Problem:
Score : 1700 points

Problem Statement

Snuke got a grid from his mother, as a birthday present. The grid has H rows and W columns. Each cell is painted black or white. All black cells are 4-connected, that is, it is possible to traverse from any black cell to any other black cell by just visiting black cells, where it is only allowed to move horizontally or vertically.

The color of the cell at the i-th row and j-th column (1 ≦ i ≦ H, 1 ≦ j ≦ W) is represented by a character s_{ij}. If s_{ij} is #, the cell is painted black. If s_{ij} is ., the cell is painted white. At least one cell is painted black.

We will define fractals as follows.
The fractal of level 0 is a 1 × 1 grid with a black cell.
The fractal of level k+1 is obtained by arranging smaller grids in H rows and W columns, following the pattern of the Snuke's grid.
At a position that corresponds to a black cell in the Snuke's grid, a copy of the fractal of level k is placed.
At a position that corresponds to a white cell in the Snuke's grid, a grid whose cells are all white, with the same dimensions as the fractal of level k, is placed.

You are given the description of the Snuke's grid, and an integer K. Find the number of connected components of black cells in the fractal of level K, modulo 10^9+7.

Constraints

1 ≦ H,W ≦ 1000

0 ≦ K ≦ 10^{18}

Each s_{ij} is either # or ..

All black cells in the given grid are 4-connected.

There is at least one black cell in the given grid.

Input

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

H W K
s_{11} .. s_{1W}
:
s_{H1} .. s_{HW}

Output

Print the number of connected components of black cells in the fractal of level K, modulo 10^9+7.

Sample Input 1

3 3 3
.#.
###
#.#

Sample Output 1

20

The fractal of level K=3 in this case is shown below. There are 20 connected components of black cells.

.............#.............
............###............
............#.#............
..........#..#..#..........
.........#########.........
.........#.##.##.#.........
..........#.....#..........
.........###...###.........
.........#.#...#.#.........
....#........#........#....
...###......###......###...
...#.#......#.#......#.#...
.#..#..#..#..#..#..#..#..#.
###########################
#.##.##.##.##.##.##.##.##.#
.#.....#..#.....#..#.....#.
###...######...######...###
#.#...#.##.#...#.##.#...#.#
....#.................#....
...###...............###...
...#.#...............#.#...
.#..#..#...........#..#..#.
#########.........#########
#.##.##.#.........#.##.##.#
.#.....#...........#.....#.
###...###.........###...###
#.#...#.#.........#.#...#.#

Sample Input 2

3 3 3
###
#.#
###

Sample Output 2

1

Sample Input 3

11 15 1000000000000000000
.....#.........
....###........
....####.......
...######......
...#######.....
..##.###.##....
..##########...
.###.....####..
.####...######.
###############
#.##..##..##..#

