Problem:
Score : 100 points

Problem Statement

You are given a grid of N rows and M columns. The square at the i-th row and j-th column will be denoted as (i,j).
Some of the squares contain an object. All the remaining squares are empty.
The state of the grid is represented by strings S_1,S_2,\cdots,S_N. The square (i,j) contains an object if S_{i,j}= # and is empty if S_{i,j}= ..

Consider placing 1 \times 2 tiles on the grid. Tiles can be placed vertically or horizontally to cover two adjacent empty squares.
Tiles must not stick out of the grid, and no two different tiles may intersect. Tiles cannot occupy the square with an object.

Calculate the maximum number of tiles that can be placed and any configulation that acheives the maximum.

Constraints

1 \leq N \leq 100

1 \leq M \leq 100

S_i is a string with length M consists of # and ..

Input

Input is given from Standard Input in the following format:

N M
S_1
S_2
\vdots
S_N

Output

On the first line, print the maximum number of tiles that can be placed.

On the next N lines, print a configulation that achieves the maximum.
Precisely, output the strings t_1,t_2,\cdots,t_N constructed by the following way.

t_i is initialized to S_i.

For each (i,j), if there is a tile that occupies (i,j) and (i+1,j), change t_{i,j}:=v, t_{i+1,j}:=^.

For each (i,j), if there is a tile that occupies (i,j) and (i,j+1), change t_{i,j}:=>, t_{i,j+1}:=<.

See samples for further information.

You may print any configulation that maximizes the number of tiles.

Sample Input 1

3 3
#..
..#
...

Sample Output 1

3
#><
vv#
^^.

The following output is also treated as a correct answer.

3
#><
v.#
^><

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Reflection.Metadata;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading;

namespace AtCoder
{
	public class Program
	{
		static void Main()
		{
			using var cin = new Scanner();
			var (n, m) = cin.Int2();
			var s = new string[n];
			for (int i = 0; i < n; i++) {
				s[i] = cin.Next();
			}

			int num = n * m + 2;
			int si = n * m;
			int ti = n * m + 1;
			var flow = new MFGraphInt(num);
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					if (s[i][j] == '#') {
						continue;
					}

					int k = i * m + j;
					if ((i + j) % 2 == 0) {
						flow.AddEdge(si, k, 1);
						Helper.DoAt4(i, j, n, m, (ii, jj) => {
							if (s[ii][jj] == '.') {
								int kk = ii * m + jj;
								flow.AddEdge(k, kk, 1);
							}
						});
					} else {
						flow.AddEdge(k, ti, 1);
					}
				}
			}

			long maxFlow = flow.Flow(si, ti);
			Console.WriteLine(maxFlow);

			var ans = new char[n][];
			for (int i = 0; i < n; i++) {
				ans[i] = new char[m];
			}

			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					ans[i][j] = s[i][j];
				}
			}

			var edges = flow.Edges();
			foreach (var e in edges) {
				if (e.From == si || e.To == ti || e.Flow == 0) {
					continue;
				}

				int i0 = e.From / m;
				int j0 = e.From % m;
				int i1 = e.To / m;
				int j1 = e.To % m;

				if (i0 == i1 + 1) {
					ans[i1][j1] = 'v';
					ans[i0][j0] = '^';
				} else if (j0 == j1 + 1) {
					ans[i1][j1] = '>';
					ans[i0][j0] = '<';
				} else if (i0 == i1 - 1) {
					ans[i0][j0] = 'v';
					ans[i1][j1] = '^';
				} else {
					ans[i0][j0] = '>';
					ans[i1][j1] = '<';
				}
			}

			for (int i = 0; i < n; i++) {
				Console.WriteLine(new string(ans[i]));
			}
		}
	}

