int[]>(graph, which);
	}
	public string ToString(Func<V, string> vertex, Func<E, string> edge)
	{
		var sb = new StringBuilder();
		sb.Append("digraph G {\n");
		foreach (var v in nodes) sb.Append($"\tv{v.Code} [label = \"{vertex(v.Information)}\"];\n");
		foreach (var e in edges) sb.Append($"\tv{e.From} -> v{e.To} [label=\"{edge(e.Information)}\"];\n");
		sb.Append("}");
		return sb.ToString();
	}
	public override string ToString() => ToString(v => v.ToString(), e => e.ToString());
}
class UnionFindTree
{
	int N;
	int[] parent, rank, size;
	public UnionFindTree(int capacity)
	{
		N = capacity;
		parent = new int[N];
		rank = new int[N];
		size = new int[N];
		for (var i = 0; i < N; i++) { parent[i] = i; size[i] = 1; }
	}
	public int GetSize(int x) => size[GetRootOf(x)];
	public int GetRootOf(int x) => parent[x] == x ? x : parent[x] = GetRootOf(parent[x]);
	public bool UniteCategory(int x, int y)
	{
		if ((x = GetRootOf(x)) == (y = GetRootOf(y))) return false;
		if (rank[x] < rank[y]) { parent[x] = y; size[y] += size[x]; }
		else
		{
			parent[y] = x; size[x] += size[y];
			if (rank[x] == rank[y]) rank[x]++;
		}
		return true;
	}
	public bool IsSameCategory(int x, int y) => GetRootOf(x) == GetRootOf(y);
}
class AVLTree<T> : IEnumerable<T>, ICollection<T>, ICollection, IEnumerable
{
	public class AVLNode : IEnumerable<T>
	{
		AVLTree<T> tree;
		int height;
		public int Height => height;
		public int Bias => Left.height - Right.height;
		public T Item;
		public AVLNode Parent;
		public AVLNode Left;
		public AVLNode Right;
		AVLNode(T x, AVLTree<T> tree) { this.tree = tree; Item = x; Left = tree.sentinel; Right = tree.sentinel; }
		public AVLNode(AVLTree<T> tree) : this(default(T), tree) { height = 0; Parent = null; }
		public AVLNode(T x, AVLNode parent, AVLTree<T> tree) : this(x, tree) { height = 1; Parent = parent; }
		public void Adjust() => height = 1 + Math.Max(Left.height, Right.height);
		public void ResetAsSentinel() { height = 0; Left = tree.sentinel; Right = tree.sentinel; }
		public IEnumerator<T> GetEnumerator()
		{
			if (this != tree.sentinel)
			{
				foreach (var x in Left) yield return x;
				yield return Item;
				foreach (var x in Right) yield return x;
			}
		}
		IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
	}
	AVLNode sentinel;
	Comparison<T> comp;
	Func<T, T, bool> equals;
	int count;
	// assumed to be comparer
	// i.e. comp(x,x)=0, and comp(x,y)>0 then comp(y,x)<0, and comp(x,y)>0 & comp(y,z)>0 then comp(x,z)>0
	public AVLTree(Comparison<T> comp)
	{
		sentinel = new AVLNode(this);
		sentinel.ResetAsSentinel();
		this.comp = comp ?? Func.DefaultComparison<T>();
		if (typeof(T).IsValueType) equals = (x, y) => x.Equals(y);
		else equals = (x, y) => ReferenceEquals(x, y);
		count = 0;
	}
	public AVLTree(IComparer<T> comp = null) : this(comp.ToComparison()) { }
	void Replace(AVLNode u, AVLNode v)
	{
		var parent = u.Parent;
		if (parent.Left == u) parent.Left = v;
		else parent.Right = v;
		v.Parent = parent;
	}
	AVLNode RotateL(AVLNode v)
	{
		var u = v.Right;
		Replace(v, u);
		v.Right = u.Left;
		u.Left.Parent = v;
		u.Left = v;
		v.Parent = u;
		v.Adjust();
		u.Adjust();
		return u;
	}
	AVLNode RotateR(AVLNode u)
	{
		var v = u.Left;
		Replace(u, v);
		u.Left = v.Right;
		v.Right.Parent = u;
		v.Right = u;
		u.Parent = v;
		u.Adjust();
		v.Adjust();
		return v;
	}
	AVLNode RotateLR(AVLNode t) { RotateL(t.Left); return RotateR(t); }
	AVLNode RotateRL(AVLNode t) { RotateR(t.Right); return RotateL(t); }
	void Adjust(bool isInsertMode, AVLNode node)
	{
		while (node.Parent != sentinel)
		{
			var parent = node.Parent;
			var height = parent.Height;
			if ((parent.Left == node) == isInsertMode)
				if (parent.Bias == 2)
					if (parent.Left.Bias >= 0) parent = RotateR(parent);
					else parent = RotateLR(parent);
				else parent.Adjust();
			else
				if (parent.Bias == -2)
				if (parent.Right.Bias <= 0) parent = RotateL(parent);
				else parent = RotateRL(parent);
			else parent.Adjust();
			if (height == parent.Height) break;
			node = parent;
		}
	}
	public void Add(T item)
	{
		var parent = sentinel;
		var pos = sentinel.Left;
		var isLeft = true;
		count++;
		while (pos != sentinel)
			if (comp(item, pos.Item) < 0) { parent = pos; pos = pos.Left; isLeft = true; }
			else { parent = pos; pos = pos.Right; isLeft = false; }
		if (isLeft)
		{
			parent.Left = new AVLNode(item, parent, this);
			Adjust(true, parent.Left);
		}
		else
		{
			parent.Right = new AVLNode(item, parent, this);
			Adjust(true, parent.Right);
		}
	}
	// if equals(x,y) holds then !(comp(x,y)<0) and !(comp(x,y)>0) must hold
	// i.e. equals(x,y) -> comp(x,y)=0
	public bool Remove(T item, AVLNode start)
	{
		var pos = start;
		while (pos != sentinel)
		{
			if (comp(item, pos.Item) < 0) pos = pos.Left;
			else if (comp(item, pos.Item) > 0) pos = pos.Right;
			else if (equals(pos.Item, item))
			{
				if (pos.Left == sentinel)
				{
					Replace(pos, pos.Right);
					Adjust(false, pos.Right);
				}
				else
				{
					var max = Max(pos.Left);
					pos.Item = max.Item;
					Replace(max, max.Left);
					Adjust(false, max.Left);
				}
				count--;
				return true;
			}
			else return Remove(item, pos.Left) || Remove(item, pos.Right);
		}
		return false;
	}
	public bool Remove(T item) => Remove(item, sentinel.Left);
	AVLNode Max(AVLNode node)
	{
		while (node.Right != sentinel) node = node.Right;
		return node;
	}
	AVLNode Min(AVLNode node)
	{
		while (node.Left != sentinel) node = node.Left;
		return node;
	}
	public bool Contains(T item)
	{
		var pos = sentinel.Left;
		while (pos != sentinel)
		{
			if (comp(item, pos.Item) < 0) pos = pos.Left;
			else if (comp(item, pos.Item) > 0) pos = pos.Right;
			else return true;
		}
		return false;
	}
	public T Find(T item)
	{
		var pos = sentinel.Left;
		while (pos != sentinel)
		{
			if (comp(item, pos.Item) < 0) pos = pos.Left;
			else if (comp(item, pos.Item) > 0) pos = pos.Right;
			else return pos.Item;
		}
		return default(T);
	}
	public AVLNode LowerBound(Predicate<T> pred) { AVLNode node; LowerBound(pred, sentinel.Left, out node); return node; }
	public AVLNode UpperBound(Predicate<T> pred) { AVLNode node; UpperBound(pred, sentinel.Left, out node); return node; }
	public AVLNode LowerBound(T item) => LowerBound(x => comp(x, item) >= 0);
	public AVLNode UpperBound(T item) => UpperBound(x => comp(x, item) <= 0);
	bool UpperBound(Predicate<T> pred, AVLNode node, out AVLNode res)
	{
		if (node == sentinel) { res = null; return false; }
		if (pred(node.Item)) { if (!UpperBound(pred, node.Right, out res)) res = node; return true; }
		else return UpperBound(pred, node.Left, out res);
	}
	bool LowerBound(Predicate<T> pred, AVLNode node, out AVLNode res)
	{
		if (node == sentinel) { res = null; return false; }
		if (pred(node.Item)) { if (!LowerBound(pred, node.Left, out res)) res = node; return true; }
		else return LowerBound(pred, node.Right, out res);
	}
	public T Min() => Min(sentinel.Left).Item;
	public AVLNode MinNode() => Min(sentinel.Left);
	public T Max() => Max(sentinel.Left).Item;
	public AVLNode MaxNode() => Max(sentinel.Left);
	public bool IsEmpty => sentinel.Left == sentinel;
	public void Clear() { sentinel.Left = sentinel; count = 0; sentinel.ResetAsSentinel(); }