Sample Output 3

301811921

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 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;
		RandomSFMT rand = Program.rand;
		int Y, X;
		long K;
		bool[,] map;
		void Read()
		{
			sc.Read(out Y, out X, out K);
			map = sc.Board(X, Y, (c, x, y) => c == '#');
		}
		void SolveOne()
		{
			if (K <= 1) { pr.WriteLine(1); return; }
			var jx = new List<int>();
			var jy = new List<int>();
			for (var x = 0; x < X; x++) if (map[x, 0] && map[x, Y - 1]) jx.Add(x);
			for (var y = 0; y < Y; y++) if (map[0, y] && map[X - 1, y]) jy.Add(y);
			var sum = 0;
			for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) if (map[x, y]) sum++;
			if (jx.Count > 0 && jy.Count > 0) pr.WriteLine(1);
			else if (jx.Count == 0 && jy.Count == 0) pr.WriteLine(Func.Pow(sum, K - 1, Mod));
			else
			{
				if (jy.Count == 0)
				{
					var tmp = new bool[Y, X];
					for (var x = 0; x < X; x++) for (var y = 0; y < Y; y++) tmp[y, x] = map[x, y];
					map = tmp;
					Func.Swap(ref X, ref Y);
				}
				var cont = 0;
				for (var x = 0; x < X - 1; x++) for (var y = 0; y < Y; y++) if (map[x, y] && map[x + 1, y]) cont++;
				var mat = new Matrix(2, 2);
				mat[0, 0] = sum;
				mat[0, 1] = -cont;
				mat[1, 1] = jy.Count;
				mat = mat.Pow((ulong)(K - 1), Mod);
				var ans = (mat[0, 0] + mat[0, 1]) % Mod;
				if (ans < 0) ans += Mod;
				pr.WriteLine(ans);
			}
		}
	}
}
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() { return (IEnumerator)GetEnumerator(); }
	}
	public int Count { get; private set; }
	Node head;
	Comparison<T> comp;
	public bool IsEmpty { get { return 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() { return 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) { return 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() { return head.GetEnumerator(); }
	IEnumerator IEnumerable.GetEnumerator() { return (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) { return Sum(item, item + 1) > 0; }
	public int Count(int item) { return 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) { return 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) { return 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) { return Sum2(0, to) + Sum2(1, to) * to; }
	public long Sum(int from, int to) { return 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) { return 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)
	{
		return 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) { return 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) { return 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) { return 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) { return 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 == null ? Func.DefaultComparison<T>() : comp; }
	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) { return 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) { return 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() { return GetEnumerator(); }
	public void CopyTo(Array array, int index) { foreach (var item in this) array.SetValue(item, index++); }
	public bool IsSynchronized { get { return false; } }
	public object SyncRoot { get { return this; } }
	public bool IsReadOnly { get { return false; } }
	public bool Contains(T item) { return 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) { return 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("{0} items, {1} buckets(size : {2})", Count, NumOfBucket, 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 = {0} <- {1}", x.Value, x.Parent);
			if (y != null) Console.WriteLine("y = {0} <- {1}", 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) { return AddAfter(node, item.Value); }
	public bool AddBefore(BucketNode<T> node, BucketNode<T> item) { return 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) { return 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) { return 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() { return 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() { return new EdgeInfo<E>(To, From, Information); }
}
class DirectedGraph<V, E> : IEnumerable<NodeInfo<V>>
{
	protected int numberOfNodes;
	public int NumberOfNodes { get { return 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) { return edgesFrom[node]; }
	public int InDegree(int node) { return edgesTo[node].Count; }
	public int OutDegree(int node) { return edgesFrom[node].Count; }
	public IEnumerable<HalfEdgeInfo<E>> EdgesTo(int node) { return edgesTo[node]; }
	public V this[int node] { get { return nodes[node].Second; } set { nodes[node].Second = value; } }
	public IEnumerable<EdgeInfo<E>> Edges { get { return 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) { return nodes.FirstOrDefault(e => e.Information.Equals(node)); }
	public EdgeInfo<E> SearchEdge(E edge) { return edges.Find(e => e.Information.Equals(edge)); }
	public IEnumerator<NodeInfo<V>> GetEnumerator() { foreach (var v in nodes) yield return v; }
	IEnumerator IEnumerable.GetEnumerator() { return 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 long[] DijkstraFromL(int from, Func<E, long> cost)
	{
		var d = Enumerable.Repeat(Func.InfL, numberOfNodes).ToArray();
		var queue = new PriorityQueue<Pair<int, long>>((x, y) => x.Second.CompareTo(y.Second));
		d[from] = 0;
		queue.Enqueue(new Pair<int, long>(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, long>(e.End, d[e.End] = tmp));
			}
		}
		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() { return 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) { return size[GetRootOf(x)]; }
	public int GetRootOf(int x) { return 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) { return 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 { get { return height; } }
		public int Bias { get { return 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) { this.height = 1; this.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() { return 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 == null ? Func.DefaultComparison<T>() : comp;
		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) { return 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) { return LowerBound(x => comp(x, item) >= 0); }
	public AVLNode UpperBound(T item) { return 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() { return Min(sentinel.Left).Item; }
	public AVLNode MinNode() { return Min(sentinel.Left); }
	public T Max() { return Max(sentinel.Left).Item; }
	public AVLNode MaxNode() { return Max(sentinel.Left); }
	public bool IsEmpty { get { return sentinel.Left == sentinel; } }
	public void Clear() { sentinel.Left = sentinel; count = 0; sentinel.ResetAsSentinel(); }
	public IEnumerator<T> GetEnumerator() { return sentinel.Left.GetEnumerator(); }
	IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
	public void CopyTo(T[] array, int arrayIndex) { foreach (var x in this) array[arrayIndex++] = x; }
	public int Count { get { return count; } }
	public bool IsReadOnly { get { return true; } }
	public void CopyTo(Array array, int index) { foreach (var x in this) array.SetValue(x, index++); }
	public bool IsSynchronized { get { return false; } }
	public object SyncRoot { get { return 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() { return IsBalanced(sentinel.Left); }
	public bool IsValidBinarySearchTree() { return IsValidBinarySearchTree(sentinel.Left); }
	bool IsBalanced(AVLNode node) { return node == sentinel || (Math.Abs(node.Bias) < 2 && IsBalanced(node.Left) && IsBalanced(node.Right)); }
	bool IsValidBinarySearchTree(AVLNode node)
	{
		return node == sentinel || (Small(node.Item, node.Left) && Large(node.Item, node.Right)
			&& IsValidBinarySearchTree(node.Left) && IsValidBinarySearchTree(node.Right));
	}
	bool Small(T item, AVLNode node) { return node == sentinel || (comp(item, node.Item) >= 0 && Small(item, node.Left) && Small(item, node.Right)); }
	bool Large(T item, AVLNode node) { return 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; }
	public bool IsEmpty { get { return Count == 0; } }
	public PriorityQueue(IEnumerable<T> source) : this((Comparison<T>)null, 0, source) { }
	public PriorityQueue(int capacity = 0, 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 = 0, 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 = 0, IEnumerable<T> source = null) { this.comp = comp == null ? Func.DefaultComparison<T>() : comp; 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() { return 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) { return list.Contains(item); }
	IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
	void ICollection.CopyTo(Array array, int index) { CopyTo(array, index); }
	bool ICollection.IsSynchronized { get { return false; } }
	object ICollection.SyncRoot { get { return 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() { return array[offset]; }
	public T PeekBack() { return 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() { return 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() { return "(" + First + ", " + Second + ")"; }
	public override int GetHashCode() { return 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 (object)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) { return new Point(p.X, p.Y); }
	public static Point operator -(Point p) { return new Point(-p.X, -p.Y); }
	public static Point operator /(Point p, int r) { return new Point(p.X / r, p.Y / r); }
	public static Point operator *(int r, Point p) { return new Point(p.X * r, p.Y * r); }
	public static Point operator *(Point p, int r) { return new Point(p.X * r, p.Y * r); }
	public static Point operator +(Point p, Point q) { return new Point(p.X + q.X, p.Y + q.Y); }
	public static Point operator -(Point p, Point q) { return 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>() { return (T)Convert(file.ReadLine(), Type.GetTypeCode(typeof(T))); }
	public int Int { get { return Get<int>(); } }
	public uint UInt { get { return Get<uint>(); } }
	public long Long { get { return Get<long>(); } }
	public ulong ULong { get { return Get<ulong>(); } }
	public double Double { get { return Get<double>(); } }
	public decimal Decimal { get { return Get<decimal>(); } }
	public char Char { get { return Get<char>(); } }
	public string String { get { return 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) { return Enumerable.Range(0, n).Select(_ => selector()).ToArray(); }
	public T[] ReadManyLines<T>(int n) { return Enumerable.Range(0, n).Select(_ => Get<T>()).ToArray(); }
	public Tuple<S, T>[] ReadManyLines<S, T>(int n) { return Enumerable.Range(0, n).Select(_ => Get<S, T>()).ToArray(); }
	public Tuple<S, T, U>[] ReadManyLines<S, T, U>(int n) { return Enumerable.Range(0, n).Select(_ => Get<S, T, U>()).ToArray(); }
	public Tuple<S, T, U, V>[] ReadManyLines<S, T, U, V>(int n) { return 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) { return 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) { return 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) { return 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) { return 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>() { return (x, y) => Comparer<T>.Default.Compare(x, y); }
	public static Comparison<T> ToComparison<T>(this IComparer<T> comp) { return 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) { return array[index.First, index.Second]; }
	public static bool InRegion(this Pair<int, int> p, int X, int Y) { return p.InRegion(0, X, 0, Y); }
	public static bool InRegion(this Pair<int, int> p, int x, int X, int y, int Y) { return 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) { return Math.Abs((n / GCD(n, m)) * m); }
	public static long Divide(long n, long m) { return (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) { return 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 System.Diagnostics.Stopwatch();
		sw.Restart();
		action();
		sw.Stop();
		return sw.ElapsedTicks * 1000m / System.Diagnostics.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> { return source.MaxElement((x, y) => x.CompareTo(y)); }