	public interface INumOperator<T> : IEqualityComparer<T>, IComparer<T> where T : struct
	{
		/// <summary>
		/// MinValue
		/// </summary>
		public T MinValue { get; }
		/// <summary>
		/// MaxValue
		/// </summary>
		public T MaxValue { get; }
		/// <summary>
		/// Addition operator +
		/// </summary>
		/// <returns><paramref name="x"/> + <paramref name="y"/></returns>
		T Add(T x, T y);
		/// <summary>
		/// Subtraction operator -
		/// </summary>
		/// <returns><paramref name="x"/> - <paramref name="y"/></returns>
		T Subtract(T x, T y);
		/// <summary>
		/// Multiplication operator *
		/// </summary>
		/// <returns><paramref name="x"/> * <paramref name="y"/></returns>
		T Multiply(T x, T y);
		/// <summary>
		/// Division operator /
		/// </summary>
		/// <returns><paramref name="x"/> / <paramref name="y"/></returns>
		T Divide(T x, T y);
		/// <summary>
		/// Remainder operator %
		/// </summary>
		/// <returns><paramref name="x"/> % <paramref name="y"/></returns>
		T Modulo(T x, T y);
		/// <summary>
		/// Greater than operator &gt;
		/// </summary>
		/// <returns><paramref name="x"/> &gt; <paramref name="y"/></returns>
		bool GreaterThan(T x, T y);
		/// <summary>
		/// Greater than or equal operator &gt;=
		/// </summary>
		/// <returns><paramref name="x"/> &gt;= <paramref name="y"/></returns>
		bool GreaterThanOrEqual(T x, T y);
		/// <summary>
		/// Less than operator &lt;
		/// </summary>
		/// <returns><paramref name="x"/> &lt; <paramref name="y"/></returns>
		bool LessThan(T x, T y);
		/// <summary>
		/// Less than or equal operator &lt;=
		/// </summary>
		/// <returns><paramref name="x"/> &lt;= <paramref name="y"/></returns>
		bool LessThanOrEqual(T x, T y);
	}
	public readonly struct IntOperator : INumOperator<int>
	{
		public int MinValue => int.MinValue;
		public int MaxValue => int.MaxValue;
		public int Add(int x, int y) => x + y;
		public int Subtract(int x, int y) => x - y;
		public int Multiply(int x, int y) => x * y;
		public int Divide(int x, int y) => x / y;
		public int Modulo(int x, int y) => x % y;
		public bool GreaterThan(int x, int y) => x > y;
		public bool GreaterThanOrEqual(int x, int y) => x >= y;
		public bool LessThan(int x, int y) => x < y;
		public bool LessThanOrEqual(int x, int y) => x <= y;
		public int Compare(int x, int y) => x.CompareTo(y);
		public bool Equals(int x, int y) => x == y;
		public int GetHashCode(int obj) => obj.GetHashCode();
	}
	public readonly struct LongOperator : INumOperator<long>
	{
		public long MinValue => long.MinValue;
		public long MaxValue => long.MaxValue;
		public long Add(long x, long y) => x + y;
		public long Subtract(long x, long y) => x - y;
		public long Multiply(long x, long y) => x * y;
		public long Divide(long x, long y) => x / y;
		public long Modulo(long x, long y) => x % y;
		public bool GreaterThan(long x, long y) => x > y;
		public bool GreaterThanOrEqual(long x, long y) => x >= y;
		public bool LessThan(long x, long y) => x < y;
		public bool LessThanOrEqual(long x, long y) => x <= y;
		public int Compare(long x, long y) => x.CompareTo(y);
		public bool Equals(long x, long y) => x == y;
		public int GetHashCode(long obj) => obj.GetHashCode();
	}
	public readonly struct UIntOperator : INumOperator<uint>
	{
		public uint MinValue => uint.MinValue;
		public uint MaxValue => uint.MaxValue;
		public uint Add(uint x, uint y) => x + y;
		public uint Subtract(uint x, uint y) => x - y;
		public uint Multiply(uint x, uint y) => x * y;
		public uint Divide(uint x, uint y) => x / y;
		public uint Modulo(uint x, uint y) => x % y;
		public bool GreaterThan(uint x, uint y) => x > y;
		public bool GreaterThanOrEqual(uint x, uint y) => x >= y;
		public bool LessThan(uint x, uint y) => x < y;
		public bool LessThanOrEqual(uint x, uint y) => x <= y;
		public int Compare(uint x, uint y) => x.CompareTo(y);
		public bool Equals(uint x, uint y) => x == y;
		public int GetHashCode(uint obj) => obj.GetHashCode();
	}
	public readonly struct ULongOperator : INumOperator<ulong>
	{
		public ulong MinValue => ulong.MinValue;
		public ulong MaxValue => ulong.MaxValue;
		public ulong Add(ulong x, ulong y) => x + y;
		public ulong Subtract(ulong x, ulong y) => x - y;
		public ulong Multiply(ulong x, ulong y) => x * y;
		public ulong Divide(ulong x, ulong y) => x / y;
		public ulong Modulo(ulong x, ulong y) => x % y;
		public bool GreaterThan(ulong x, ulong y) => x > y;
		public bool GreaterThanOrEqual(ulong x, ulong y) => x >= y;
		public bool LessThan(ulong x, ulong y) => x < y;
		public bool LessThanOrEqual(ulong x, ulong y) => x <= y;
		public int Compare(ulong x, ulong y) => x.CompareTo(y);
		public bool Equals(ulong x, ulong y) => x == y;
		public int GetHashCode(ulong obj) => obj.GetHashCode();
	}
	public readonly struct DoubleOperator : INumOperator<double>
	{
		public double MinValue => double.MinValue;
		public double MaxValue => double.MaxValue;
		public double Add(double x, double y) => x + y;
		public double Subtract(double x, double y) => x - y;
		public double Multiply(double x, double y) => x * y;
		public double Divide(double x, double y) => x / y;
		public double Modulo(double x, double y) => x % y;
		public bool GreaterThan(double x, double y) => x > y;
		public bool GreaterThanOrEqual(double x, double y) => x >= y;
		public bool LessThan(double x, double y) => x < y;
		public bool LessThanOrEqual(double x, double y) => x <= y;
		public int Compare(double x, double y) => x.CompareTo(y);
		public bool Equals(double x, double y) => x == y;
		public int GetHashCode(double obj) => obj.GetHashCode();
	}