	public IEnumerator<T> GetEnumerator() => sentinel.Left.GetEnumerator();
	IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
	public void CopyTo(T[] array, int arrayIndex) { foreach (var x in this) array[arrayIndex++] = x; }
	public int Count => count;
	public bool IsReadOnly => true;
	public void CopyTo(Array array, int index) { foreach (var x in this) array.SetValue(x, index++); }
	public bool IsSynchronized => false;
	public object SyncRoot => this;
	public override string ToString()
	{
		var nodes = new StringBuilder();
		var edges = new StringBuilder();
		ConcatSubTree(nodes, edges, sentinel.Left, "L");
		return $"digraph G {{\n{nodes.ToString()}{edges.ToString()}}}";
	}
	void ConcatSubTree(StringBuilder nodes, StringBuilder edges, AVLNode node, string code)
	{
		if (node == sentinel) return;
		nodes.Append($"\tv{code} [label = \"{node.Height}:{node.Item}\"];\n");
		if (node.Left != sentinel) edges.Append($"\tv{code} -> v{code}L;\n");
		if (node.Right != sentinel) edges.Append($"\tv{code} -> v{code}R;\n");
		ConcatSubTree(nodes, edges, node.Left, $"{code}L");
		ConcatSubTree(nodes, edges, node.Right, $"{code}R");
	}
	public bool IsBalanced() => IsBalanced(sentinel.Left);
	public bool IsValidBinarySearchTree() => IsValidBinarySearchTree(sentinel.Left);
	bool IsBalanced(AVLNode node) => node == sentinel || (Math.Abs(node.Bias) < 2 && IsBalanced(node.Left) && IsBalanced(node.Right));
	bool IsValidBinarySearchTree(AVLNode node)
		=> node == sentinel || (Small(node.Item, node.Left) && Large(node.Item, node.Right)
			&& IsValidBinarySearchTree(node.Left) && IsValidBinarySearchTree(node.Right));
	bool Small(T item, AVLNode node) => node == sentinel || (comp(item, node.Item) >= 0 && Small(item, node.Left) && Small(item, node.Right));
	bool Large(T item, AVLNode node) => node == sentinel || (comp(item, node.Item) <= 0 && Large(item, node.Left) && Large(item, node.Right));
	public static void CheckAVL(Random rand, int N)
	{
		Comparison<double> comp = (x, y) => x.CompareTo(y);
		var avl = new AVLTree<double>(comp);
		var toBeLeft = new double[N];
		var toBeRemoved = new double[N];
		for (var i = 0; i < N; i++) avl.Add(toBeRemoved[i] = rand.NextDouble());
		for (var i = 0; i < N; i++) avl.Add(toBeLeft[i] = rand.NextDouble());
		for (var i = 0; i < N; i++) Console.Write(avl.Remove(toBeRemoved[i]) ? "" : "!!!NOT REMOVED!!! => " + toBeRemoved[i] + "\n");
		var insertErrors = toBeLeft.All(x => avl.Contains(x));
		var deleteErrors = avl.Count == N;
		//Console.WriteLine("【AVL木の構造】");
		//Console.WriteLine(avl);
		if (insertErrors && deleteErrors) Console.WriteLine("○\t挿入, 削除操作が正しく行われています.");
		else if (insertErrors) Console.WriteLine("×\t挿入(または削除)操作に問題があります.");
		else Console.WriteLine("×\t削除(または挿入)操作に問題があります.");
		if (avl.IsBalanced()) Console.WriteLine("○\tAVL木は平衡条件を保っています.");
		else Console.WriteLine("×\tAVL木の平衡条件が破れています.");
		if (avl.IsValidBinarySearchTree()) Console.WriteLine("○\tAVL木は二分探索木になっています.");
		else Console.WriteLine("×\tAVL木は二分探索木になっていません.");
		Array.Sort(toBeLeft, comp);
		Console.WriteLine($"最小値 : {avl.Min()} ≡ {toBeLeft.First()}");
		Console.WriteLine($"最大値 : {avl.Max()} ≡ {toBeLeft.Last()}");
		Console.WriteLine($"要素数 : {avl.Count} 個");
	}
}
class PriorityQueue<T> : IEnumerable<T>, ICollection, IEnumerable, ICloneable
{
	Comparison<T> comp;
	List<T> list;
	public int Count { get; private set; } = 0;
	public bool IsEmpty => Count == 0;
	public PriorityQueue(IEnumerable<T> source) : this((Comparison<T>)null, 0, source) { }
	public PriorityQueue(int capacity = 4, IEnumerable<T> source = null) : this((Comparison<T>)null, capacity, source) { }
	public PriorityQueue(IComparer<T> comp, IEnumerable<T> source) : this(comp.ToComparison(), source) { }
	public PriorityQueue(IComparer<T> comp, int capacity = 4, IEnumerable<T> source = null) : this(comp.ToComparison(), source) { list.Capacity = capacity; }
	public PriorityQueue(Comparison<T> comp, IEnumerable<T> source) : this(comp, 0, source) { }
	public PriorityQueue(Comparison<T> comp, int capacity = 4, IEnumerable<T> source = null) { this.comp = comp ?? Func.DefaultComparison<T>(); list = new List<T>(capacity); if (source != null) foreach (var x in source) Enqueue(x); }
	/// <summary>
	/// add an item
	/// this is an O(log n) operation
	/// </summary>
	/// <param name="x">item</param>
	public void Enqueue(T x)
	{
		var pos = Count++;
		list.Add(x);
		while (pos > 0)
		{
			var p = (pos - 1) / 2;
			if (comp(list[p], x) <= 0) break;
			list[pos] = list[p];
			pos = p;
		}
		list[pos] = x;
	}
	/// <summary>
	/// return the minimum element and remove it
	/// this is an O(log n) operation
	/// </summary>
	/// <returns>the minimum</returns>
	public T Dequeue()
	{
		var value = list[0];
		var x = list[--Count];
		list.RemoveAt(Count);
		if (Count == 0) return value;
		var pos = 0;
		while (pos * 2 + 1 < Count)
		{
			var a = 2 * pos + 1;
			var b = 2 * pos + 2;
			if (b < Count && comp(list[b], list[a]) < 0) a = b;
			if (comp(list[a], x) >= 0) break;
			list[pos] = list[a];
			pos = a;
		}
		list[pos] = x;
		return value;
	}
	/// <summary>
	/// look at the minimum element
	/// this is an O(1) operation
	/// </summary>
	/// <returns>the minimum</returns>
	public T Peek() => list[0];
	public IEnumerator<T> GetEnumerator() { var x = (PriorityQueue<T>)Clone(); while (x.Count > 0) yield return x.Dequeue(); }
	void CopyTo(Array array, int index) { foreach (var x in this) array.SetValue(x, index++); }
	public object Clone() { var x = new PriorityQueue<T>(comp, Count); x.list.AddRange(list); return x; }
	public void Clear() { list = new List<T>(); Count = 0; }
	public void TrimExcess() => list.TrimExcess();
	/// <summary>
	/// check whether item is in this queue
	/// this is an O(n) operation
	/// </summary>
	public bool Contains(T item) => list.Contains(item);
	IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
	void ICollection.CopyTo(Array array, int index) => CopyTo(array, index);
	bool ICollection.IsSynchronized => false;
	object ICollection.SyncRoot => this;
}
class Deque<T>
{
	T[] array;
	int offset, capacity;
	public int Count { get; protected set; }
	public Deque(int capacity) { array = new T[this.capacity = capacity]; Count = 0; offset = 0; }
	public Deque() : this(16) { }
	public T this[int index] { get { return array[GetIndex(index)]; } set { array[GetIndex(index)] = value; } }
	int GetIndex(int index) { var tmp = index + offset; return tmp >= capacity ? tmp - capacity : tmp; }
	public T PeekFront() => array[offset];
	public T PeekBack() => array[GetIndex(Count - 1)];
	public void PushFront(T item)
	{
		if (Count == capacity) Extend();
		if (--offset < 0) offset += array.Length;
		array[offset] = item;
		Count++;
	}
	public T PopFront()
	{
		Count--;
		var tmp = array[offset++];
		if (offset >= capacity) offset -= capacity;
		return tmp;
	}
	public void PushBack(T item)
	{
		if (Count == capacity) Extend();
		var id = (Count++) + offset;
		if (id >= capacity) id -= capacity;
		array[id] = item;
	}
	public T PopBack() => array[GetIndex(--Count)];
	public void Insert(int index, T item)
	{
		PushFront(item);
		for (var i = 0; i < index; i++) this[i] = this[i + 1];
		this[index] = item;