	public static int MinElement<T>(IEnumerable<T> source, Comparison<T> comp) { return source.MaxElement((x, y) => comp(y, x)); }
	public static int MinElement<T>(IEnumerable<T> source) where T : IComparable<T> { return 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) { return (char)(rand.Next(0, 'z' - 'a' + 1) + 'a'); }
	public static char NextChar(this RandomSFMT rand) { return (char)(rand.Next(0, 'z' - 'a' + 1) + 'a'); }
	public static string NextString(this Random rand, int length) { return new string(Enumerable.Range(0, length).Select(_ => rand.NextChar()).ToArray()); }
	public static string NextString(this RandomSFMT rand, int length) { return 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) { return flag ? "YES" : "NO"; }
	public static int SetToMin(ref int min, int other) { return min = Math.Min(min, other); }
	public static int SetToMax(ref int max, int other) { return max = Math.Max(max, other); }
	public static long SetToMin(ref long min, long other) { return min = Math.Min(min, other); }
	public static long SetToMax(ref long max, long other) { return 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) { return Math.Abs(n); }
	public static long Abs(this long n) { return Math.Abs(n); }
	public static double Abs(this double n) { return Math.Abs(n); }
	public static float Abs(this float n) { return Math.Abs(n); }
	public static decimal Abs(this decimal n) { return Math.Abs(n); }
	public static short Abs(this short n) { return Math.Abs(n); }
	public static sbyte Abs(this sbyte n) { return 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; }
}
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) { return (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) { return Next(0, maxValue); }
	protected override double Sample() { return NextUnif(); }
	public override double NextDouble() { return NextUnif(); }
	public override int Next() { return 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) { return 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_;
			}
		}
	}
}
namespace GeometryLong
{
	class Vector3D : IEnumerable<long>, IEquatable<Vector3D>, IComparable<Vector3D>, IComparable
	{
		public long X;
		public long Y;
		public long Z;
		public Vector3D() : this(0, 0, 0) { }
		public Vector3D(long x, long y, long z) { X = x; Y = y; Z = z; }
		public Vector3D(long[] components) : this(components[0], components[1], components[2]) { }
		public Vector3D(Func<int, long> components) : this(components(0), components(1), components(2)) { }
		public long this[int n] { get { return n == 0 ? X : n == 1 ? Y : Z; } set { if (n == 0) X = value; else if (n == 1) Y = value; else Z = value; } }
		public double Abs { get { return Math.Sqrt(this * this); } }
		public static Vector3D operator +(Vector3D p) { return new Vector3D(p.X, p.Y, p.Z); }
		public static Vector3D operator -(Vector3D p) { return new Vector3D(-p.X, -p.Y, -p.Z); }
		public static Vector3D operator -(Vector3D p, Vector3D q) { return new Vector3D(p.X - q.X, p.Y - q.Y, p.Z - q.Z); }
		public static Vector3D operator +(Vector3D p, Vector3D q) { return new Vector3D(p.X + q.X, p.Y + q.Y, p.Z + q.Z); }
		public static Vector3D operator *(Vector3D p, long r) { return new Vector3D(p.X * r, p.Y * r, p.Z * r); }
		public static Vector3D operator /(Vector3D p, long r) { return new Vector3D(p.X / r, p.Y / r, p.Z / r); }
		public static Vector3D operator *(long r, Vector3D p) { return p * r; }
		public static long operator *(Vector3D p, Vector3D q) { return p.X * q.X + p.Y * q.Y + p.Z * q.Z; }
		public static bool operator ==(Vector3D p, Vector3D q) { return ReferenceEquals(p, q) || ((object)p != null && (object)q != null && p.X == q.X && p.Y == q.Y && p.Z == q.Z); }
		public static bool operator !=(Vector3D p, Vector3D q) { return !ReferenceEquals(p, q) && ((object)p == null || (object)q == null || p.X != q.X || p.Y != q.Y || p.Z != q.Z); }
		public static bool operator <(Vector3D p, Vector3D q) { return p.X == q.X ? p.Y == q.Y ? p.Z < q.Z : p.Y < q.Y : p.X < q.X; }
		public static bool operator <=(Vector3D p, Vector3D q) { return p.X == q.X ? p.Y == q.Y ? p.Z <= q.Z : p.Y < q.Y : p.X < q.X; }
		public static bool operator >(Vector3D p, Vector3D q) { return p.X == q.X ? p.Y == q.Y ? p.Z > q.Z : p.Y > q.Y : p.X > q.X; }
		public static bool operator >=(Vector3D p, Vector3D q) { return p.X == q.X ? p.Y == q.Y ? p.Z >= q.Z : p.Y > q.Y : p.X > q.X; }
		public static Vector3D operator %(Vector3D p, Vector3D q) { return new Vector3D(p.Y * q.Z - p.Z * q.Y, p.Z * q.X - p.X * q.Z, p.X * q.Y - p.Y * q.X); }
		public bool OnLine(Line3D line) { return ((this - line.P) % (this - line.Q)).Abs <= Geometry.EPS; }
		public bool OnSegment(Line3D line) { return OnLine(line) && (this - line.P) * (line.Q - this) >= Geometry.EPS; }
		public static explicit operator Vector(Vector3D p) { return new Vector(3, i => p[i]); }
		public static explicit operator Vector3D(Vector p) { return new Vector3D(p[0], p[1], p[2]); }
		public IEnumerator<long> GetEnumerator() { yield return X; yield return Y; yield return Z; }
		IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
		public Vector3D ProjectTo(Vector3D direction) { return ((this * direction) / (direction * direction)) * direction; }
		public override bool Equals(object obj)
		{
			if (obj == null) return false;
			var p = obj as Vector3D;
			return (object)p != null && (X == p.X) && (Y == p.Y) && (Z == p.Z);
		}
		public override int GetHashCode() { return X.GetHashCode() ^ Y.GetHashCode() ^ Z.GetHashCode(); }
		public bool Equals(Vector3D other) { return this == other; }
		public int CompareTo(Vector3D other) { return other == null || this > other ? 1 : this < other ? -1 : 0; }
		public int CompareTo(object obj) { return obj == null ? 1 : CompareTo((Vector3D)obj); }
	}
	class Line3D : IEnumerable<Vector3D>
	{
		public Vector3D P;
		public Vector3D Q;
		public Vector3D this[int n] { get { return n == 0 ? P : Q; } set { if (n == 0) P = value; else Q = value; } }
		public Vector3D D { get { return Q - P; } }
		public Line3D(Vector3D p, Vector3D q) { P = p; Q = q; }
		public IEnumerator<Vector3D> GetEnumerator() { yield return P; yield return Q; }
		IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
	}
	class Line2D : IEnumerable<Vector2D>
	{
		public Vector2D P;
		public Vector2D Q;
		public Vector2D this[int n] { get { return n == 0 ? P : Q; } set { if (n == 0) P = value; else Q = value; } }
		public Vector2D D { get { return Q - P; } }
		public Line2D(Vector2D p, Vector2D q) { P = p; Q = q; }
		public IEnumerator<Vector2D> GetEnumerator() { yield return P; yield return Q; }
		IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
	}
	// points assumed to be counter clockwise
	class Polygon : List<Vector2D>
	{
		public Polygon() : base() { }
		public Vector2D Next(int i) { return this[(i + 1) % Count]; }
		public Vector2D Prev(int i) { return this[(i + Count - 1) % Count]; }
		private Vector2D Diff(int i) { return Next(i) - this[i]; }
		public long Area2 { get { var area = 0L; for (var i = 0; i < Count; i++) area += this[i] % Next(i); return area; } }
		public long Area { get { return Area2 / 2; } }
		public enum State { Out = -1, On, In };
		public State Includes(Vector2D p)
		{
			var isIn = false;
			for (var i = 0; i < Count; i++)
			{
				var a = this[i] - p;
				var b = Next(i) - p;
				if (a.Y > b.Y) { var tmp = a; a = b; b = tmp; }
				var cross = a % b;
				if (a.Y <= 0 && 0 < b.Y && cross < 0) isIn = !isIn;
				if (cross == 0 && a * b <= 0) return State.On;
			}
			return isIn ? State.In : State.Out;
		}
		public bool IsConvex() { for (var i = 0; i < Count; i++) if (Geometry.CCW(Prev(i), this[i], Next(i)) > 0) return false; return true; }// only applicable to a convex polygon
		public long Diameter2()
		{
			int i0 = 0, j0 = 0;
			for (int k = 1; k < Count; ++k) { if (this[k].Y > this[i0].Y) i0 = k; if (this[k].Y < this[j0].Y) j0 = k; }
			var max = (this[i0] - this[j0]).Norm2;
			int i, maxi, j, maxj;
			i = maxi = i0;
			j = maxj = j0;
			do
			{
				if (Diff(i) % Diff(j) >= 0) j = (j + 1) % Count;
				else i = (i + 1) % Count;
				var tmp = (this[i] - this[j]).Norm2;
				if (tmp > max) { max = tmp; maxi = i; maxj = j; }
			} while (i != i0 || j != j0);
			return max;
		}
	}
	class Circle2D
	{
		public Vector2D C;
		public long R;
		public Circle2D() : this(new Vector2D(), 0) { }
		public Circle2D(Vector2D c, long r) { C = c; R = r; }
		public double Area { get { return R * R * Math.PI; } }
	}
	class Vector2D : IEnumerable<long>, IEquatable<Vector2D>, IComparable<Vector2D>, IComparable
	{
		public long X;
		public long Y;
		public Vector2D() : this(0, 0) { }
		public Vector2D(long x, long y) { X = x; Y = y; }
		public Vector2D(long[] components) : this(components[0], components[1]) { }
		public Vector2D(Func<int, long> components) : this(components(0), components(1)) { }
		public long this[int n] { get { return n == 0 ? X : Y; } set { if (n == 0) X = value; else Y = value; } }
		public double Abs { get { return Math.Sqrt(this * this); } }
		public double Norm { get { return Math.Sqrt(this * this); } }
		public long Norm2 { get { return this * this; } }
		public static Vector2D operator +(Vector2D p) { return new Vector2D(p.X, p.Y); }
		public static Vector2D operator -(Vector2D p) { return new Vector2D(-p.X, -p.Y); }
		public static Vector2D operator -(Vector2D p, Vector2D q) { return new Vector2D(p.X - q.X, p.Y - q.Y); }
		public static Vector2D operator +(Vector2D p, Vector2D q) { return new Vector2D(p.X + q.X, p.Y + q.Y); }
		public static Vector2D operator *(Vector2D p, long r) { return new Vector2D(p.X * r, p.Y * r); }
		public static Vector2D operator /(Vector2D p, long r) { return new Vector2D(p.X / r, p.Y / r); }
		public static Vector2D operator *(long r, Vector2D p) { return p * r; }
		public static long operator *(Vector2D p, Vector2D q) { return p.X * q.X + p.Y * q.Y; }
		public static explicit operator Vector(Vector2D p) { return new Vector(2, i => p[i]); }
		public static explicit operator Vector2D(Vector p) { return new Vector2D(p[0], p[1]); }
		public static explicit operator Vector3D(Vector2D p) { return new Vector3D(p.X, p.Y, 0); }
		public static explicit operator Vector2D(Vector3D p) { return new Vector2D(p.X, p.Y); }
		public static bool operator ==(Vector2D p, Vector2D q) { return ReferenceEquals(p, q) || ((object)p != null && (object)q != null && p.X == q.X && p.Y == q.Y); }
		public static bool operator !=(Vector2D p, Vector2D q) { return !ReferenceEquals(p, q) && ((object)p == null || (object)q == null || p.X != q.X || p.Y != q.Y); }
		public static bool operator <(Vector2D p, Vector2D q) { return p.X == q.X ? p.Y < q.Y : p.X < q.X; }
		public static bool operator <=(Vector2D p, Vector2D q) { return p.X == q.X ? p.Y <= q.Y : p.X < q.X; }
		public static bool operator >(Vector2D p, Vector2D q) { return p.X == q.X ? p.Y > q.Y : p.X > q.X; }
		public static bool operator >=(Vector2D p, Vector2D q) { return p.X == q.X ? p.Y >= q.Y : p.X > q.X; }
		// cross product ('%' was chosen because it has a good priority as operator)
		public static long operator %(Vector2D p, Vector2D q) { return p.X * q.Y - p.Y * q.X; }
		public bool OnLine(Line2D line) { return (this - line.P) % (this - line.Q) == 0; }
		public bool OnSegment(Line2D line) { return OnLine(line) && (this - line.P) * (line.Q - this) >= 0; }
		public IEnumerator<long> GetEnumerator() { yield return X; yield return Y; }
		IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
		public Vector2D ProjectTo(Vector2D direction) { return ((this * direction) / (direction * direction)) * direction; }
		public override bool Equals(object obj)
		{
			if (obj == null) return false;
			var p = obj as Vector2D;
			return (object)p != null && (X == p.X) && (Y == p.Y);
		}
		public override int GetHashCode() { return X.GetHashCode() ^ Y.GetHashCode(); }
		public bool Equals(Vector2D other) { return this == other; }
		public int CompareTo(Vector2D other) { return other == null || this > other ? 1 : this < other ? -1 : 0; }
		public int CompareTo(object obj) { return obj == null ? 1 : CompareTo((Vector2D)obj); }
	}
	static class Geometry