	/// <summary>
	/// 最大フロー問題 を解くライブラリ(int版)です。
	/// </summary>
	public class MFGraphInt : MFGraph<int, IntOperator> { public MFGraphInt(int n) : base(n) { } }

	/// <summary>
	/// 最大フロー問題 を解くライブラリ(long版)です。
	/// </summary>
	public class MFGraphLong : MFGraph<long, LongOperator> { public MFGraphLong(int n) : base(n) { } }

	/// <summary>
	/// 最大フロー問題 を解くライブラリです。
	/// </summary>
	/// <typeparam name="TValue">容量の型</typeparam>
	/// <typeparam name="TOp"><typeparamref name="TValue"/>に対応する演算を提要する型</typeparam>
	/// <remarks>
	/// <para>制約: <typeparamref name="TValue"/> は int, long。</para>
	/// <para>
	/// 内部では各辺 e について 2 つの変数、流量 f_e と容量 c_e を管理しています。
	/// 頂点 v から出る辺の集合を out(v)、入る辺の集合を in(v)、
	/// また頂点 v について g(v, f) = (Σ_in(v) f_e) - (Σ_out(v) f_e) とします。 
	/// </para>
	/// </remarks>
	public class MFGraph<TValue, TOp>
		 where TValue : struct
		 where TOp : struct, INumOperator<TValue>
	{
		static readonly TOp op = default;

		/// <summary>
		/// <paramref name="n"/> 頂点 0 辺のグラフを作ります。
		/// </summary>
		/// <remarks>
		/// <para>制約: 0 ≤ <paramref name="n"/> ≤ 10^8</para>
		/// <para>計算量: O(<paramref name="n"/>)</para>
		/// </remarks>
		public MFGraph(int n)
		{
			_n = n;
			_g = new List<EdgeInternal>[n];
			for (int i = 0; i < n; i++) {
				_g[i] = new List<EdgeInternal>();
			}
			_pos = new List<(int first, int second)>();
		}