	}
	public T RemoveAt(int index)
	{
		var tmp = this[index];
		for (var i = index; i > 0; i--) this[i] = this[i - 1];
		PopFront();
		return tmp;
	}
	void Extend()
	{
		var newArray = new T[capacity << 1];
		if (offset > capacity - Count)
		{
			var length = array.Length - offset;
			Array.Copy(array, offset, newArray, 0, length);
			Array.Copy(array, 0, newArray, length, Count - length);
		}
		else Array.Copy(array, offset, newArray, 0, Count);
		array = newArray;
		offset = 0;
		capacity <<= 1;
	}
}
class PairComparer<S, T> : IComparer<Pair<S, T>>
	where S : IComparable<S>
	where T : IComparable<T>
{
	public PairComparer() { }
	public int Compare(Pair<S, T> x, Pair<S, T> y)
	{
		var p = x.First.CompareTo(y.First);
		if (p != 0) return p;
		else return x.Second.CompareTo(y.Second);
	}
}
class Pair<S, T>
{
	public S First;
	public T Second;
	public Pair() { First = default(S); Second = default(T); }
	public Pair(S s, T t) { First = s; Second = t; }
	public override string ToString() => $"({First}, {Second})";
	public override int GetHashCode() => First.GetHashCode() ^ Second.GetHashCode();
	public override bool Equals(object obj)
	{
		if (ReferenceEquals(this, obj)) return true;
		else if (obj == null) return false;
		var tmp = obj as Pair<S, T>;
		return tmp != null && First.Equals(tmp.First) && Second.Equals(tmp.Second);
	}
}
class Point : Pair<int, int>
{
	public int X { get { return First; } set { First = value; } }
	public int Y { get { return Second; } set { Second = value; } }
	public Point() : base(0, 0) { }
	public Point(int x, int y) : base(x, y) { }
	public IEnumerable<Point> Neighbors4()
	{
		yield return new Point(X - 1, Y);
		yield return new Point(X, Y - 1);
		yield return new Point(X, Y + 1);
		yield return new Point(X + 1, Y);
	}
	public IEnumerable<Point> Neighbors8()
	{
		yield return new Point(X - 1, Y - 1);
		yield return new Point(X - 1, Y);
		yield return new Point(X - 1, Y + 1);
		yield return new Point(X, Y - 1);
		yield return new Point(X, Y + 1);
		yield return new Point(X + 1, Y - 1);
		yield return new Point(X + 1, Y);
		yield return new Point(X + 1, Y + 1);
	}
	public static Point operator +(Point p) => new Point(p.X, p.Y);
	public static Point operator -(Point p) => new Point(-p.X, -p.Y);
	public static Point operator /(Point p, int r) => new Point(p.X / r, p.Y / r);
	public static Point operator *(int r, Point p) => new Point(p.X * r, p.Y * r);
	public static Point operator *(Point p, int r) => new Point(p.X * r, p.Y * r);
	public static Point operator +(Point p, Point q) => new Point(p.X + q.X, p.Y + q.Y);
	public static Point operator -(Point p, Point q) => new Point(p.X - q.X, p.Y - q.Y);
}
class Printer : IDisposable
{
	bool isConsole;
	TextWriter file;
	public Printer() { file = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; isConsole = true; }
	public Printer(string path) { file = new StreamWriter(path, false) { AutoFlush = false }; isConsole = false; }
	public void Write<T>(T value) => file.Write(value);
	public void Write(bool b) => file.Write(b ? "YES" : "NO");
	public void Write(string str, params object[] args) => file.Write(str, args);
	public void WriteLine() => file.WriteLine();
	public void WriteLine<T>(T value) => file.WriteLine(value);
	public void WriteLine(bool b) => file.WriteLine(b ? "YES" : "NO");
	public void WriteLine<T>(IEnumerable<T> list) { foreach (var x in list) file.WriteLine(x); }
	public void WriteLine<T>(List<T> list) { foreach (var x in list) file.WriteLine(x); }
	public void WriteLine<T>(T[] list) { foreach (var x in list) file.WriteLine(x); }
	public void WriteLine(string str, params object[] args) => file.WriteLine(str, args);
	public void Dispose() { file.Flush(); if (!isConsole) file.Dispose(); }
}
class Scanner : IDisposable
{
	bool isConsole;
	TextReader file;
	public Scanner() { file = Console.In; }
	public Scanner(string path) { file = new StreamReader(path); isConsole = false; }
	public void Dispose() { if (!isConsole) file.Dispose(); }
	public T Get<T>() => (T)Convert(file.ReadLine(), Type.GetTypeCode(typeof(T)));
	public int Int => Get<int>();
	public uint UInt => Get<uint>();
	public long Long => Get<long>();
	public ulong ULong => Get<ulong>();
	public double Double => Get<double>();
	public decimal Decimal => Get<decimal>();
	public char Char => Get<char>();
	public string String => Get<string>();
	public Tuple<S, T> Get<S, T>() { S s; T t; Read(out s, out t); return new Tuple<S, T>(s, t); }
	public Tuple<S, T, U> Get<S, T, U>() { S s; T t; U u; Read(out s, out t, out u); return new Tuple<S, T, U>(s, t, u); }
	public Tuple<S, T, U, V> Get<S, T, U, V>() { S s; T t; U u; V v; Read(out s, out t, out u, out v); return new Tuple<S, T, U, V>(s, t, u, v); }
	public Tuple<S, T, U, V, W> Get<S, T, U, V, W>() { S s; T t; U u; V v; W w; Read(out s, out t, out u, out v, out w); return new Tuple<S, T, U, V, W>(s, t, u, v, w); }
	public Tuple<S, T, U, V, W, X> Get<S, T, U, V, W, X>() { S s; T t; U u; V v; W w; X x; Read(out s, out t, out u, out v, out w, out x); return new Tuple<S, T, U, V, W, X>(s, t, u, v, w, x); }
	public Tuple<S, T, U, V, W, X, Y> Get<S, T, U, V, W, X, Y>() { S s; T t; U u; V v; W w; X x; Y y; Read(out s, out t, out u, out v, out w, out x, out y); return new Tuple<S, T, U, V, W, X, Y>(s, t, u, v, w, x, y); }
	public Tuple<S, T, U, V, W, X, Y, Z> Get<S, T, U, V, W, X, Y, Z>() { S s; T t; U u; V v; W w; X x; Y y; Z z; Read(out s, out t, out u, out v, out w, out x, out y, out z); return new Tuple<S, T, U, V, W, X, Y, Z>(s, t, u, v, w, x, y, z); }
	public Pair<S, T> Pair<S, T>() { S s; T t; Read(out s, out t); return new Pair<S, T>(s, t); }
	object Convert(string str, TypeCode type)
	{
		if (type == TypeCode.Int32) return int.Parse(str);
		else if (type == TypeCode.UInt32) return uint.Parse(str);
		else if (type == TypeCode.Int64) return long.Parse(str);
		else if (type == TypeCode.UInt64) return ulong.Parse(str);
		else if (type == TypeCode.Double) return double.Parse(str);
		else if (type == TypeCode.Decimal) return decimal.Parse(str);
		else if (type == TypeCode.Char) return str[0];
		else if (type == TypeCode.String) return str;
		else if (type == Type.GetTypeCode(typeof(Point))) { int s, t; Read(out s, out t); return new Point(s, t); }
		else throw new Exception();
	}
	public T[] ReadMany<T>() { var type = Type.GetTypeCode(typeof(T)); return file.ReadLine().Split(sep, StringSplitOptions.RemoveEmptyEntries).Select(str => (T)Convert(str, type)).ToArray(); }
	public T[] ReadMany<T>(int n) { var type = Type.GetTypeCode(typeof(T)); return file.ReadLine().Split(sep, StringSplitOptions.RemoveEmptyEntries).Take(n).Select(str => (T)Convert(str, type)).ToArray(); }
	public T[] ReadManyLines<T>(int n, Func<T> selector) => Enumerable.Range(0, n).Select(_ => selector()).ToArray();
	public T[] ReadManyLines<T>(int n) => Enumerable.Range(0, n).Select(_ => Get<T>()).ToArray();
	public Tuple<S, T>[] ReadManyLines<S, T>(int n) => Enumerable.Range(0, n).Select(_ => Get<S, T>()).ToArray();
	public Tuple<S, T, U>[] ReadManyLines<S, T, U>(int n) => Enumerable.Range(0, n).Select(_ => Get<S, T, U>()).ToArray();
	public Tuple<S, T, U, V>[] ReadManyLines<S, T, U, V>(int n) => Enumerable.Range(0, n).Select(_ => Get<S, T, U, V>()).ToArray();