	{
		public static double EPS = 1e-5;
		public static int CCW(Vector2D a, Vector2D b, Vector2D c)
		{
			var p = b - a;
			var q = c - a;
			var cross = p % q;
			if (cross > 0) return +1;       // counter clockwise
			if (cross < 0) return -1;       // clockwise
			if (p * q < 0) return +2;       // c--a--b on line
			if (b * b < c * c) return -2;   // a--b--c on line (or a=b)
			return 0;                       // a--c--b on line (or b=c)
		}
		public static bool IntersectsLL(Line2D l, Line2D m) { var dl = l.D; return dl % m.D != 0 || dl % (m.P - l.P) == 0; }
		public static bool IntersectsLS(Line2D l, Line2D s) { var dl = l.D; return (dl % (s.P - l.P)) * (dl % (s.Q - l.P)) <= 0; }
		public static bool IntersectsLP(Line2D l, Vector2D p) { return (l.Q - p) % (l.P - p) == 0; }
		public static bool IntersectsSS(Line2D s, Line2D t) { return CCW(s.P, s.Q, t.P) * CCW(s.P, s.Q, t.Q) <= 0 && CCW(t.P, t.Q, s.P) * CCW(t.P, t.Q, s.Q) <= 0; }
		public static bool IntersectsSP(Line2D s, Vector2D p) { return (s.P - p).Abs + (s.Q - p).Abs - s.D.Abs < EPS; }
		public static bool IntersectsPP(Vector2D p, Vector2D q) { return p.X == q.X && p.Y == q.Y; }
		public static Vector2D Project(Vector2D p, Line2D l) { return l.P + p.ProjectTo(l.D); }
		public static Vector2D Reflect(Vector2D p, Line2D l) { return 2 * Project(p, l) - p; }
		public static double DistanceLP(Line2D l, Vector2D p) { return (p - Project(p, l)).Abs; }
		public static double DistanceLL(Line2D l, Line2D m) { return IntersectsLL(l, m) ? 0 : DistanceLP(l, m.P); }
		public static double DistanceLS(Line2D l, Line2D s) { return IntersectsLS(l, s) ? 0 : Math.Min(DistanceLP(l, s.P), DistanceLP(l, s.Q)); }
		public static double DistanceSP(Line2D s, Vector2D p)
		{
			var r = Project(p, s);
			return IntersectsSP(s, r) ? (r - p).Abs : Math.Min((s.P - p).Abs, (s.Q - p).Abs);
		}
		public static double DistanceSS(Line2D s, Line2D t)
		{
			return IntersectsSS(s, t) ? 0 : Func.Min(DistanceSP(s, t.P), DistanceSP(s, t.Q), DistanceSP(t, s.P), DistanceSP(t, s.Q));
		}
		public static Vector2D CrossPoint(Line2D l, Line2D m)
		{
			var dl = l.D;
			var A = dl % m.D;
			var B = dl % (l.Q - m.P);
			if (A == 0 && B == 0) return m.P;
			return m.P + B / A * m.D;
		}
		public static List<Vector2D> ConvexHull(List<Vector2D> points)
		{
			int n = points.Count, k = 0;
			points.Sort((x, y) => x.CompareTo(y));
			var convex = new List<Vector2D>(2 * n);
			for (var i = 0; i < 2 * n; i++) convex.Add(null);
			for (var i = 0; i < n; convex[k++] = points[i++]) while (k >= 2 && CCW(convex[k - 2], convex[k - 1], points[i]) <= 0) --k;
			for (int i = n - 2, t = k + 1; i >= 0; convex[k++] = points[i--]) while (k >= t && CCW(convex[k - 2], convex[k - 1], points[i]) <= 0) --k;
			convex.RemoveRange(k - 1, convex.Count - k + 1);
			return convex;
		}
	}
	class Vector : IEnumerable<long>
	{
		public int Dimension { get; protected set; }
		protected long[] a;
		public IEnumerable<long> Components { get { return a; } }
		public long this[int n] { get { return a[n]; } set { a[n] = value; } }
		public Vector(int dim) { Dimension = dim; a = new long[Dimension]; }
		public Vector(long[] components) : this(components.Length, i => components[i]) { }
		public Vector(int dim, Func<int, long> components) : this(dim) { for (var i = 0; i < dim; i++) this.a[i] = components(i); }
		public static Vector operator +(Vector p) { return new Vector(p.a); }
		public static Vector operator -(Vector p) { return new Vector(p.Dimension, i => -p.a[i]); }
		public static Vector operator -(Vector p, Vector q) { return new Vector(p.Dimension, i => p.a[i] - q.a[i]); }
		public static Vector operator +(Vector p, Vector q) { return new Vector(p.Dimension, i => p.a[i] + q.a[i]); }
		public static Vector operator *(Vector p, long r) { return new Vector(p.Dimension, i => p.a[i] * r); }
		public static Vector operator /(Vector p, long r) { return new Vector(p.Dimension, i => p.a[i] / r); }
		public static Vector operator %(Vector p, long mod) { return new Vector(p.Dimension, i => p.a[i] % mod); }
		public static Vector operator *(long r, Vector p) { return new Vector(p.Dimension, i => p.a[i] * r); }
		public static long operator *(Vector p, Vector q) { var sum = 0L; for (var i = 0; i < p.Dimension; i++) sum += p.a[i] * q.a[i]; return sum; }
		public override string ToString()
		{
			var sb = new StringBuilder();
			sb.Append("(");
			for (var i = 0; i < Dimension; i++) sb.Append(a[i] + (i == Dimension - 1 ? ")" : ", "));
			return sb.ToString();
		}
		public Vector ProjectTo(Vector direction) { return ((this * direction) / (direction * direction)) * direction; }
		public IEnumerator<long> GetEnumerator() { return (IEnumerator<long>)a.GetEnumerator(); }
		IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
		public Vector SolveLinearEquation(Matrix A, long mod)
		{
			var inv = new Vector(Dimension, i => { var x = a[i] % mod; return x >= 0 ? x : x + mod; });
			var M = new Matrix(Dimension, Dimension, (r, c) => { var x = A[r, c] % mod; return x >= 0 ? x : x + mod; });
			for (var j = 0; j < Dimension; j++)
			{
				var pos = j;
				while (pos < Dimension && M[pos, j] == 0) pos++;
				if (pos == Dimension) return null;
				M.SwitchRow(j, pos, j);
				inv.a.Swap(j, pos);
				var inv2 = Func.Inverse(M[j, j], mod);
				inv.a[j] *= inv2;
				M.MultiplicateRow(j, inv2, j + 1);
				M.ModRow(j, mod, j + 1);
				M[j, j] = 1;
				for (var i = j + 1; i < Dimension; i++) { inv.a[i] -= (inv.a[j] * M[i, j]) % mod; inv.a[i] %= mod; if (inv.a[i] < 0) inv.a[i] += mod; M.AddRow(j, i, -M[i, j], j + 1); M.ModRow(i, mod, j + 1); }
			}
			for (var j = Dimension - 1; j > 0; j--) for (var i = j - 1; i >= 0; i--) { inv.a[i] -= (inv.a[j] * M[i, j]) % mod; inv.a[i] %= mod; if (inv.a[i] < 0) inv.a[i] += mod; }
			return inv;
		}
	}
	class Matrix
	{
		public int Row { get; protected set; }
		public int Column { get; protected set; }
		private long[,] a;
		public long this[int r, int c] { get { return a[r, c]; } set { a[r, c] = value; } }
		public Matrix(int r, int c) { Row = r; Column = c; a = new long[r, c]; }
		public Matrix(long[,] components) : this(components.GetLength(0), components.GetLength(1), (i, j) => components[i, j]) { }
		public Matrix(int r, int c, Func<int, int, long> components) : this(r, c) { for (var i = 0; i < r; i++) for (var j = 0; j < c; j++) this.a[i, j] = components(i, j); }
		public long Trace { get { var sum = 0L; for (var i = 0; i < Row; i++) sum += a[i, i]; return sum; } }
		public static Matrix operator +(Matrix p) { return new Matrix(p.a); }
		public static Matrix operator -(Matrix p) { return new Matrix(p.Row, p.Column, (i, j) => -p.a[i, j]); }
		public static Matrix operator -(Matrix p, Matrix q) { return new Matrix(p.Row, p.Column, (i, j) => p.a[i, j] - q.a[i, j]); }
		public static Matrix operator +(Matrix p, Matrix q) { return new Matrix(p.Row, p.Column, (i, j) => p.a[i, j] + q.a[i, j]); }
		public static Matrix operator *(Matrix p, long r) { return new Matrix(p.Row, p.Column, (i, j) => p.a[i, j] * r); }
		public static Matrix operator /(Matrix p, long r) { return new Matrix(p.Row, p.Column, (i, j) => p.a[i, j] / r); }
		public static Matrix operator %(Matrix p, long mod) { return new Matrix(p.Row, p.Column, (i, j) => p.a[i, j] % mod); }
		public static Matrix operator *(long r, Matrix p) { return new Matrix(p.Row, p.Column, (i, j) => p.a[i, j] * r); }
		public static Matrix operator *(Matrix p, Matrix q)
		{
			return new Matrix(p.Row, q.Column, (i, j) => { var sum = 0L; for (var k = 0; k < p.Column; k++) sum = (sum + p.a[i, k] * q.a[k, j]); return sum; });
		}
		public static Vector operator *(Matrix p, Vector v)
		{
			return new Vector(p.Row, i => { var sum = 0L; for (var j = 0; j < p.Column; j++) sum = (sum + p.a[i, j] * v[j]); return sum; });
		}
		public static Vector operator *(Vector v, Matrix p)
		{
			return new Vector(p.Column, j => { var sum = 0L; for (var i = 0; i < p.Row; i++) sum = (sum + p.a[i, j] * v[i]); return sum; });
		}
		public Matrix Transpose() { return new Matrix(Column, Row, (i, j) => a[j, i]); }
		public override string ToString()
		{
			var sb = new StringBuilder();
			for (var i = 0; i < Row; i++) for (var j = 0; j < Column; j++) sb.Append(a[i, j] + (j == Column - 1 ? "\n" : " "));
			sb.Remove(sb.Length - 1, 1);
			return sb.ToString();
		}
		public static Matrix One(int size) { return new Matrix(size, size, (i, j) => i == j ? 1 : 0); }
		public static Matrix Zero(int size) { return new Matrix(size, size); }
		public static Matrix Diagonal(long[] diag) { return new Matrix(diag.Length, diag.Length, (i, j) => i == j ? diag[i] : 0); }
		public Matrix Pow(ulong n, long mod)
		{
			var p = One(Row);
			var x = this % mod;
			while (n > 0)
			{
				if ((n & 1) == 1) p = (p * x) % mod;
				n >>= 1;
				x = (x * x) % mod;
			}
			return p % mod;
		}
		public Matrix Pow(ulong n)
		{
			var p = One(Row);
			var x = +this;
			while (n > 0)
			{
				if ((n & 1) == 1) p *= x;
				n >>= 1;
				x *= x;
			}
			return p;
		}
		public static Matrix Pow(long[] coeff, long N, long mod)
		{
			var dim = coeff.Length;
			var vec = new Vector(dim, i => i == 0 ? 1 : 0);
			var mat = new Matrix(dim, dim, (i, j) => j == i + 1 ? 1 : i == dim - 1 ? coeff[j] % mod : 0);
			for (var m = N; m > 0; m >>= 1)
			{
				if ((m & 1) == 1) vec = vec * mat;
				vec %= mod;
				var a = new Vector(dim, i => mat.a[0, i]);
				a = a * mat;
				a %= mod;
				var tmp = new Vector[dim];
				tmp[0] = a;
				for (var k = 1; k < dim; k++)
				{
					var w = tmp[k - 1][dim - 1];
					tmp[k] = new Vector(dim, i => i == 0 ? coeff[0] * w : tmp[k - 1][i - 1] + w * coeff[i]);
					tmp[k] %= mod;
				}
				mat = new Matrix(dim, dim, (i, j) => tmp[i][j] % mod);
			}
			var hoge = new Vector[dim];
			hoge[0] = vec % mod;
			for (var k = 1; k < dim; k++)
			{
				var w = hoge[k - 1][dim - 1];
				hoge[k] = new Vector(dim, i => i == 0 ? coeff[0] * w : hoge[k - 1][i - 1] + w * coeff[i]);
				hoge[k] %= mod;
			}
			return new Matrix(dim, dim, (i, j) => hoge[i][j] % mod);
		}
		public static Matrix Pow(long[] coeff, long N)
		{
			var dim = coeff.Length;
			var vec = new Vector(dim, i => i == 0 ? 1 : 0);
			var mat = new Matrix(dim, dim, (i, j) => j == i + 1 ? 1 : i == dim - 1 ? coeff[j] : 0);
			for (var m = N; m > 0; m >>= 1)
			{
				if ((m & 1) == 1) vec = vec * mat;
				var a = new Vector(dim, i => mat.a[0, i]);
				a = a * mat;
				var tmp = new Vector[dim];
				tmp[0] = a;
				for (var k = 1; k < dim; k++)
				{
					var w = tmp[k - 1][dim - 1];
					tmp[k] = new Vector(dim, i => i == 0 ? coeff[0] * w : tmp[k - 1][i - 1] + w * coeff[i]);
				}
				mat = new Matrix(dim, dim, (i, j) => tmp[i][j]);
			}
			var hoge = new Vector[dim];
			hoge[0] = vec;
			for (var k = 1; k < dim; k++)
			{
				var w = hoge[k - 1][dim - 1];
				hoge[k] = new Vector(dim, i => i == 0 ? coeff[0] * w : hoge[k - 1][i - 1] + w * coeff[i]);
			}
			return new Matrix(dim, dim, (i, j) => hoge[i][j]);
		}
		public long Determinant(long mod)
		{
			if (Row == 0) return 1;
			if (Row == 1) { var tmp = a[0, 0] % mod; return tmp >= 0 ? tmp : tmp + mod; }
			if (Row == 2) { var tmp = ((a[0, 0] % mod) * (a[1, 1] % mod) - (a[0, 1] % mod) * (a[1, 0] % mod)) % mod; return tmp >= 0 ? tmp : tmp + mod; }
			var mat = new Matrix(Row, Column, (r, c) => { var tmp = a[r, c] % mod; return tmp >= 0 ? tmp : tmp + mod; });
			var prod = 1L;
			for (var j = 0; j < Row; j++)
			{
				var pos = j;
				while (pos < Row && mat.a[pos, j] == 0) pos++;
				if (pos == Row) return 0;
				if (j != pos) { mat.SwitchRow(j, pos, j); prod = mod - prod; }
				prod = (prod * mat.a[j, j]) % mod;
				var inv = Func.Inverse(mat.a[j, j], mod);
				mat.MultiplicateRow(j, inv, j);
				mat.ModRow(j, mod, j);
				for (var i = j + 1; i < Row; i++) { mat.AddRow(j, i, -mat.a[i, j], j); mat.ModRow(i, mod, j); }
			}
			return prod;
		}
		public Matrix Invert(long mod)
		{
			var inv = One(Row);
			var tmp = new Matrix(Row, Column, (r, c) => { var x = a[r, c] % mod; return x >= 0 ? x : x + mod; });
			for (var j = 0; j < Row; j++)
			{
				var pos = j;