		/// <summary>
		/// <paramref name="from"/> から <paramref name="to"/> へ
		/// 最大容量 <paramref name="cap"/>、流量 0 の辺を追加し、何番目に追加された辺かを返します。
		/// </summary>
		/// <remarks>
		/// 制約: 
		/// <list type="bullet">
		/// <item>
		/// <description>0 ≤ <paramref name="from"/>, <paramref name="to"/> &lt; n</description>
		/// </item>
		/// <item>
		/// <description>0 ≤ <paramref name="cap"/></description>
		/// </item>
		/// </list>
		/// <para>計算量: ならしO(1)</para>
		/// </remarks>
		public int AddEdge(int from, int to, TValue cap)
		{
			int m = _pos.Count;
			Debug.Assert(0 <= from && from < _n);
			Debug.Assert(0 <= to && to < _n);
			Debug.Assert(default(TOp).LessThanOrEqual(default, cap));
			_pos.Add((from, _g[from].Count));
			_g[from].Add(new EdgeInternal(to, _g[to].Count, cap));
			_g[to].Add(new EdgeInternal(from, _g[from].Count - 1, default));
			return m;
		}

		/// <summary>
		/// 今の内部の辺の状態を返します。
		/// </summary>
		/// <remarks>
		/// <para>AddEdge で <paramref name="i"/> 番目 (0-indexed) に追加された辺を返す。</para>
		/// <para>計算量: O(1)</para>
		/// </remarks>
		public Edge GetEdge(int i)
		{
			int m = _pos.Count;
			Debug.Assert(0 <= i && i < m);
			var _e = _g[_pos[i].first][_pos[i].second];
			var _re = _g[_e.To][_e.Rev];
			return new Edge(_pos[i].first, _e.To, default(TOp).Add(_e.Cap, _re.Cap), _re.Cap);
		}

		/// <summary>
		/// 今の内部の辺の状態を返します。
		/// </summary>
		/// <remarks>
		/// <para>辺の順番はadd_edgeで追加された順番と同一。</para>
		/// <para>計算量: O(n)</para>
		/// </remarks>
		public List<Edge> Edges()
		{
			int m = _pos.Count;
			var result = new List<Edge>();
			for (int i = 0; i < m; i++) {
				result.Add(GetEdge(i));
			}
			return result;
		}

		/// <summary>
		/// <paramref name="i"/> 番目に追加された辺の容量、流量を
		/// <paramref name="newCap"/>, <paramref name="newFlow"/> に変更します。
		/// </summary>
		/// <remarks>
		/// <para>
		/// 他の辺の容量、流量は変更しません。
		/// 辺 <paramref name="i"/> の流量、容量のみを
		/// <paramref name="newCap"/>, <paramref name="newFlow"/> へ変更します。
		/// </para>
		/// </remarks>
		public void ChangeEdge(int i, TValue newCap, TValue newFlow)
		{
			int m = _pos.Count;
			Debug.Assert(0 <= i && i < m);
			Debug.Assert(default(TOp).LessThanOrEqual(default, newFlow) && default(TOp).LessThanOrEqual(newFlow, newCap));
			var _e = _g[_pos[i].first][_pos[i].second];
			var _re = _g[_e.To][_e.Rev];
			_e.Cap = default(TOp).Subtract(newCap, newFlow);
			_re.Cap = newFlow;
		}