	public Tuple<S, T, U, V, W>[] ReadManyLines<S, T, U, V, W>(int n) => Enumerable.Range(0, n).Select(_ => Get<S, T, U, V, W>()).ToArray();
	public Tuple<S, T, U, V, W, X>[] ReadManyLines<S, T, U, V, W, X>(int n) => Enumerable.Range(0, n).Select(_ => Get<S, T, U, V, W, X>()).ToArray();
	public Tuple<S, T, U, V, W, X, Y>[] ReadManyLines<S, T, U, V, W, X, Y>(int n) => Enumerable.Range(0, n).Select(_ => Get<S, T, U, V, W, X, Y>()).ToArray();
	public Tuple<S, T, U, V, W, X, Y, Z>[] ReadManyLines<S, T, U, V, W, X, Y, Z>(int n) => Enumerable.Range(0, n).Select(_ => Get<S, T, U, V, W, X, Y, Z>()).ToArray();
	public T[,] ReadManyManyLines<T>(int X, int Y)
	{
		var array = new T[X, Y];
		for (var y = 0; y < Y; y++) { var tmp = ReadMany<T>(X); for (var x = 0; x < X; x++) array[x, y] = tmp[x]; }
		return array;
	}
	public void Read<S>(out S s)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S))).ToArray();
		s = (S)read[0];
	}
	public void Read<S, T>(out S s, out T t)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S)), Type.GetTypeCode(typeof(T))).ToArray();
		s = (S)read[0];
		t = (T)read[1];
	}
	public void Read<S, T, U>(out S s, out T t, out U u)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S)), Type.GetTypeCode(typeof(T)), Type.GetTypeCode(typeof(U))).ToArray();
		s = (S)read[0];
		t = (T)read[1];
		u = (U)read[2];
	}
	public void Read<S, T, U, V>(out S s, out T t, out U u, out V v)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S)), Type.GetTypeCode(typeof(T)), Type.GetTypeCode(typeof(U)), Type.GetTypeCode(typeof(V))).ToArray();
		s = (S)read[0];
		t = (T)read[1];
		u = (U)read[2];
		v = (V)read[3];
	}
	public void Read<S, T, U, V, W>(out S s, out T t, out U u, out V v, out W w)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S)), Type.GetTypeCode(typeof(T)),
			Type.GetTypeCode(typeof(U)), Type.GetTypeCode(typeof(V)), Type.GetTypeCode(typeof(W))).ToArray();
		s = (S)read[0];
		t = (T)read[1];
		u = (U)read[2];
		v = (V)read[3];
		w = (W)read[4];
	}
	public void Read<S, T, U, V, W, X>(out S s, out T t, out U u, out V v, out W w, out X x)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S)), Type.GetTypeCode(typeof(T)),
			Type.GetTypeCode(typeof(U)), Type.GetTypeCode(typeof(V)), Type.GetTypeCode(typeof(W)), Type.GetTypeCode(typeof(X))).ToArray();
		s = (S)read[0];
		t = (T)read[1];
		u = (U)read[2];
		v = (V)read[3];
		w = (W)read[4];
		x = (X)read[5];
	}
	public void Read<S, T, U, V, W, X, Y>(out S s, out T t, out U u, out V v, out W w, out X x, out Y y)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S)), Type.GetTypeCode(typeof(T)),
			Type.GetTypeCode(typeof(U)), Type.GetTypeCode(typeof(V)), Type.GetTypeCode(typeof(W)), Type.GetTypeCode(typeof(X)), Type.GetTypeCode(typeof(Y))).ToArray();
		s = (S)read[0];
		t = (T)read[1];
		u = (U)read[2];
		v = (V)read[3];
		w = (W)read[4];
		x = (X)read[5];
		y = (Y)read[6];
	}
	public void Read<S, T, U, V, W, X, Y, Z>(out S s, out T t, out U u, out V v, out W w, out X x, out Y y, out Z z)
	{
		var read = ReadMulti(Type.GetTypeCode(typeof(S)), Type.GetTypeCode(typeof(T)),
			Type.GetTypeCode(typeof(U)), Type.GetTypeCode(typeof(V)), Type.GetTypeCode(typeof(W)),
			Type.GetTypeCode(typeof(X)), Type.GetTypeCode(typeof(Y)), Type.GetTypeCode(typeof(Z))).ToArray();
		s = (S)read[0];
		t = (T)read[1];
		u = (U)read[2];
		v = (V)read[3];
		w = (W)read[4];
		x = (X)read[5];
		y = (Y)read[6];
		z = (Z)read[7];
	}
	static char[] sep = new char[] { ' ', '/' };
	IEnumerable<object> ReadMulti(params TypeCode[] types)
	{
		var input = file.ReadLine().Split(sep, StringSplitOptions.RemoveEmptyEntries);
		for (var i = 0; i < types.Length; i++) yield return Convert(input[i], types[i]);
	}
	public T[,] Board<T>(int X, int Y, Func<char, int, int, T> selector)
	{
		var array = new T[X, Y];
		for (var y = 0; y < Y; y++)
		{
			var str = Get<string>();
			for (var x = 0; x < X; x++) array[x, y] = selector(str[x], x, y);
		}
		return array;
	}
}
static class Func
{
	public const int Inf = 1073741789;  // 2 * Inf < int.MaxValue, and Inf is a prime number
	public const long InfL = 4011686018427387913L;  // 2 * InfL < long.MaxValue, and InfL is a prime number
	public static Comparison<T> DefaultComparison<T>() => (x, y) => Comparer<T>.Default.Compare(x, y);
	public static Comparison<T> ToComparison<T>(this IComparer<T> comp) => comp == null ? DefaultComparison<T>() : (x, y) => comp.Compare(x, y);
	/// <summary>
	/// Find the first number x such that pred(x) is true
	/// if pred(x) is false for all min&lt;=x&lt;max, then return max
	/// in other words, pred(max) is assumed to be true
	/// </summary>
	/// <param name="min">inclusive lower limit</param>
	/// <param name="max">exclusive upper limit</param>
	/// <param name="pred">monotonous predicate, i.e. if pred(a) and a&lt;b, then pred(b)</param>
	/// <returns>first number such that satisfy pred</returns>
	public static long FirstBinary(long min, long max, Predicate<long> pred)
	{
		while (min < max)
		{
			var mid = (min + max) / 2;
			if (pred(mid)) max = mid;
			else min = mid + 1;
		}
		return min;
	}
	/// <summary>
	/// Find the first number x such that pred(x) is true
	/// if pred(x) is false for all min&lt;=x&lt;max, then return max
	/// in other words, pred(max) is assumed to be true
	/// </summary>
	/// <param name="min">inclusive lower limit</param>
	/// <param name="max">exclusive upper limit</param>
	/// <param name="pred">monotonous predicate, i.e. if pred(a) and a&lt;b, then pred(b)</param>
	/// <returns>first number such that satisfy pred</returns>
	public static int FirstBinary(int min, int max, Predicate<int> pred)
	{
		while (min < max)
		{
			var mid = (min + max) / 2;
			if (pred(mid)) max = mid;
			else min = mid + 1;
		}
		return min;
	}
	public static Dictionary<T, S> Reverse<S, T>(this IDictionary<S, T> dict)
	{
		var r = new Dictionary<T, S>();
		foreach (var t in dict) r.Add(t.Value, t.Key);
		return r;
	}
	public static void Swap<T>(this IList<T> array, int i, int j) { var tmp = array[i]; array[i] = array[j]; array[j] = tmp; }
	public static void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
	public static T IndexAt<T>(this T[,] array, Pair<int, int> index) => array[index.First, index.Second];
	public static bool InRegion(this Pair<int, int> p, int X, int Y) => p.InRegion(0, X, 0, Y);
	public static bool InRegion(this Pair<int, int> p, int x, int X, int y, int Y) => p.First >= x && p.Second >= y && p.First < X && p.Second < Y;
	/// <summary>
	/// get all permutation of 0, 1, ..., n - 1
	/// </summary>
	/// <param name="n">length of array</param>
	/// <param name="func">if you want to change the elements of the array, you must take a copy</param>
	public static void Permutation(int n, Action<int[]> func)
	{
		var array = new int[n];
		var unused = new bool[n];
		for (var i = 0; i < n; i++) unused[i] = true;
		Permutation(n, 0, array, unused, func);
	}
	static void Permutation(int n, int i, int[] array, bool[] unused, Action<int[]> func)
	{
		if (i == n) func(array);
		else
			for (var x = 0; x < n; x++)
				if (unused[x])
				{
					array[i] = x;
					unused[x] = false;
					Permutation(n, i + 1, array, unused, func);