				while (pos < Row && tmp.a[pos, j] == 0) pos++;
				if (pos == Row) return null;
				tmp.SwitchRow(j, pos);
				inv.SwitchRow(j, pos);
				var inv2 = Func.Inverse(tmp.a[j, j], mod);
				inv.MultiplicateRow(j, inv2);
				tmp.MultiplicateRow(j, inv2, j);
				inv.ModRow(j, mod);
				tmp.ModRow(j, mod, j);
				for (var i = j + 1; i < Row; i++) { inv.AddRow(j, i, -tmp.a[i, j]); inv.ModRow(i, mod); tmp.AddRow(j, i, -tmp.a[i, j], j); inv.ModRow(i, mod, j); }
			}
			for (int j = Row - 1; j > 0; j--) for (int i = j - 1; i >= 0; i--)
				{
					inv.AddRow(j, i, -tmp.a[i, j]); inv.ModRow(i, mod);
					tmp.AddRow(j, i, -tmp.a[i, j]); tmp.ModRow(i, mod);
				}
			return inv;
		}
		public void ModRow(int i, long mod, int start = 0) { for (var y = start; y < Column; y++) { a[i, y] %= mod; if (a[i, y] < 0) a[i, y] += mod; } }
		public void SwitchRow(int i, int j, int start = 0) { for (var y = start; y < Column; y++) { var tmp = a[i, y]; a[i, y] = a[j, y]; a[j, y] = tmp; } }
		public void MultiplicateRow(int i, long r, int start = 0) { for (var y = start; y < Column; y++) a[i, y] *= r; }
		public void AddRow(int from, int to, long r, int start = 0) { for (var y = start; y < Column; y++) a[to, y] += a[from, y] * r; }
	}
	class Fraction : IComparable, IComparable<Fraction>, IEquatable<Fraction>
	{
		public long Num { get; set; }
		public long Denom { get; set; }
		public bool IsInteger { get { return Denom == 1; } }
		public Fraction Inverse { get { return Num > 0 ? Init(Denom, Num) : Init(-Denom, -Num); } }
		public Fraction Abs { get { return Num > 0 ? this : -this; } }
		public Fraction() : this(0) { }
		public Fraction(long n) { Num = n; Denom = 1; }
		public Fraction(long numerator, long denominator) { Num = numerator; Denom = denominator; Reduce(); }
		private static Fraction Init(long num, long denom) { var r = new Fraction(); r.Num = num; r.Denom = denom; return r; }
		protected void Reduce() { var tmp = Func.GCD(Num, Denom) * (Denom < 0 ? -1 : 1); Num /= tmp; Denom /= tmp; }
		public static Fraction operator +(Fraction a, Fraction b) { var gcd = Func.GCD(a.Denom, b.Denom); return new Fraction(a.Num * (b.Denom / gcd) + (a.Denom / gcd) * b.Num, (a.Denom / gcd) * b.Denom); }
		public static Fraction operator -(Fraction a, Fraction b) { var gcd = Func.GCD(a.Denom, b.Denom); return new Fraction(a.Num * (b.Denom / gcd) - (a.Denom / gcd) * b.Num, (a.Denom / gcd) * b.Denom); }
		public static Fraction operator *(Fraction a, Fraction b)
		{
			var g = Func.GCD(a.Num, b.Denom);
			var h = Func.GCD(a.Denom, b.Num);
			return Init((a.Num / g) * (b.Num / h), (a.Denom / h) * (b.Denom / g));
		}
		public static Fraction operator /(Fraction a, Fraction b) { return a * b.Inverse; }
		public static Fraction operator +(Fraction a) { return Init(a.Num, a.Denom); }
		public static Fraction operator -(Fraction a) { return Init(-a.Num, a.Denom); }
		public static bool operator ==(Fraction a, Fraction b)
		{
			return ReferenceEquals(a, b) || ((object)a != null && (object)b != null && a.Num == b.Num && a.Denom == b.Denom);
		}
		public static bool operator !=(Fraction a, Fraction b)
		{
			return !ReferenceEquals(a, b) && ((object)a == null || (object)b == null || a.Num != b.Num || a.Denom != b.Denom);
		}
		public static bool operator >(Fraction a, Fraction b) { if (b.Num == 0) return a.Num > 0; var t = a / b; return b.Num > 0 ? t.Num > t.Denom : t.Num < t.Denom; }
		public static bool operator >=(Fraction a, Fraction b) { if (b.Num == 0) return a.Num >= 0; var t = a / b; return b.Num > 0 ? t.Num >= t.Denom : t.Num <= t.Denom; }
		public static bool operator <(Fraction a, Fraction b) { if (b.Num == 0) return a.Num < 0; var t = a / b; return b.Num > 0 ? t.Num < t.Denom : t.Num > t.Denom; }
		public static bool operator <=(Fraction a, Fraction b) { if (b.Num == 0) return a.Num <= 0; var t = a / b; return b.Num > 0 ? t.Num <= t.Denom : t.Num >= t.Denom; }
		public static explicit operator int(Fraction a) { return (int)(a.Num / a.Denom); }
		public static explicit operator long(Fraction a) { return (long)(a.Num / a.Denom); }
		public static explicit operator double(Fraction a) { return (double)a.Num / a.Denom; }
		public static explicit operator decimal(Fraction a) { return (decimal)a.Num / a.Denom; }
		public static implicit operator Fraction(int a) { return new Fraction(a); }
		public static implicit operator Fraction(long a) { return new Fraction(a); }
		public override string ToString() { return IsInteger ? Num.ToString() : Num + "/" + Denom; }
		public override int GetHashCode() { return Num.GetHashCode() ^ Denom.GetHashCode(); }
		public override bool Equals(object x) { if (x == null) return false; var p = x as Fraction; return (object)p != null && Num * p.Denom == p.Num * Denom; }
		public int CompareTo(object x) { return x == null ? 1 : CompareTo((Fraction)x); }
		public int CompareTo(Fraction r) { if (r.Num == 0) return Num.CompareTo(0); var t = this / r; return (r.Num > 0 ? -1 : 1) * t.Num.CompareTo(t.Denom); }
		public bool Equals(Fraction r) { return this == r; }
		public Fraction Pow(ulong n)
		{
			var np = 1L;
			var dp = 1L;
			var nx = Num;
			var dx = Denom;
			while (n > 0)
			{
				if ((n & 1) == 1) { np *= nx; dp *= dx; }
				n >>= 1;
				nx *= nx;
				dx *= dx;
			}
			return new Fraction(np, dp);
		}
		// when parse failed, return null
		public static Fraction Parse(string s)
		{
			long tmp;
			if (long.TryParse(s, out tmp)) return tmp;
			var result = new Fraction();
			if (s.Length < 3) return null;
			else
			{
				var flag = 1;
				if (s[0] == '-' || s[0] == '+')
				{
					flag = s[0] == '+' ? +1 : -1;
					s = s.Substring(1);
				}
				var split = 0;
				while (split < s.Length && s[split] != '/' && s[split] != '÷' && s[split] != '／') split++;
				if (split == s.Length || !char.IsDigit(s[split + 1])) return null;
				string num_s = s.Substring(0, split), denom_s = s.Substring(split + 1, s.Length - split - 1);
				long num, denom;
				if (!long.TryParse(num_s, out num) || !long.TryParse(denom_s, out denom) || denom == 0) return null;
				result.Num = num * flag;
				result.Denom = denom;
				result.Reduce();
				return result;
			}
		}
	}
	class BigFraction : IComparable, IComparable<BigFraction>, IEquatable<BigFraction>
	{
		public BigInteger Num { get; set; }
		public BigInteger Denom { get; set; }
		public bool IsInteger { get { return Denom == 1; } }
		public BigFraction Inverse { get { return Num > 0 ? Init(Denom, Num) : Init(-Denom, -Num); } }
		public BigFraction Abs { get { return Num > 0 ? this : -this; } }
		public BigFraction() : this(0) { }
		public BigFraction(BigInteger n) { Num = n; Denom = 1; }
		public BigFraction(BigInteger numerator, BigInteger denominator) { Num = numerator; Denom = denominator; Reduce(); }
		private static BigFraction Init(BigInteger num, BigInteger denom) { var r = new BigFraction(); r.Num = num; r.Denom = denom; return r; }
		protected void Reduce() { var tmp = Func.GCD(Num, Denom) * (Denom < 0 ? -1 : 1); Num /= tmp; Denom /= tmp; }
		public static BigFraction operator +(BigFraction a, BigFraction b) { var gcd = Func.GCD(a.Denom, b.Denom); return new BigFraction(a.Num * (b.Denom / gcd) + (a.Denom / gcd) * b.Num, (a.Denom / gcd) * b.Denom); }
		public static BigFraction operator -(BigFraction a, BigFraction b) { var gcd = Func.GCD(a.Denom, b.Denom); return new BigFraction(a.Num * (b.Denom / gcd) - (a.Denom / gcd) * b.Num, (a.Denom / gcd) * b.Denom); }
		public static BigFraction operator *(BigFraction a, BigFraction b)
		{
			var g = Func.GCD(a.Num, b.Denom);
			var h = Func.GCD(a.Denom, b.Num);
			return Init((a.Num / g) * (b.Num / h), (a.Denom / h) * (b.Denom / g));
		}
		public static BigFraction operator /(BigFraction a, BigFraction b) { return a * b.Inverse; }
		public static BigFraction operator +(BigFraction a) { return Init(a.Num, a.Denom); }
		public static BigFraction operator -(BigFraction a) { return Init(-a.Num, a.Denom); }
		public static bool operator ==(BigFraction a, BigFraction b)
		{
			return ReferenceEquals(a, b) || ((object)a != null && (object)b != null && a.Num == b.Num && a.Denom == b.Denom);
		}
		public static bool operator !=(BigFraction a, BigFraction b)
		{
			return !ReferenceEquals(a, b) && ((object)a == null || (object)b == null || a.Num != b.Num || a.Denom != b.Denom);
		}
		public static bool operator >(BigFraction a, BigFraction b) { if (b.Num == 0) return a.Num > 0; var t = a / b; return b.Num > 0 ? t.Num > t.Denom : t.Num < t.Denom; }
		public static bool operator >=(BigFraction a, BigFraction b) { if (b.Num == 0) return a.Num >= 0; var t = a / b; return b.Num > 0 ? t.Num >= t.Denom : t.Num <= t.Denom; }
		public static bool operator <(BigFraction a, BigFraction b) { if (b.Num == 0) return a.Num < 0; var t = a / b; return b.Num > 0 ? t.Num < t.Denom : t.Num > t.Denom; }
		public static bool operator <=(BigFraction a, BigFraction b) { if (b.Num == 0) return a.Num <= 0; var t = a / b; return b.Num > 0 ? t.Num <= t.Denom : t.Num >= t.Denom; }
		public static explicit operator int(BigFraction a) { return (int)(a.Num / a.Denom); }
		public static explicit operator long(BigFraction a) { return (long)(a.Num / a.Denom); }
		public static explicit operator BigInteger(BigFraction a) { return a.Num / a.Denom; }
		public static explicit operator double(BigFraction a) { return (double)a.Num / (double)a.Denom; }
		public static explicit operator decimal(BigFraction a) { return (decimal)a.Num / (decimal)a.Denom; }
		public static implicit operator BigFraction(int a) { return new BigFraction(a); }
		public static implicit operator BigFraction(long a) { return new BigFraction(a); }
		public static implicit operator BigFraction(BigInteger a) { return new BigFraction(a); }
		public override string ToString() { return IsInteger ? Num.ToString() : Num + "/" + Denom; }
		public override int GetHashCode() { return Num.GetHashCode() ^ Denom.GetHashCode(); }
		public override bool Equals(object x) { if (x == null) return false; var p = x as BigFraction; return (object)p != null && Num * p.Denom == p.Num * Denom; }
		public int CompareTo(object x) { return x == null ? 1 : CompareTo((BigFraction)x); }
		public int CompareTo(BigFraction r) { if (r.Num == 0) return Num.CompareTo(0); var t = this / r; return (r.Num > 0 ? -1 : 1) * t.Num.CompareTo(t.Denom); }
		public bool Equals(BigFraction r) { return this == r; }
		public BigFraction Pow(ulong n)
		{
			var np = BigInteger.One;
			var dp = BigInteger.One;
			var nx = Num;
			var dx = Denom;
			while (n > 0)
			{
				if ((n & 1) == 1) { np *= nx; dp *= dx; }
				n >>= 1;
				nx *= nx;
				dx *= dx;
			}
			return new BigFraction(np, dp);
		}
		// when parse failed, return null
		public static BigFraction Parse(string s)
		{
			if (!s.Contains("/") && !s.Contains("／") && !s.Contains("÷") && !s.Contains("."))
			{
				BigInteger tmp;
				if (BigInteger.TryParse(s, out tmp)) return tmp;
				return null;
			}
			if (s.Length < 3) return null;
			else if (!s.Contains("."))
			{
				var flag = 1;
				if (s[0] == '-' || s[0] == '+')
				{
					flag = s[0] == '+' ? +1 : -1;
					s = s.Substring(1);
				}
				var split = 0;
				while (split < s.Length && s[split] != '/' && s[split] != '÷' && s[split] != '／') split++;
				if (split >= s.Length || (split + 1 < s.Length && !char.IsDigit(s[split + 1]))) return null;
				string num_s = s.Substring(0, split), denom_s = s.Substring(split + 1, s.Length - split - 1);
				BigInteger num, denom;
				if (!BigInteger.TryParse(num_s, out num) || !BigInteger.TryParse(denom_s, out denom) || denom == 0) return null;
				var result = new BigFraction();
				result.Num = num * flag;
				result.Denom = denom;
				result.Reduce();
				return result;
			}
			else
			{
				var flag = 1;
				if (s[0] == '-' || s[0] == '+')
				{
					flag = s[0] == '+' ? +1 : -1;
					s = s.Substring(1);
				}
				var split = 0;
				while (split < s.Length && s[split] != '.' && s[split] != '．') split++;
				if (split + 1 >= s.Length || !char.IsDigit(s[split + 1])) return null;
				var exp10 = s.Length - split - 1;
				s = s.Remove(split, 1);
				BigInteger num;