		/// <summary>
		/// 頂点 <paramref name="s"/> から <paramref name="t"/> へ流せる限り流し、
		/// 流せた量を返します。
		/// </summary>
		/// <remarks>
		/// <para>
		/// 複数回呼ぶことも可能で、その時の挙動は
		/// 変更前と変更後の流量を f_e, f'_e として、以下の条件を満たすように変更します。
		/// </para>
		/// <list type="bullet">
		/// <item>
		/// <description>0 ≤ f'_e ≤ C_e</description>
		/// </item>
		/// <item>
		/// <description>
		/// <paramref name="s"/>, <paramref name="t"/> 以外の頂天 v について、
		/// g(v, f) = g(v, f')
		/// </description>
		/// </item>
		/// <item>
		/// <description>
		/// (flowLimit を指定した場合) g(t, f') - g(t, f) ≤ flowLimit
		/// </description>
		/// </item>
		/// <item>
		/// <description>
		/// g(t, f') - g(t, f) が委譲の条件を満たすうち最大。この g(t, f') - g(t, f) を返す。
		/// </description>
		/// </item>
		/// </list>
		/// <para>制約: 返値が <typeparamref name="TValue"/> に収まる。</para>
		/// 計算量: m を追加された辺数として、
		/// <list type="bullet">
		/// <item>
		/// <description>O(min(n^(2/3) m, m^(3/2))) (辺の容量が全部 1 の時)</description>
		/// </item>
		/// <item>
		/// <description>O(n^2 m)</description>
		/// </item>
		/// </list>
		/// </remarks>
		public TValue Flow(int s, int t)
		{
			return Flow(s, t, default(TOp).MaxValue);
		}

		/// <summary>
		/// 頂点 <paramref name="s"/> から <paramref name="t"/> へ
		/// 流量 <paramref name="flowLimit"/> に達するまで流せる限り流し、
		/// 流せた量を返します。
		/// </summary>
		/// <remarks>
		/// <para>
		/// 複数回呼ぶことも可能で、その時の挙動は
		/// 変更前と変更後の流量を f_e, f'_e として、以下の条件を満たすように変更します。
		/// </para>
		/// <list type="bullet">
		/// <item>
		/// <description>0 ≤ f'_e ≤ C_e</description>
		/// </item>
		/// <item>
		/// <description>
		/// <paramref name="s"/>, <paramref name="t"/> 以外の頂天 v について、
		/// g(v, f) = g(v, f')
		/// </description>
		/// </item>
		/// <item>
		/// <description>
		/// (<paramref name="flowLimit"/> を指定した場合) g(t, f') - g(t, f) ≤ <paramref name="flowLimit"/>
		/// </description>
		/// </item>
		/// <item>
		/// <description>
		/// g(t, f') - g(t, f) が委譲の条件を満たすうち最大。この g(t, f') - g(t, f) を返す。
		/// </description>
		/// </item>
		/// </list>
		/// <para>制約: 返値が <typeparamref name="TValue"/> に収まる。</para>
		/// 計算量: m を追加された辺数として、
		/// <list type="bullet">
		/// <item>
		/// <description>O(min(n^(2/3) m, m^(3/2))) (辺の容量が全部 1 の時)</description>
		/// </item>
		/// <item>
		/// <description>O(n^2 m)</description>
		/// </item>
		/// </list>
		/// </remarks>
		public TValue Flow(int s, int t, TValue flowLimit)
		{
			Debug.Assert(0 <= s && s < _n);
			Debug.Assert(0 <= t && t < _n);

			var level = new int[_n];
			var iter = new int[_n];
			var que = new Queue<int>();

			void Bfs()
			{
				for (int i = 0; i < _n; i++) {
					level[i] = -1;
				}

				level[s] = 0;
				que.Clear();
				que.Enqueue(s);
				while (que.Count > 0) {
					int v = que.Dequeue();
					foreach (var e in _g[v]) {
						if (default(TOp).Equals(e.Cap, default) || level[e.To] >= 0) continue;
						level[e.To] = level[v] + 1;
						if (e.To == t) return;
						que.Enqueue(e.To);
					}
				}
			}