					unused[x] = true;
				}
	}
	public static long Fact(int n)
	{
		var fact = 1L;
		for (var i = 2; i <= n; i++) fact *= i;
		return fact;
	}
	public static Dictionary<long, int> Factorize(this long n, List<int> primes)
	{
		var d = new Dictionary<long, int>();
		for (var j = 0; j < primes.Count; j++)
		{
			var i = primes[j];
			if (i * i > n) break;
			if (n % i == 0)
			{
				d.Add(i, 0);
				while (n % i == 0) { n /= i; d[i]++; }
			}
		}
		if (n > 1) d.Add(n, 1);
		return d;
	}
	public static Dictionary<long, int> Factorize(this long n)
	{
		var d = new Dictionary<long, int>();
		for (var i = 2L; i * i <= n; i++)
			if (n % i == 0)
			{
				d.Add(i, 0);
				while (n % i == 0) { n /= i; d[i]++; }
			}
		if (n > 1) d.Add(n, 1);
		return d;
	}
	public static long LCM(long n, long m) => Math.Abs((n / GCD(n, m)) * m);
	public static long Divide(long n, long m) => (n - Remainder(n, m)) / m;
	public static long Remainder(long n, long m)
	{
		if (m == 0) throw new DivideByZeroException();
		else if (m < 0) return Remainder(n, -m);
		else
		{
			var r = n % m;
			return r < 0 ? r + m : r;
		}
	}
	public static long Recurrence(long[] coeff, long[] init, long N, long mod)
	{
		var K = init.Length;
		if (N < 0)
		{
			var inv = Inverse(coeff[0], mod);
			var rc = new long[K];
			for (var i = 1; i < K; i++) rc[K - i] = -coeff[i] * inv % mod;
			rc[0] = inv;
			var ri = new long[K];
			for (var i = 0; i < K; i++) ri[K - 1 - i] = init[i];
			return Recurrence(rc, ri, K - 1 - N, mod);
		}
		var tmp = new long[K];
		Recurrence(coeff, init, tmp, N, mod);
		var sum = 0L;
		for (var i = 0; i < K; i++) sum += init[i] * tmp[i] % mod;
		sum %= mod;
		if (sum < 0) sum += mod;
		return sum;
	}
	public static void Recurrence(long[] coeff, long[] init, long[] state, long N, long mod)
	{
		var K = init.Length;
		if (N < K) state[N] = init[N];
		else if ((N & 1) == 0)
		{
			var tmp = new long[K][];
			for (var i = 0; i < K; i++) tmp[i] = new long[K];
			Recurrence(coeff, init, tmp[0], N / 2, mod);
			for (var i = 1; i < K; i++) tmp[i] = Next(coeff, tmp[i - 1], mod);
			for (var i = 0; i < K; i++)
			{
				state[i] = 0;
				for (var j = 0; j < K; j++) state[i] += tmp[0][j] * tmp[j][i] % mod;
				state[i] %= mod;
			}
		}
		else if (N < 2 * K || (N & 2) == 0)
		{
			var tmp = new long[K];
			Recurrence(coeff, init, tmp, N - 1, mod);
			tmp = Next(coeff, tmp, mod);
			for (var i = 0; i < K; i++) state[i] = tmp[i];
		}
		else
		{
			var tmp = new long[K];
			Recurrence(coeff, init, tmp, N + 1, mod);
			tmp = Prev(coeff, tmp, mod);
			for (var i = 0; i < K; i++) state[i] = tmp[i];
		}
	}
	static long[] Next(long[] coeff, long[] state, long mod)
	{
		var K = coeff.Length;
		var tmp = new long[K];
		for (var i = 0; i < K; i++) tmp[i] = coeff[i] * state[K - 1] % mod;
		for (var i = 1; i < K; i++) tmp[i] = (tmp[i] + state[i - 1]) % mod;
		return tmp;
	}
	static long[] Prev(long[] coeff, long[] state, long mod)
	{
		var K = coeff.Length;
		var tmp = new long[K];
		var inv = Inverse(coeff[0], mod);
		tmp[K - 1] = state[0] * inv % mod;
		for (var i = 1; i < K; i++) tmp[i - 1] = (state[i] - coeff[i] * tmp[K - 1] % mod) % mod;
		return tmp;
	}
	// get all primes less than or equal to n
	public static List<int> GetPrimes(int n)
	{
		if (n < 3) n = 3;
		var m = (n - 1) >> 1;
		var primes = new List<int>((int)(n / Math.Log(n)));
		primes.Add(2);
		var composites = new bool[m];
		composites[0] = false;
		for (var p = 0; p < m; p++)
		{
			if (!composites[p])
			{
				var pnum = 2 * p + 3;
				primes.Add(pnum);
				for (var k = 3 * p + 3; k < m; k += pnum) composites[k] = true;
			}
		}
		return primes;
	}
	/// <summary>
	/// solve nx+my=1 and returns (x,y)
	/// </summary>
	/// <param name="n">assumed to be with m</param>
	/// <param name="m">assumed to be with n</param>
	/// <returns>(x,y) where nx+my=1</returns>
	public static Tuple<long, long> SolveLinear(long n, long m)
	{
		if (n < 0) { var p = SolveLinear(-n, m); return p == null ? p : new Tuple<long, long>(-p.Item1, p.Item2); }
		if (m < 0) { var p = SolveLinear(n, -m); return p == null ? p : new Tuple<long, long>(p.Item1, -p.Item2); }
		if (n < m) { var p = SolveLinear(m, n); return p == null ? p : new Tuple<long, long>(p.Item2, p.Item1); }
		long a = 1, b = 0, c = 0, d = 1;
		while (m > 0)
		{
			var r = n % m;
			var q = n / m;
			n = m;
			m = r;
			var tmp = a;
			a = -a * q + b;
			b = tmp;
			tmp = c;
			c = -c * q + d;
			d = tmp;
		}
		return n != 1 ? null : new Tuple<long, long>(d, b);
	}
	public static int GCD(int n, int m)
	{
		var a = Math.Abs(n);
		var b = Math.Abs(m);
		if (a < b) { var c = a; a = b; b = c; }
		while (b > 0)
		{
			var c = a % b;
			a = b;
			b = c;
		}
		return a;
	}
	/*public static long GCD(long n, long m)
	{
		var a = Math.Abs(n);
		var b = Math.Abs(m);
		if (a < b) { var c = a; a = b; b = c; }
		while (b > 0)
		{
			var c = a % b;
			a = b;
			b = c;
		}
		return a;
	}*/
	public static long GCD(long a, long b)
	{
		var n = (ulong)Math.Abs(a); var m = (ulong)Math.Abs(b);
		if (n == 0) return (long)m; if (m == 0) return (long)n;
		int zm = 0, zn = 0;
		while ((n & 1) == 0) { n >>= 1; zn++; }
		while ((m & 1) == 0) { m >>= 1; zm++; }
		while (m != n)
		{
			if (m > n) { m -= n; while ((m & 1) == 0) m >>= 1; }
			else { n -= m; while ((n & 1) == 0) n >>= 1; }
		}
		return (long)n << Math.Min(zm, zn);
	}
	public static BigInteger GCD(BigInteger a, BigInteger b) => BigInteger.GreatestCommonDivisor(a, b);
	public static long Inverse(long a, long mod)
	{
		if (a < 0) { a %= mod; if (a < 0) a += mod; }
		var t = SolveLinear(a, mod);
		return t.Item1 > 0 ? t.Item1 : t.Item1 + mod;
	}
	public static ulong Pow(ulong a, ulong b, ulong mod)
	{
		var p = 1uL;
		var x = a;
		while (b > 0)
		{
			if ((b & 1) == 1) p = (p * x) % mod;
			b >>= 1;
			x = (x * x) % mod;
		}
		return p;
	}
	public static long Pow(long a, long b, long mod)
	{
		var p = 1L;
		var x = a;
		while (b > 0)
		{
			if ((b & 1) == 1) p = (p * x) % mod;
			b >>= 1;
			x = (x * x) % mod;
		}
		return p;
	}
	public static long Pow(long a, long b)
	{
		if (a == 1) return 1;
		else if (a == 0) { if (b >= 0) return 0; else throw new DivideByZeroException(); }
		else if (b < 0) return 0;
		var p = 1L;
		var x = a;
		while (b > 0)
		{
			if ((b & 1) == 1) p *= x;
			b >>= 1;
			x *= x;
		}
		return p;
	}
	public static ulong Pow(ulong a, ulong b)
	{
		var p = 1ul;
		var x = a;
		while (b > 0)
		{
			if ((b & 1) == 1) p *= x;
			b >>= 1;
			x *= x;
		}
		return p;
	}
	public static long ChineseRemainder(Tuple<long, long> modRemainder1, Tuple<long, long> modRemainder2)
	{
		var m1 = modRemainder1.Item1;
		var m2 = modRemainder2.Item1;
		var a1 = modRemainder1.Item2;
		var a2 = modRemainder2.Item2;
		var t = SolveLinear(m1, m2);
		var n1 = t.Item1;
		var n2 = t.Item2;
		return (m1 * n1 * a2 + m2 * n2 * a1) % (m1 * m2);
	}
	public static long ChineseRemainder(params Tuple<long, long>[] modRemainder)
	{
		if (modRemainder.Length == 0) throw new DivideByZeroException();
		else if (modRemainder.Length == 1) return modRemainder[0].Item2;
		else if (modRemainder.Length == 2) return ChineseRemainder(modRemainder[0], modRemainder[1]);
		else
		{
			var tuple = new Tuple<long, long>(1, 0);