				if (!BigInteger.TryParse(s, out num)) return null;
				var result = new BigFraction();
				result.Num = num * flag;
				result.Denom = BigInteger.Pow(10, exp10);
				result.Reduce();
				return result;
			}
		}
	}
	struct ModInteger : IEquatable<ModInteger>, IFormattable
	{
		const long Mod = 1000000007;
		public long x;
		public ModInteger(long x) { this.x = x % Mod; }
		public static ModInteger operator +(ModInteger a) { return new ModInteger(a.x); }
		public static ModInteger operator -(ModInteger a) { return new ModInteger(-a.x); }
		public static ModInteger operator +(ModInteger a, ModInteger b) { return (a.x + b.x) % Mod; }
		public static ModInteger operator -(ModInteger a, ModInteger b) { return (a.x - b.x) % Mod; }
		public static ModInteger operator *(ModInteger a, ModInteger b) { return (a.x * b.x) % Mod; }
		public static ModInteger operator /(ModInteger a, ModInteger b) { return a * b.Inverse; }
		public static bool operator ==(ModInteger a, ModInteger b) { return (a.x - b.x) % Mod == 0; }
		public static bool operator !=(ModInteger a, ModInteger b) { return (a.x - b.x) % Mod != 0; }
		public static explicit operator long(ModInteger a) { return a.x < 0 ? a.x + Mod : a.x; }
		public static implicit operator ModInteger(long a) { return new ModInteger(a); }
		public override bool Equals(object x) { if (x == null) return false; var p = (ModInteger)x; return p == this; }
		public override int GetHashCode() { return x.GetHashCode(); }
		public override string ToString() { return x.ToString(); }
		public bool Equals(ModInteger other) { return this == other; }
		public string ToString(string format, IFormatProvider formatProvider) { return ((long)this).ToString(format, formatProvider); }
		public ModInteger Pow(long exponent) { return Func.Pow(x, exponent, Mod); }
		public ModInteger Inverse { get { return new ModInteger(Func.Inverse(x, Mod)); } }
	}
	class MyInteger : IEquatable<MyInteger>, IComparable<MyInteger>, IComparable
	{
		public const int Base = 1000000000;    // 2*Base < int.MaxValue
											   //public const int Base = 200;
		private const int MaxDigit = Base - 1;
		public const int LogBase = 9;
		// 0 -> isPositive = true, digits = { }
		// digits[digits.Length - 1] != 0
		// 0 <= digits[i] < Base
		// Number = (isPositive ? 1 : -1) * (digits[0] + Base * (digits[1] + Base * (digits[2] + ...)))
		public bool isPositive;
		public int[] digits;
		public MyInteger() { digits = new int[0]; isPositive = true; }
		public MyInteger(long x) { var y = (MyInteger)x; isPositive = y.isPositive; digits = y.digits; }
		public MyInteger(ulong x) { var y = (MyInteger)x; isPositive = y.isPositive; digits = y.digits; }
		public MyInteger(bool isPositive, int[] digits) { this.isPositive = isPositive; this.digits = new int[digits.Length]; digits.CopyTo(this.digits, 0); }
		public bool IsEven { get { return digits.Length == 0 || digits[0] % 2 == 0; } }
		public bool IsZero { get { return digits.Length == 0; } }
		public bool IsOne { get { return digits.Length == 1 && isPositive && digits[0] == 1; } }
		public int Sign { get { return isPositive ? digits.Length == 0 ? 0 : 1 : -1; } }
		public MyInteger Abs { get { return new MyInteger(true, digits); } }
		public void Negate() { if (digits.Length > 0) isPositive = !isPositive; }
		public static MyInteger Max(MyInteger a, MyInteger b) { return a >= b ? a : b; }
		public static MyInteger Min(MyInteger a, MyInteger b) { return a <= b ? a : b; }
		public static readonly MyInteger MinusOne = new MyInteger(false, new int[] { 1 });
		public static readonly MyInteger Zero = new MyInteger();
		public static readonly MyInteger One = new MyInteger(true, new int[] { 1 });
		public static readonly MyInteger Two = new MyInteger(true, new int[] { 2 });
		public bool Check
		{
			get
			{
				if (IsZero) return isPositive;
				if (digits[digits.Length - 1] == 0) return false;
				for (var i = 0; i < digits.Length; i++) if (digits[i] < 0 || digits[i] >= Base) return false;
				return true;
			}
		}
		public void Increment()
		{
			var N = digits.Length;
			if (isPositive)
			{
				if (N == 0) digits = new int[] { 1 };
				else
				{
					var i = 0;
					while (i < N && digits[i] == MaxDigit) i++;
					if (i == N) { digits = new int[N + 1]; digits[N] = 1; }
					else
					{
						digits[i--]++;
						for (; i >= 0; i--) digits[i] = 0;
					}
				}
			}
			else
			{
				if (N == 1 && digits[0] == 1) { isPositive = true; digits = new int[0]; }
				else
				{
					var i = 0;
					while (digits[i] == 0) i++;
					if (i == N - 1 && digits[i] == 1)
					{
						digits = new int[N - 1];
						for (i = 0; i < N - 1; i++) digits[i] = MaxDigit;
					}
					else
					{
						digits[i--]--;
						for (; i >= 0; i--) digits[i] = MaxDigit;
					}
				}
			}
		}
		public void Decrement() { Negate(); Increment(); Negate(); }
		public void Add(MyInteger a)
		{
			var N = digits.Length;
			var M = a.digits.Length;
			if (M == 0) return;
			else if (N == 0) { isPositive = a.isPositive; digits = new int[M]; a.digits.CopyTo(digits, 0); }
			else if (isPositive == a.isPositive)
			{
				if (N < M) digits = Add(a.digits, digits).ToArray();
				else
				{
					var k = 0;
					for (var i = 0; i < M; i++)
					{
						digits[i] += k + a.digits[i];
						if (digits[i] >= Base) { digits[i] -= Base; k = 1; }
						else k = 0;
					}
					for (var i = M; i < N; i++)
					{
						digits[i] += k;
						if (digits[i] >= Base) { digits[i] -= Base; k = 1; }
						else k = 0;
					}
					if (k > 0) { var tmp = new int[N + 1]; for (var i = 0; i < N; i++) tmp[i] = digits[i]; tmp[N] = 1; digits = tmp; }
				}
			}
			else
			{
				if (N < M) { var c = this + a; isPositive = c.isPositive; digits = c.digits; }
				else if (N > M)
				{
					var k = 0;
					var z = -1;
					for (var i = 0; i < M; i++)
					{
						digits[i] -= (a.digits[i] + k);
						if (digits[i] < 0) { digits[i] += Base; k = 1; }
						else k = 0;
						if (digits[i] != 0) z = i;
					}
					for (var i = M; i < N; i++)
					{
						digits[i] -= k;
						if (digits[i] < 0) { digits[i] += Base; k = 1; }
						else k = 0;
						if (digits[i] != 0) z = i;
					}
					if (digits[N - 1] == 0)
					{
						var tmp = new int[z + 1];
						for (var i = 0; i <= z; i++) tmp[i] = digits[i];
						digits = tmp;
					}
				}
				else
				{
					var j = N - 1;
					while (j >= 0 && digits[j] == a.digits[j]) j--;
					if (j < 0) { isPositive = true; digits = new int[0]; }
					else
					{
						var flag = digits[j] >= a.digits[j];
						var p = flag ? digits : a.digits;
						var q = !flag ? digits : a.digits;
						var k = 0;
						var z = -1;
						for (var i = 0; i <= j; i++)
						{
							digits[i] = p[i] - (q[i] + k);
							if (digits[i] < 0) { digits[i] += Base; k = 1; }
							else k = 0;
							if (digits[i] != 0) z = i;
						}
						if (j < N - 1 || digits[j] == 0)
						{
							var tmp = new int[z + 1];
							for (var i = 0; i <= z; i++) tmp[i] = digits[i];
							digits = tmp;
						}
						isPositive ^= !flag;
					}
				}
			}
		}
		public void Subtract(MyInteger a) { a.Negate(); Add(a); a.Negate(); }
		public static MyInteger Add(MyInteger a, MyInteger b)
		{
			if (a.IsZero) return +b;
			else if (b.IsZero) return +a;
			else if (a.isPositive == b.isPositive)
			{
				var c = Add(a.digits, b.digits);
				return new MyInteger(a.isPositive, c.ToArray());
			}
			else
			{
				var c = Subtract(a.digits, b.digits);
				if (c.Item2.Count == 0) return new MyInteger();
				return new MyInteger(!a.isPositive ^ c.Item1, c.Item2.ToArray());
			}
		}
		public static MyInteger Subtract(MyInteger a, MyInteger b) { b.Negate(); var res = a + b; b.Negate(); return res; }
		static List<int> Add(int[] a, int[] b)
		{
			var N = a.Length;
			var M = b.Length;
			if (N < M) return Add(b, a);
			var c = new List<int>(N);
			var k = 0;
			for (var i = 0; i < M; i++)
			{
				var x = k + a[i] + b[i];
				if (x >= Base) { x -= Base; k = 1; }
				else k = 0;
				c.Add(x);
			}
			for (var i = M; i < N; i++)
			{
				var x = k + a[i];
				if (x >= Base) { x -= Base; k = 1; }
				else k = 0;
				c.Add(x);
			}
			if (k > 0) c.Add(k);
			return c;
		}
		public static Tuple<bool, List<int>> Subtract(int[] a, int[] b)
		{
			var N = a.Length;
			var M = b.Length;
			if (N < M) { var tmp = Subtract(b, a); return new Tuple<bool, List<int>>(!tmp.Item1, tmp.Item2); }
			else if (N > M)
			{
				var c = new List<int>(N);
				var k = 0;
				var z = -1;
				for (var i = 0; i < M; i++)
				{
					var x = a[i] - (b[i] + k);
					if (x < 0) { x += Base; k = 1; }
					else k = 0;
					c.Add(x);
					if (x != 0) z = i;
				}
				for (var i = M; i < N; i++)
				{
					var x = a[i] - k;
					if (x < 0) { x += Base; k = 1; }
					else k = 0;
					c.Add(x);
					if (x != 0) z = i;
				}
				if (c[c.Count - 1] == 0) c.RemoveRange(z + 1, c.Count - z - 1);
				return new Tuple<bool, List<int>>(true, c);
			}
			else
			{
				var c = new List<int>(N);
				var j = N - 1;
				while (j >= 0 && a[j] == b[j]) j--;
				if (j < 0) return new Tuple<bool, List<int>>(true, new List<int>());
				var flag = a[j] >= b[j];
				if (!flag) { var tmp = a; a = b; b = tmp; }
				var k = 0;
				var z = -1;
				for (var i = 0; i <= j; i++)
				{
					var x = a[i] - (b[i] + k);
					if (x < 0) { x += Base; k = 1; }
					else k = 0;
					c.Add(x);
					if (x != 0) z = i;
				}
				if (c[c.Count - 1] == 0) c.RemoveRange(z + 1, c.Count - z - 1);
				return new Tuple<bool, List<int>>(flag, c);
			}
		}
		public static int[] Multiply(int[] a, int[] b)
		{
			var N = a.Length;
			var M = b.Length;
			var L = N + M;
			var c = new long[L];
			for (var i = 0; i < N; i++)
				for (var j = 0; j < M; j++)
				{
					long q;
					var p = (int)Math.DivRem((long)a[i] * b[j], Base, out q);
					c[i + j] += (int)q;
					c[i + j + 1] += p;
				}
			var k = 0L;
			for (var i = 0; i < L; i++)
			{
				c[i] += k;
				if (c[i] >= Base) { k = c[i] / Base; c[i] %= Base; }
				else k = 0;
			}
			var z = c.Length - 1;
			while (c[z] == 0) z--;
			var res = new int[z + 1];
			for (var i = 0; i <= z; i++) res[i] = (int)c[i];
			return res;
		}
		public static MyInteger Divide(MyInteger a, MyInteger b)
		{
			if (b.IsZero) throw new DivideByZeroException();
			else if (a.IsZero) return new MyInteger();
			else if (a.isPositive) return new MyInteger(b.isPositive, Divide(a.digits, b.digits));
			else
			{
				var tmp = DivRem(a.digits, b.digits);
				var x = new MyInteger(true, tmp.Item1);
				if (tmp.Item2.Length != 0) x.Increment();
				if (!b.isPositive) x.Negate();
				return x;
			}
		}
		public static MyInteger Remainder(MyInteger a, MyInteger b)
		{
			if (b.IsZero) throw new DivideByZeroException();
			else if (a.IsZero) return new MyInteger();
			else if (a.isPositive) return new MyInteger(b.isPositive, DivRem(a.digits, b.digits).Item2);
			else
			{
				var tmp = DivRem(a.digits, b.digits);
				var x = new MyInteger(true, tmp.Item2);
				if (tmp.Item2.Length != 0) x.Add(b);
				if (!b.isPositive) x.Negate();
				return x;
			}
		}
		public static MyInteger DivRem(MyInteger a, MyInteger b, out MyInteger r) { var t = DivRem(a, b); r = t.Item2; return t.Item1; }
		public static Tuple<MyInteger, MyInteger> DivRem(MyInteger a, MyInteger b)
		{
			if (b.IsZero) throw new DivideByZeroException();
			else if (a.IsZero) return new Tuple<MyInteger, MyInteger>(new MyInteger(), new MyInteger());
			else if (a.isPositive)
			{
				var t = DivRem(a.digits, b.digits);
				return new Tuple<MyInteger, MyInteger>(new MyInteger(b.isPositive, t.Item1), new MyInteger(b.isPositive, t.Item2));
			}
			else
			{
				var tmp = DivRem(a.digits, b.digits);
				var q = new MyInteger(true, tmp.Item1);
				var r = new MyInteger(true, tmp.Item2);
				if (tmp.Item2.Length != 0) { q.Increment(); r.Add(b); }
				if (!b.isPositive) { q.Negate(); r.Negate(); }
				return new Tuple<MyInteger, MyInteger>(q, r);
			}
		}
		private static int[] Divide(int[] a, int[] b)
		{
			var N = a.Length;
			var M = b.Length;
			if (N < M) return new int[0];
			else if (N == M)
			{
				var k = a[N - 1] / b[N - 1];
			}
			else
			{