			TValue Dfs(int v, TValue up)
			{
				if (v == s) return up;
				var res = default(TValue);
				int level_v = level[v];
				for (; iter[v] < _g[v].Count; iter[v]++) {
					EdgeInternal e = _g[v][iter[v]];
					if (level_v <= level[e.To] || default(TOp).Equals(_g[e.To][e.Rev].Cap, default)) continue;
					var up1 = default(TOp).Subtract(up, res);
					var up2 = _g[e.To][e.Rev].Cap;
					var d = Dfs(e.To, default(TOp).LessThan(up1, up2) ? up1 : up2);
					if (default(TOp).Compare(d, default) <= 0) continue;
					_g[v][iter[v]].Cap = default(TOp).Add(_g[v][iter[v]].Cap, d);
					_g[e.To][e.Rev].Cap = default(TOp).Subtract(_g[e.To][e.Rev].Cap, d);
					res = default(TOp).Add(res, d);
					if (res.Equals(up)) break;
				}

				return res;
			}

			TValue flow = default;
			while (default(TOp).LessThan(flow, flowLimit)) {
				Bfs();
				if (level[t] == -1) break;
				for (int i = 0; i < _n; i++) {
					iter[i] = 0;
				}
				while (default(TOp).LessThan(flow, flowLimit)) {
					var f = Dfs(t, default(TOp).Subtract(flowLimit, flow));
					if (default(TOp).Equals(f, default)) break;
					flow = default(TOp).Add(flow, f);
				}
			}
			return flow;
		}

		/// <summary>
		/// 長さ n の配列を返します。
		/// i 番目の要素には、頂点 <paramref name="s"/> から i へ残余グラフで到達可能なとき、
		/// またその時のみ true を返します。
		/// Flow(s, t) を flowLimit なしでちょうど一回呼んだ後に呼ぶと、
		/// 返り値はs, t 間の mincut に対応します。
		/// </summary>
		/// <remarks>
		/// <para>
		/// 各辺 e = (u, v, f_e, c_e) について、 f_e &lt; c_e ならば辺 (u, v) を張り、
		/// 0 &lt; f_e ならば辺 (u, v) を張ったと仮定したとき、
		/// 頂点 ss から到達可能な頂点の集合を返します。
		/// </para>
		/// 計算量: m を追加された辺数として、
		/// <list type="bullet">
		/// <item>
		/// <description>O(n + m)</description>
		/// </item>
		/// </list>
		/// </remarks>
		public bool[] MinCut(int s)
		{
			var visited = new bool[_n];
			var que = new Queue<int>();
			que.Enqueue(s);
			while (que.Count > 0) {
				int p = que.Dequeue();
				visited[p] = true;
				foreach (var e in _g[p]) {
					if (!default(TOp).Equals(e.Cap, default) && !visited[e.To]) {
						visited[e.To] = true;
						que.Enqueue(e.To);
					}
				}
			}

			return visited;
		}

		public struct Edge
		{
			public int From { get; set; }
			public int To { get; set; }
			public TValue Cap { get; set; }
			public TValue Flow { get; set; }
			public Edge(int from, int to, TValue cap, TValue flow)
			{
				From = from;
				To = to;
				Cap = cap;
				Flow = flow;
			}
		};

		private class EdgeInternal
		{
			public int To { get; set; }
			public int Rev { get; set; }
			public TValue Cap { get; set; }
			public EdgeInternal(int to, int rev, TValue cap)
			{
				To = to;
				Rev = rev;
				Cap = cap;
			}
		};

		private readonly int _n;
		private readonly List<(int first, int second)> _pos;
		private readonly List<EdgeInternal>[] _g;
	}

	public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>
	{
		private readonly Func<TKey, TValue> initialzier_;
		public HashMap(Func<TKey, TValue> initialzier)
			: base()
		{
			initialzier_ = initialzier;
		}

		public HashMap(Func<TKey, TValue> initialzier, int capacity)
			: base(capacity)
		{
			initialzier_ = initialzier;
		}

		new public TValue this[TKey key]
		{
			get
			{
				if (ContainsKey(key) == false) {
					base[key] = initialzier_(key);
				}