			for (var i = 0; i < modRemainder.Length; i++)
			{
				var tmp = ChineseRemainder(tuple, modRemainder[i]);
				tuple = new Tuple<long, long>(tuple.Item1 * modRemainder[i].Item1, tmp);
			}
			return tuple.Item2;
		}
	}
	// forward transform -> theta= 2*PI/n
	// reverse transform -> theta=-2*PI/n, and use a[i]/n instead of a
	// O(n*log(n))
	public static void FastFourierTransform(int n, double theta, Complex[] a)
	{
		for (var m = n; m >= 2; m >>= 1)
		{
			var mh = m >> 1;
			for (var i = 0; i < mh; i++)
			{
				var w = Complex.Exp(i * theta * Complex.ImaginaryOne);
				for (var j = i; j < n; j += m)
				{
					var k = j + mh;
					var x = a[j] - a[k];
					a[j] += a[k];
					a[k] = w * x;
				}
			}
			theta *= 2;
		}
		var s = 0;
		for (var j = 1; j < n - 1; j++)
		{
			for (var k = n >> 1; k > (s ^= k); k >>= 1) ;
			if (j < s) a.Swap(s, j);
		}
	}
	// get table of Euler function
	// let return value f, f[i]=phi(i) for 0<=i<=n
	// nearly O(n)
	public static long[] EulerFunctionTable(long n)
	{
		if (n < 2) n = 2;
		var f = new long[n + 1];
		for (var i = 0L; i <= n; i++) f[i] = i;
		for (var i = 2L; i <= n; i++) if (f[i] == i) for (var j = i; j <= n; j += i) f[j] = f[j] / i * (i - 1);
		return f;
	}
	// O(sqrt(n))
	public static long EulerFunction(long n)
	{
		var res = n;
		for (var i = 2L; i * i <= n; i++)
			if (n % i == 0)
			{
				res = res / i * (i - 1);
				do n /= i; while (n % i == 0);
			}
		if (n != 1) res = res / n * (n - 1);
		return res;
	}
	// get moebius function of d s.t. 0<=d<=n
	// O(n)
	public static int[] MoebiusFunctionTable(long n)
	{
		if (n < 2) n = 2;
		var f = new int[n + 1];
		var p = new bool[n + 1];
		for (var i = 0L; i <= n; i++) f[i] = 1;
		for (var i = 2L; i <= n; i++) if (!p[i])
			{
				for (var j = i; j <= n; j += i) { f[j] *= -1; p[j] = true; }
				for (var j = i * i; j <= n; j += i * i) f[j] = 0;
			}
		return f;
	}
	// get moebius function of d s.t. d|n
	// if dict.ContainsKey(d), dict[d]!=0, otherwise moebius function of d is 0
	// O(sqrt(n))
	public static Dictionary<long, int> MoebiusFunctionOfDivisors(long n)
	{
		var ps = new List<long>();
		for (var i = 2L; i * i <= n; i++)
			if (n % i == 0)
			{
				ps.Add(i);
				do n /= i; while (n % i == 0);
			}
		if (n != 1) ps.Add(n);
		var dict = new Dictionary<long, int>();
		var m = ps.Count;
		for (var i = 0; i < (1 << m); i++)
		{
			var mu = 1;
			var k = 1L;
			for (var j = 0; j < m; j++) if ((i & (1 << j)) != 0) { mu *= -1; k *= ps[j]; }
			dict.Add(k, mu);
		}
		return dict;
	}
	// O(sqrt(n))
	public static int MoebiusFunction(long n)
	{
		var mu = 1;
		for (var i = 2L; i * i <= n; i++)
			if (n % i == 0)
			{
				mu *= -1;
				if ((n /= i) % i == 0) return 0;
			}
		return n == 1 ? mu : -mu;
	}
	// O(sqrt(n))
	public static long CarmichaelFunction(long n)
	{
		var lambda = 1L;
		var c = 0;
		while (n % 2 == 0) { n /= 2; c++; }
		if (c == 2) lambda = 2; else if (c > 2) lambda = 1 << (c - 2);
		for (var i = 3L; i * i <= n; i++)
			if (n % i == 0)
			{
				var tmp = i - 1;
				n /= i;
				while (n % i == 0) { n /= i; tmp *= i; }
				lambda = LCM(lambda, tmp);
			}
		if (n != 1) lambda = LCM(lambda, n - 1);
		return lambda;
	}
	// a+bi is Gaussian prime or not
	public static bool IsGaussianPrime(ulong a, ulong b)
	{
		if (a == 0) return b % 4 == 3 && IsPrime(b);
		else if (b == 0) return a % 4 == 3 && IsPrime(a);
		else return IsPrime(a * a + b * b);
	}
	// nearly O(200)
	public static bool IsPrime(ulong n)
	{
		if (n <= 1 || (n > 2 && n % 2 == 0)) return false;
		var test = new uint[] { 2, 3, 5, 7, 11, 13, 17, 19, 23, 111 };
		var d = n - 1;
		int s = 0;
		while (d % 2 == 0) { ++s; d /= 2; }
		Predicate<ulong> f = t =>
		{
			var x = Pow(t, d, n);
			if (x == 1) return true;
			for (var r = 0L; r < s; r++)
			{
				if (x == n - 1) return true;
				x = (x * x) % n;
			}
			return false;
		};
		for (var i = 0; test[i] < n && test[i] != 111; i++) if (!f(test[i])) return false;
		return true;
	}
	public static decimal MeasureTime(Action action)
	{
		var sw = new Stopwatch();
		sw.Restart();
		action();
		sw.Stop();
		return sw.ElapsedTicks * 1000m / Stopwatch.Frequency;
	}
	static readonly double GoldenRatio = 2 / (3 + Math.Sqrt(5));
	// assume f is 凹
	// find c s.t. a<=c<=b and for all a<=x<=b, f(c)<=f(x)
	public static double GoldenSectionSearch(double a, double b, Func<double, double> f)
	{
		double c = a + GoldenRatio * (b - a), d = b - GoldenRatio * (b - a);
		double fc = f(c), fd = f(d);
		while (d - c > 1e-9)
		{
			if (fc > fd)
			{
				a = c; c = d; d = b - GoldenRatio * (b - a);
				fc = fd; fd = f(d);
			}
			else
			{
				b = d; d = c; c = a + GoldenRatio * (b - a);
				fd = fc; fc = f(c);
			}
		}
		return c;
	}
	// O(NW)
	public static int KnapsackW(int[] w, int[] v, int W)
	{
		var N = w.Length;
		var dp = new int[W + 1];
		for (var i = 0; i < N; i++) for (var j = W; j >= w[i]; j--)
				dp[j] = Math.Max(dp[j], v[i] + dp[j - w[i]]);
		return dp[W];
	}
	// O(NV)
	public static int KnapsackV(int[] w, int[] v, int W)
	{
		var N = w.Length;
		var V = v.Sum();
		var dp = new int[V + 1];
		for (var i = 1; i <= V; i++) dp[i] = Inf;
		for (var i = 0; i < N; i++) for (var j = V; j >= v[i]; j--)
				dp[j] = Math.Min(dp[j], w[i] + dp[j - v[i]]);
		for (var j = V; j >= 0; j--) if (dp[j] <= W) return j;
		return 0;
	}
	// O(N*2^(N/2))
	public static long KnapsackN(long[] w, long[] v, int W)
	{
		var N = w.Length;
		var half = N / 2;
		var items = new Tuple<long, long>[N];
		for (var i = 0; i < N; i++) items[i] = new Tuple<long, long>(w[i], v[i]);
		Array.Sort(items, (x, y) => x.Item1.CompareTo(y.Item1));
		Func<int, int, List<Pair<long, long>>> gen = (start, end) =>
		{
			if (start >= end) return new List<Pair<long, long>>();
			var lim = 1 << (end - start);
			var list = new List<Pair<long, long>>();
			for (var i = 0; i < lim; i++)
			{
				var weight = 0L;
				var value = 0L;
				var tmp = i;
				for (var j = start; j < end; j++)
				{
					if ((tmp & 1) == 1) { weight += items[j].Item1; value += items[j].Item2; }
					tmp >>= 1;
				}
				if (weight <= W) list.Add(new Pair<long, long>(weight, value));
			}
			list.Sort((x, y) => { var c = x.First.CompareTo(y.First); return c == 0 ? x.Second.CompareTo(y.Second) : c; });
			var n = list.Count;
			if (n == 0) return list;
			for (var i = list.Count - 2; i >= 0; i--) if (list[i].First == list[i + 1].First) list[i].Second = Math.Max(list[i].Second, list[i + 1].Second);
			var small = new List<Pair<long, long>>();
			var last = -1;
			while (last + 1 < n)
			{
				var tmp = list[last + 1].First;
				last = Func.FirstBinary(last + 1, n, x => list[x].First > tmp) - 1;
				if (small.Count == 0 || list[last].Second > small[small.Count - 1].Second) small.Add(list[last]);
			}
			return small;
		};
		var first = gen(0, half);
		var second = gen(half, N);
		var max = 0L;
		var last2 = second.Count;
		foreach (var item in first)
		{
			last2 = Func.FirstBinary(0, last2, x => second[x].First > W - item.First) - 1;
			if (last2 < 0) break;
			if (second[last2].First <= W - item.First) Func.SetToMax(ref max, item.Second + second[last2].Second);
			last2++;
		}
		return max;
	}
	// nums[i] が counts[i] 個
	// K is partial sum?