			}
			return null;
		}
		private static Tuple<int[], int[]> DivRem(int[] a, int[] b)
		{
			return null;
		}
		public static MyInteger RightShift(MyInteger a, int x)
		{
			if (x < 0) return LeftShift(a, -x);
			else if (x == 0) return +a;
			else return new MyInteger(a.isPositive, RightShift(a.digits, x));
		}
		public static MyInteger LeftShift(MyInteger a, int x)
		{
			if (x < 0) return RightShift(a, -x);
			else if (x == 0) return +a;
			else return new MyInteger(a.isPositive, LeftShift(a.digits, x));
		}
		private static int[] RightShift(int[] a, int x)
		{
			var b = new int[a.Length - x];
			for (var i = 0; i < b.Length; i++) b[i] = a[i + x];
			return b;
		}
		private static int[] LeftShift(int[] a, int x)
		{
			var b = new int[a.Length + x];
			for (var i = 0; i < a.Length; i++) b[i + x] = a[i];
			return b;
		}
		public static MyInteger Pow(MyInteger a, MyInteger b)
		{
			if (a.IsZero)
			{
				if (!b.isPositive || b.IsZero) throw new DivideByZeroException();
				else return new MyInteger();
			}
			else if (a.digits.Length == 1 && a.digits[0] == 1)
			{
				if (a.isPositive) return +One;
				else return new MyInteger(b.IsEven, new int[] { 1 });
			}
			else if (b.IsZero) return +One;
			else if (!b.isPositive) return new MyInteger();
			var p = +One;
			var x = a;
			while (b.digits.Length > 0)
			{
				if ((b.digits[0] & 1) == 1) p *= x;
				b /= Two;
				x *= x;
			}
			return p;
		}
		public static MyInteger Multiply(MyInteger a, MyInteger b)
		{
			return a.IsZero || b.IsZero ? new MyInteger() : new MyInteger(a.isPositive == b.isPositive, Multiply(a.digits, b.digits));
		}