				return base[key];
			}

			set { base[key] = value; }
		}
	}

	public struct ModInt
	{
		public const long P = 1000000007;
		//public const long P = 998244353;

		public static ModInt Inverse(ModInt value) => Pow(value, P - 2);
		public static ModInt Pow(ModInt value, long k) => Pow(value.value_, k);
		public static ModInt Pow(long value, long k)
		{
			long ret = 1;
			for (k %= P - 1; k > 0; k >>= 1, value = value * value % P) {
				if ((k & 1) == 1) {
					ret = ret * value % P;
				}
			}
			return new ModInt(ret);
		}

		private long value_;

		public ModInt(long value)
			=> value_ = value;
		public ModInt(long value, bool mods)
		{
			if (mods) {
				value %= P;
				if (value < 0) {
					value += P;
				}
			}

			value_ = value;
		}

		public static ModInt operator +(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = (lhs.value_ + rhs.value_) % P;
			return lhs;
		}
		public static ModInt operator +(long lhs, ModInt rhs)
		{
			rhs.value_ = (lhs + rhs.value_) % P;
			return rhs;
		}
		public static ModInt operator +(ModInt lhs, long rhs)
		{
			lhs.value_ = (lhs.value_ + rhs) % P;
			return lhs;
		}

		public static ModInt operator -(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = (P + lhs.value_ - rhs.value_) % P;
			return lhs;
		}
		public static ModInt operator -(long lhs, ModInt rhs)
		{
			rhs.value_ = (P + lhs - rhs.value_) % P;
			return rhs;
		}
		public static ModInt operator -(ModInt lhs, long rhs)
		{
			lhs.value_ = (P + lhs.value_ - rhs) % P;
			return lhs;
		}

		public static ModInt operator *(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = lhs.value_ * rhs.value_ % P;
			return lhs;
		}
		public static ModInt operator *(long lhs, ModInt rhs)
		{
			rhs.value_ = lhs * rhs.value_ % P;
			return rhs;
		}
		public static ModInt operator *(ModInt lhs, long rhs)
		{
			lhs.value_ = lhs.value_ * rhs % P;
			return lhs;
		}

		public static ModInt operator /(ModInt lhs, ModInt rhs)
		{
			long exp = P - 2;
			while (exp > 0) {
				if (exp % 2 > 0) {
					lhs *= rhs;
				}

				rhs *= rhs;
				exp /= 2;
			}

			return lhs;
		}

		public static implicit operator ModInt(long n) => new ModInt(n, true);
		public long ToLong() => value_;
		public override string ToString() => value_.ToString();
	}

	public static class Helper
	{
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMin<T>(ref T target, T value) where T : IComparable<T>
			=> target = target.CompareTo(value) > 0 ? value : target;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMin<T>(ref T target, T value, Action<T> onUpdated) where T : IComparable<T>
		{
			if (target.CompareTo(value) > 0) {
				target = value;
				onUpdated(value);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMax<T>(ref T target, T value) where T : IComparable<T>
			=> target = target.CompareTo(value) < 0 ? value : target;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMax<T>(ref T target, T value, Action<T> onUpdated) where T : IComparable<T>
		{
			if (target.CompareTo(value) < 0) {
				target = value;
				onUpdated(value);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[] Array<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToArray();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static List<T> List<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToList();

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,] Array2<T>(int n, int m, T init)
			where T : struct
		{
			var array = new T[n, m];
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					array[i, j] = init;
				}
			}

			return array;
		}
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,] Array2<T>(int n, int m, Func<int, int, T> initializer)
		{
			var array = new T[n, m];
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					array[i, j] = initializer(i, j);
				}
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,] Array3<T>(int n1, int n2, int n3, T init)
			where T : struct
		{
			var array = new T[n1, n2, n3];
			for (int i1 = 0; i1 < n1; i1++) {
				for (int i2 = 0; i2 < n2; i2++) {
					for (int i3 = 0; i3 < n3; i3++) {
						array[i1, i2, i3] = init;
					}
				}
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,,] Array4<T>(int n1, int n2, int n3, int n4, T init)
			where T : struct
		{
			var array = new T[n1, n2, n3, n4];
			for (int i1 = 0; i1 < n1; i1++) {
				for (int i2 = 0; i2 < n2; i2++) {
					for (int i3 = 0; i3 < n3; i3++) {
						for (int i4 = 0; i4 < n4; i4++) {
							array[i1, i2, i3, i4] = init;
						}
					}
				}
			}