	// O(NK)
	public static bool PartialSum(int[] nums, int[] counts, int K)
	{
		var N = nums.Length;
		var memo = new int[K + 1];
		for (var s = 1; s <= K; s++) memo[s] = -1;
		for (var n = 0; n < N; n++) for (var s = 0; s <= K; s++) memo[s] = memo[s] >= 0 ? counts[n] : s < nums[n] ? -1 : memo[s - nums[n]] - 1;
		return memo[K] >= 0;
	}
	// O(N log(N))
	public static int LongestIncreasingSubsequence(int[] a)
	{
		var N = a.Length;
		var memo = new int[N];
		for (var n = 0; n < N; n++) memo[n] = Inf;
		for (var n = 0; n < N; n++)
		{
			var k = Func.FirstBinary(0, N, x => a[n] <= memo[x]);
			memo[k] = a[n];
		}
		return Func.FirstBinary(0, N, x => memo[x] == Inf);
	}
	// O(nm)
	public static int LongestCommonSubsequence(string s, string t)
	{
		var n = s.Length;
		var m = t.Length;
		var memo = new int[n + 1, m + 1];
		for (int i = n - 1; i >= 0; i--)
			for (int j = m - 1; j >= 0; j--)
				if (s[i] == t[j]) memo[i, j] = memo[i + 1, j + 1] + 1;
				else memo[i, j] = Math.Max(memo[i + 1, j], memo[i, j + 1]);
		return memo[0, 0];
	}
	// the number of ways of dividing N to M numbers
	// O(NM)
	public static int Partition(int N, int M, int Mod)
	{
		var memo = new long[N + 1, M + 1];
		for (var m = 0; m <= M; m++) memo[0, m] = 1;
		for (var n = 1; n <= N; n++)
		{
			memo[n, 0] = 0;
			for (var m = 1; m <= M; m++) memo[n, m] = (memo[n, m - 1] + (n - m >= 0 ? memo[n - m, m] : 0)) % Mod;
		}
		return (int)memo[N, M];
	}
	// max{f(a)+...+f(b-1) | from<=a<b<=to}
	// O(to-from)
	public static long MaxIntervalSum(int from, int to, Func<long, long> f)
	{
		long max, dp;
		max = dp = f(from);
		for (var i = from + 1; i < to; i++)
		{
			var tmp = f(i);
			dp = tmp + Math.Max(0, dp);
			max = Math.Max(max, dp);
		}
		return max;
	}
	public static int MaxElement<T>(this IEnumerable<T> source, Comparison<T> comp)
	{
		var p = source.GetEnumerator();
		if (!p.MoveNext()) return -1;
		var max = p.Current;
		var mi = 0;
		var i = 0;
		while (p.MoveNext())
		{
			i++;
			if (comp(max, p.Current) < 0) { max = p.Current; mi = i; }
		}
		return mi;
	}
	public static int MaxElement<T>(this IEnumerable<T> source) where T : IComparable<T> => source.MaxElement((x, y) => x.CompareTo(y));
	public static int MinElement<T>(IEnumerable<T> source, Comparison<T> comp) => source.MaxElement((x, y) => comp(y, x));
	public static int MinElement<T>(IEnumerable<T> source) where T : IComparable<T> => source.MaxElement((x, y) => y.CompareTo(x));
	public static void Shuffle<T>(IList<T> source, Random rand) { for (var i = source.Count - 1; i >= 0; --i) source.Swap(i, rand.Next(0, i + 1)); }
	public static void Shuffle<T>(IList<T> source, RandomSFMT rand) { for (var i = source.Count - 1; i >= 0; --i) source.Swap(i, rand.Next(0, i + 1)); }
	public static char NextChar(this Random rand) => (char)(rand.Next(0, 'z' - 'a' + 1) + 'a');
	public static char NextChar(this RandomSFMT rand) => (char)(rand.Next(0, 'z' - 'a' + 1) + 'a');
	public static string NextString(this Random rand, int length) => new string(Enumerable.Range(0, length).Select(_ => rand.NextChar()).ToArray());
	public static string NextString(this RandomSFMT rand, int length) => new string(Enumerable.Range(0, length).Select(_ => rand.NextChar()).ToArray());
	public static IEnumerable<T> Rotate<T>(this IEnumerable<T> source)
	{
		var e = source.GetEnumerator();
		if (e.MoveNext())
		{
			var f = e.Current;
			while (e.MoveNext()) yield return e.Current;
			yield return f;
		}
	}
	public static T Apply<T>(this Func<T, T> func, T x, int n)
	{
		var a = x;
		for (var i = 0; i < n; i++) a = func(a);
		return a;
	}
	public static void MemberSet<T>(this T[] array, T value)
	{
		var X = array.Length;
		for (var x = 0; x < X; x++) array[x] = value;
	}
	public static void MemberSet<T>(this T[,] array, T value)
	{
		var X = array.GetLength(0); var Y = array.GetLength(1);
		for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) array[x, y] = value;
	}
	public static void MemberSet<T>(this T[,,] array, T value)
	{
		var X = array.GetLength(0); var Y = array.GetLength(1); var Z = array.GetLength(2);
		for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) for (var z = 0; z < Z; z++) array[x, y, z] = value;
	}
	public static void MemberSet<T>(this T[,,,] array, T value)
	{
		var X = array.GetLength(0); var Y = array.GetLength(1); var Z = array.GetLength(2); var W = array.GetLength(3);
		for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) for (var z = 0; z < Z; z++) for (var w = 0; w < W; w++) array[x, y, z, w] = value;
	}
	public static string ToYesNo(this bool flag) => flag ? "YES" : "NO";
	public static int SetToMin(ref int min, int other) => min = Math.Min(min, other);
	public static int SetToMax(ref int max, int other) => max = Math.Max(max, other);
	public static long SetToMin(ref long min, long other) => min = Math.Min(min, other);
	public static long SetToMax(ref long max, long other) => max = Math.Max(max, other);
	public static Tuple<SortedDictionary<int, int>, SortedDictionary<int, int>> Compress(IEnumerable<int> coord, int width, int X)
	{
		var tmp = new SortedSet<int>();
		foreach (var x in coord)
		{
			for (var w = -width; w <= width; w++)
				if (x + w < 0 || x + w >= X) continue;
				else if (tmp.Contains(x + w)) continue;
				else tmp.Add(x + w);
		}
		var index = 0;
		var inverse = new SortedDictionary<int, int>();
		var dict = new SortedDictionary<int, int>();
		foreach (var pair in tmp)
		{
			dict.Add(pair, index);
			inverse.Add(index++, pair);
		}
		return new Tuple<SortedDictionary<int, int>, SortedDictionary<int, int>>(dict, inverse);
	}
	public static int MSB(uint n)
	{
		n |= (n >> 1);
		n |= (n >> 2);
		n |= (n >> 4);
		n |= (n >> 8);
		n |= (n >> 16);
		return BitCount(n) - 1;
	}
	public static int BitCount(uint n)
	{
		n = (n & 0x55555555) + ((n >> 1) & 0x55555555);
		n = (n & 0x33333333) + ((n >> 2) & 0x33333333);
		n = (n & 0x0f0f0f0f) + ((n >> 4) & 0x0f0f0f0f);
		n = (n & 0x00ff00ff) + ((n >> 8) & 0x00ff00ff);
		return (int)((n & 0x0000ffff) + ((n >> 16) & 0x0000ffff));
	}
	public static int LSB(uint n)
	{
		n |= (n << 1);
		n |= (n << 2);
		n |= (n << 4);
		n |= (n << 8);
		n |= (n << 16);
		return 32 - BitCount(n);
	}
	public static int MSB(ulong n)
	{
		n |= (n >> 1);
		n |= (n >> 2);
		n |= (n >> 4);
		n |= (n >> 8);
		n |= (n >> 16);
		n |= (n >> 32);
		return BitCount(n) - 1;
	}
	public static int BitCount(ulong n)
	{
		n = (n & 0x5555555555555555) + ((n >> 1) & 0x5555555555555555);
		n = (n & 0x3333333333333333) + ((n >> 2) & 0x3333333333333333);
		n = (n & 0x0f0f0f0f0f0f0f0f) + ((n >> 4) & 0x0f0f0f0f0f0f0f0f);
		n = (n & 0x00ff00ff00ff00ff) + ((n >> 8) & 0x00ff00ff00ff00ff);
		n = (n & 0x0000ffff0000ffff) + ((n >> 16) & 0x0000ffff0000ffff);
		return (int)((n & 0x00000000ffffffff) + ((n >> 32) & 0x00000000ffffffff));
	}
	public static int LSB(ulong n)
	{
		n |= (n << 1);
		n |= (n << 2);
		n |= (n << 4);
		n |= (n << 8);
		n |= (n << 16);
		n |= (n << 32);
		return 64 - BitCount(n);
	}
	public static int Abs(this int n) => Math.Abs(n);
	public static long Abs(this long n) => Math.Abs(n);
	public static double Abs(this double n) => Math.Abs(n);
	public static float Abs(this float n) => Math.Abs(n);
	public static decimal Abs(this decimal n) => Math.Abs(n);
	public static short Abs(this short n) => Math.Abs(n);
	public static sbyte Abs(this sbyte n) => Math.Abs(n);
	public static int Min(params int[] nums) { var min = int.MaxValue; foreach (var n in nums) min = Math.Min(min, n); return min; }
	public static long Min(params long[] nums) { var min = long.MaxValue; foreach (var n in nums) min = Math.Min(min, n); return min; }
	public static uint Min(params uint[] nums) { var min = uint.MaxValue; foreach (var n in nums) min = Math.Min(min, n); return min; }
	public static ulong Min(params ulong[] nums) { var min = ulong.MaxValue; foreach (var n in nums) min = Math.Min(min, n); return min; }
	public static double Min(params double[] nums) { var min = double.MaxValue; foreach (var n in nums) min = Math.Min(min, n); return min; }