		#region 演算子のオーバーロード

		public static MyInteger operator +(MyInteger a) { return new MyInteger(a.isPositive, a.digits); }
		public static MyInteger operator -(MyInteger a) { return a.IsZero ? new MyInteger() : new MyInteger(!a.isPositive, a.digits); }
		public static MyInteger operator ++(MyInteger c) { var d = +c; d.Increment(); return d; }
		public static MyInteger operator --(MyInteger c) { var d = +c; d.Decrement(); return d; }
		public static MyInteger operator +(MyInteger a, MyInteger b) { return Add(a, b); }
		public static MyInteger operator -(MyInteger a, MyInteger b) { return Subtract(a, b); }
		public static MyInteger operator *(MyInteger a, MyInteger b) { return Multiply(a, b); }
		public static MyInteger operator /(MyInteger a, MyInteger b) { return Divide(a, b); }
		public static MyInteger operator %(MyInteger a, MyInteger b) { return Remainder(a, b); }
		public static bool operator ==(MyInteger a, MyInteger b) { return Compare(a, b) == 0; }
		public static bool operator !=(MyInteger a, MyInteger b) { return Compare(a, b) != 0; }
		public static bool operator <(MyInteger a, MyInteger b) { return Compare(a, b) < 0; }
		public static bool operator >(MyInteger a, MyInteger b) { return Compare(a, b) > 0; }
		public static bool operator <=(MyInteger a, MyInteger b) { return Compare(a, b) <= 0; }
		public static bool operator >=(MyInteger a, MyInteger b) { return Compare(a, b) >= 0; }
		public static MyInteger operator >>(MyInteger a, int x) { return RightShift(a, x); }
		public static MyInteger operator <<(MyInteger a, int x) { return LeftShift(a, x); }
		public static MyInteger operator ^(MyInteger a, MyInteger b) { return Pow(a, b); }

		#region キャスト

		public static explicit operator ulong(MyInteger a)
		{
			if (!a.isPositive) throw new OverflowException();
			var x = 0uL;
			var e = 1uL;
			var lim = a.digits.Length - 1;
			for (var i = 0; i < lim; i++) { x += e * (ulong)a.digits[i]; e *= Base; }
			x += e * (ulong)a.digits[lim];
			return x;
		}
		public static explicit operator long(MyInteger a)
		{
			if (a.IsZero) return 0;
			var x = 0L;
			var e = 1L;
			var lim = a.digits.Length - 1;
			for (var i = 0; i < lim; i++) { x += e * a.digits[i]; e *= Base; }
			x += e * a.digits[lim];
			return x * (a.isPositive ? 1 : -1);
		}
		public static explicit operator uint(MyInteger a)
		{
			if (!a.isPositive) throw new OverflowException();
			var x = 0u;
			var e = 1u;
			var lim = a.digits.Length - 1;
			for (var i = 0; i < lim; i++) { x += e * (uint)a.digits[i]; e *= Base; }
			x += e * (uint)a.digits[lim];
			return x;
		}
		public static explicit operator int(MyInteger a)
		{
			if (a.IsZero) return 0;
			var x = 0;
			var e = 1;
			var lim = a.digits.Length - 1;
			for (var i = 0; i < lim; i++) { x += e * a.digits[i]; e *= Base; }
			x += e * a.digits[lim];
			return x * (a.isPositive ? 1 : -1);
		}
		public static implicit operator MyInteger(ulong x)
		{
			if (x == 0) return new MyInteger();
			var a = new List<int>();
			while (x > 0) { a.Add((int)(x % Base)); x /= Base; }
			return new MyInteger(true, a.ToArray());
		}
		public static implicit operator MyInteger(long x)
		{
			if (x == 0) return new MyInteger();
			var flag = x > 0 ? 1 : -1;
			x *= flag;
			var a = new List<int>();
			while (x > 0) { a.Add((int)(x % Base)); x /= Base; }
			return new MyInteger(flag == 1, a.ToArray());
		}
		public static implicit operator MyInteger(uint x)
		{
			if (x == 0) return new MyInteger();
			var a = new List<int>();
			while (x > 0) { a.Add((int)(x % Base)); x /= Base; }
			return new MyInteger(true, a.ToArray());
		}
		public static implicit operator MyInteger(int x)
		{
			if (x == 0) return new MyInteger();
			var flag = x > 0 ? 1 : -1;
			x *= flag;
			var a = new List<int>();
			while (x > 0) { a.Add(x % Base); x /= Base; }
			return new MyInteger(flag == 1, a.ToArray());
		}

		#endregion

		#endregion

		#region 比較、文字列

		public override bool Equals(object x) { if (x == null) return false; var p = x as MyInteger; return (object)p != null && this == p; }
		public override int GetHashCode()
		{
			var h = isPositive.GetHashCode();
			h ^= digits.Length.GetHashCode();
			for (var i = 0; i < digits.Length; i++) h ^= digits[i];
			return h;
		}
		public override string ToString()
		{
			var N = digits.Length;
			if (N == 0) return "0";
			var sb = new StringBuilder();
			if (!isPositive) sb.Append('-');
			var i = N - 1;
			sb.Append(digits[i--]);
			for (; i >= 0; i--) sb.Append(digits[i].ToString().PadLeft(LogBase, '0'));
			return sb.ToString();
		}
		public MyInteger Parse(string str)
		{
			if (str.Length == 0) throw new FormatException();
			var flag = true;
			var i = 0;
			var last = str.Length - 1;
			while (last >= 0 && (str[last] == ' '||str[last]=='\t'||str[last]=='\n')) last--;
			if (str[i] == '-') { flag = false; i++; }
			else if (str[i] == '+') i++;
			else if (str[i] < '0' || str[i] > '9') throw new FormatException();
			while (i < str.Length && str[i] == '0') i++;
			if (i == str.Length) return new MyInteger();
			var x = new int[(str.Length - i + LogBase - 1) / LogBase];
			var rem = (str.Length - i) - x.Length * LogBase;
			for (; i < str.Length; i += LogBase)
			{

			}
			return new MyInteger(flag, x);
		}
		public bool Equals(MyInteger other) { return this == other; }
		public static int Compare(MyInteger x, MyInteger y)
		{
			if (ReferenceEquals(x, y)) return 0;
			else if ((object)x == null) return -1;
			else if ((object)y == null) return 1;
			var N = x.digits.Length;
			var M = y.digits.Length;
			if (x.isPositive && y.isPositive)
			{
				if (N < M) return -1;
				else if (N > M) return 1;
				else if (x.IsZero) return 0;
				var j = N - 1;
				while (j >= 0 && x.digits[j] == y.digits[j]) j--;
				return j >= 0 ? x.digits[j].CompareTo(y.digits[j]) : 0;
			}
			else if (x.isPositive) return 1;
			else if (y.isPositive) return -1;
			else
			{
				if (N < M) return 1;
				else if (N > M) return -1;
				var j = N - 1;
				while (j >= 0 && x.digits[j] == y.digits[j]) j--;
				return j >= 0 ? y.digits[j].CompareTo(x.digits[j]) : 0;
			}
		}
		public int CompareTo(MyInteger x) { return Compare(this, x); }
		public int CompareTo(object x) { return x == null ? 1 : Compare(this, x as MyInteger); }

		#endregion
	}
}