	public static decimal Min(params decimal[] nums) { var min = decimal.MaxValue; foreach (var n in nums) min = Math.Min(min, n); return min; }
	public static int Max(params int[] nums) { var min = int.MinValue; foreach (var n in nums) min = Math.Max(min, n); return min; }
	public static long Max(params long[] nums) { var min = long.MinValue; foreach (var n in nums) min = Math.Max(min, n); return min; }
	public static uint Max(params uint[] nums) { var min = uint.MinValue; foreach (var n in nums) min = Math.Max(min, n); return min; }
	public static ulong Max(params ulong[] nums) { var min = ulong.MinValue; foreach (var n in nums) min = Math.Max(min, n); return min; }
	public static double Max(params double[] nums) { var min = double.MinValue; foreach (var n in nums) min = Math.Max(min, n); return min; }
	public static decimal Max(params decimal[] nums) { var min = decimal.MinValue; foreach (var n in nums) min = Math.Max(min, n); return min; }
	public static void MultiKeySort(this string[] list) => new MultiSorter(list).QuickSort();
	class MultiSorter
	{
		const int MIN = 0;
		string[] a;
		int max;
		public MultiSorter(string[] l) { a = l; max = a.Max(s => s.Length); }
		public void QuickSort() { if (a.Length <= 1) return; QuickSort(0, a.Length, 0); }
		public int At(int i, int z) => z < a[i].Length ? a[i][z] : MIN;
		public int At(string s, int z) => z < s.Length ? s[z] : MIN;
		public void QuickSort(int l, int r, int z)
		{
			int w = r - l, pl = l, pm = l + w / 2, pn = r - 1, c;
			if (w > 30)
			{
				var d = w / 8;
				pl = Median(pl, pl + d, pl + 2 * d, z);
				pm = Median(pm - d, pm, pm + d, z);
				pn = Median(pn - 2 * d, pn - d, pn, z);
			}
			pm = Median(pl, pm, pn, z);
			var s = a[pm]; a[pm] = a[l]; a[l] = s;
			var pivot = At(l, z);
			int i = l + 1, x = l + 1, j = r - 1, y = r - 1;
			while (true)
			{
				while (i <= j && (c = At(i, z) - pivot) <= 0)
				{
					if (c == 0) { if (i != x) { s = a[i]; a[i] = a[x]; a[x] = s; } x++; }
					i++;
				}
				while (i <= j && (c = At(j, z) - pivot) >= 0)
				{
					if (c == 0) { if (j != y) { s = a[j]; a[j] = a[y]; a[y] = s; } y--; }
					j--;
				}
				if (i > j) break;
				s = a[i]; a[i] = a[j]; a[j] = s;
				i++; j--;
			}
			j++; y++;
			var m = Min(x - l, i - x); SwapRegion(l, i - m, m);
			m = Min(y - j, r - y); SwapRegion(i, r - m, m);
			i += l - x;
			j += r - y;
			if (i - l >= 10) QuickSort(l, i, z); else InsertSort(l, i, z);
			if (pivot != MIN) if (j - i >= 10) QuickSort(i, j, z + 1); else InsertSort(i, j, z + 1);
			if (r - j >= 10) QuickSort(j, r, z); else InsertSort(j, r, z);
		}
		private void SwapRegion(int p, int q, int n)
		{
			string s;
			while (n-- > 0) { s = a[p]; a[p++] = a[q]; a[q++] = s; }
		}
		private void InsertSort(int l, int r, int z)
		{
			string s;
			for (int i = l + 1; i < r; i++)
			{
				var tmp = a[i];
				int x = z, y = z, p, q;
				s = a[i - 1];
				while ((p = At(tmp, x++)) == (q = At(s, y++)) && p != MIN) ;
				if (q > p)
				{
					int j = i;
					while (true)
					{
						a[j] = a[j - 1];
						--j;
						if (j <= l) break;
						x = y = z;
						s = a[j - 1];
						while ((p = At(tmp, x++)) == (q = At(s, y++)) && p != MIN) ;
						if (q <= p) break;
					}
					a[j] = tmp;
				}
			}
		}
		private int Median(int a, int b, int c, int z)
		{
			int p = At(a, z), q = At(b, z);
			if (p == q) return a;
			var r = At(c, z);
			if (r == p || r == q) return c;
			return p < q ?
					(q < r ? b : (p < r ? c : a))
					: (q > r ? b : (p < r ? a : c));
		}
	}
}
class RandomSFMT : Random
{
	int index, coin_bits, byte_pos, range, shift;
	uint coin_save, byte_save, bse;
	protected uint[] x = new uint[40];
	static uint[] ParityData = { 0x00000001U, 0x00000000U, 0x00000000U, 0x20000000U };
	public virtual void gen_rand_all()
	{
		int a = 0, b = 28, c = 32, d = 36; uint y; uint[] p = x;
		do
		{
			y = p[a + 3] ^ (p[a + 3] << 24) ^ (p[a + 2] >> 8) ^ ((p[b + 3] >> 5) & 0xb5ffff7fU);
			p[a + 3] = y ^ (p[c + 3] >> 8) ^ (p[d + 3] << 14);
			y = p[a + 2] ^ (p[a + 2] << 24) ^ (p[a + 1] >> 8) ^ ((p[b + 2] >> 5) & 0xaff3ef3fU);
			p[a + 2] = y ^ ((p[c + 2] >> 8) | (p[c + 3] << 24)) ^ (p[d + 2] << 14);
			y = p[a + 1] ^ (p[a + 1] << 24) ^ (p[a] >> 8) ^ ((p[b + 1] >> 5) & 0x7fefcfffU);
			p[a + 1] = y ^ ((p[c + 1] >> 8) | (p[c + 2] << 24)) ^ (p[d + 1] << 14);
			y = p[a] ^ (p[a] << 24) ^ ((p[b] >> 5) & 0xf7fefffdU);
			p[a] = y ^ ((p[c] >> 8) | (p[c + 1] << 24)) ^ (p[d] << 14);
			c = d; d = a; a += 4; b += 4;
			if (b == 40) b = 0;
		} while (a != 40);
	}
	void period_certification()
	{
		uint work, inner = 0; int i, j;
		index = 40; range = 0; coin_bits = 0; byte_pos = 0;
		for (i = 0; i < 4; i++) inner ^= x[i] & ParityData[i];
		for (i = 16; i > 0; i >>= 1) inner ^= inner >> i;
		inner &= 1;
		if (inner == 1) return;
		for (i = 0; i < 4; i++) for (j = 0, work = 1; j < 32; j++, work <<= 1) if ((work & ParityData[i]) != 0) { x[i] ^= work; return; }
	}
	public void InitMt(uint s)
	{
		unchecked
		{
			x[0] = s;
			for (uint p = 1; p < 40; p++) x[p] = s = 1812433253 * (s ^ (s >> 30)) + p;
			period_certification();
		}
	}
	public RandomSFMT(uint s) { InitMt(s); }
	public void InitMtEx(uint[] init_key)
	{
		uint r, i, j, c, key_len = (uint)init_key.Length;
		unchecked
		{
			for (i = 0; i < 40; i++) x[i] = 0x8b8b8b8b;
			if (key_len + 1 > 40) c = key_len + 1; else c = 40;
			r = x[0] ^ x[17] ^ x[39]; r = (r ^ (r >> 27)) * 1664525;
			x[17] += r; r += key_len; x[22] += r; x[0] = r; c--;
			for (i = 1, j = 0; j < c && j < key_len; j++)
			{
				r = x[i] ^ x[(i + 17) % 40] ^ x[(i + 39) % 40];
				r = (r ^ (r >> 27)) * 1664525; x[(i + 17) % 40] += r;
				r += init_key[j] + i; x[(i + 22) % 40] += r;
				x[i] = r; i = (i + 1) % 40;
			}
			for (; j < c; j++)
			{
				r = x[i] ^ x[(i + 17) % 40] ^ x[(i + 39) % 40];
				r = (r ^ (r >> 27)) * 1664525; x[(i + 17) % 40] += r; r += i;
				x[(i + 22) % 40] += r; x[i] = r; i = (i + 1) % 40;
			}
			for (j = 0; j < 40; j++)
			{
				r = x[i] + x[(i + 17) % 40] + x[(i + 39) % 40];
				r = (r ^ (r >> 27)) * 1566083941; x[(i + 17) % 40] ^= r;
				r -= i; x[(i + 22) % 40] ^= r; x[i] = r; i = (i + 1) % 40;
			}
			period_certification();
		}
	}
	public RandomSFMT(uint[] init_key) { InitMtEx(init_key); }
	public RandomSFMT() : this((uint)(DateTime.Now.Ticks & 0xffffffff)) { }
	public uint NextMt() { if (index == 40) { gen_rand_all(); index = 0; } return x[index++]; }
	public int NextInt(int n) => (int)(n * (1.0 / 4294967296.0) * NextMt());
	public double NextUnif() { uint z = NextMt() >> 11, y = NextMt(); return (y * 2097152.0 + z) * (1.0 / 9007199254740992.0); }
	public int NextBit() { if (--coin_bits == -1) { coin_bits = 31; return (int)(coin_save = NextMt()) & 1; } else return (int)(coin_save >>= 1) & 1; }
	public int NextByte() { if (--byte_pos == -1) { byte_pos = 3; return (int)(byte_save = NextMt()) & 255; } else return (int)(byte_save >>= 8) & 255; }
	public override int Next(int maxValue) => Next(0, maxValue);
	protected override double Sample() => NextUnif();
	public override double NextDouble() => NextUnif();
	public override int Next() => 1 + NextIntEx(int.MaxValue);
	public override void NextBytes(byte[] buffer) { for (var i = 0; i < buffer.Length; i++) buffer[i] = (byte)NextByte(); }
	public override int Next(int min, int max) => min + NextIntEx(max - min);
	public int NextIntEx(int range_)
	{
		uint y_, base_, remain_; int shift_;
		if (range_ <= 0) return 0;
		if (range_ != range)
		{
			bse = (uint)(range = range_);
			for (shift = 0; bse <= (1UL << 30); shift++) bse <<= 1;
		}
		while (true)
		{
			y_ = NextMt() >> 1;
			if (y_ < bse) return (int)(y_ >> shift);
			base_ = bse; shift_ = shift; y_ -= base_;
			remain_ = (1U << 31) - base_;
			for (; remain_ >= (uint)range_; remain_ -= base_)
			{
				for (; base_ > remain_; base_ >>= 1) shift_--;
				if (y_ < base_) return (int)(y_ >> shift_);
				else y_ -= base_;
			}
		}
	}
}