			return array;
		}

		private static readonly int[] delta4_ = { 1, 0, -1, 0, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void DoAt4(int i, int j, int imax, int jmax, Action<int, int> action)
		{
			for (int dn = 0; dn < 4; dn++) {
				int d4i = i + delta4_[dn];
				int d4j = j + delta4_[dn + 1];
				if ((uint)d4i < (uint)imax && (uint)d4j < (uint)jmax) {
					action(d4i, d4j);
				}
			}
		}

		private static readonly int[] delta8_ = { 1, 0, -1, 0, 1, 1, -1, -1, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void DoAt8(int i, int j, int imax, int jmax, Action<int, int> action)
		{
			for (int dn = 0; dn < 8; dn++) {
				int d8i = i + delta8_[dn];
				int d8j = j + delta8_[dn + 1];
				if ((uint)d8i < (uint)imax && (uint)d8j < (uint)jmax) {
					action(d8i, d8j);
				}
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void ForEachSubBit(int bit, Action<int> action)
		{
			for (int sub = bit; sub >= 0; sub--) {
				sub &= bit;
				action(sub);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string Reverse(string src)
		{
			var chars = src.ToCharArray();
			for (int i = 0, j = chars.Length - 1; i < j; i++, j--) {
				var tmp = chars[i];
				chars[i] = chars[j];
				chars[j] = tmp;
			}
			return new string(chars);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string Join<T>(this IEnumerable<T> values, string separator = "")
			=> string.Join(separator, values);
	}

	public class Scanner : IDisposable
	{
		private const int BUFFER_SIZE = 1024;
		private const int ASCII_CHAR_BEGIN = 33;
		private const int ASCII_CHAR_END = 126;
		private readonly string filePath_;
		private readonly Stream stream_;
		private readonly byte[] buf_ = new byte[BUFFER_SIZE];
		private int length_ = 0;
		private int index_ = 0;
		private bool isEof_ = false;

		public Scanner(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				stream_ = Console.OpenStandardInput();
			} else {
				filePath_ = file;
				stream_ = new FileStream(file, FileMode.Open);
			}

			Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) {
				AutoFlush = false
			});
		}

		public void Dispose()
		{
			Console.Out.Flush();
			stream_.Dispose();
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public char Char()
		{
			byte b;
			do {
				b = Read();
			} while (b < ASCII_CHAR_BEGIN || ASCII_CHAR_END < b);

			return (char)b;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public string Next()
		{
			var sb = new StringBuilder();
			for (var b = Char(); b >= ASCII_CHAR_BEGIN && b <= ASCII_CHAR_END; b = (char)Read()) {
				sb.Append(b);
			}

			return sb.ToString();
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int() => (int)Long();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int(int offset) => Int() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int) Int2(int offset = 0)
			=> (Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int, int) Int3(int offset = 0)
			=> (Int(offset), Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int, int, int) Int4(int offset = 0)
			=> (Int(offset), Int(offset), Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int[] ArrayInt(int length, int offset = 0)
		{
			int[] Array = new int[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Int(offset);
			}
			return Array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long()
		{
			long ret = 0;
			byte b;
			bool ng = false;
			do {
				b = Read();
			} while (b != '-' && (b < '0' || '9' < b));

			if (b == '-') {
				ng = true;
				b = Read();
			}

			for (; true; b = Read()) {
				if (b < '0' || '9' < b) {
					return ng ? -ret : ret;
				} else {
					ret = ret * 10 + b - '0';
				}
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long(long offset) => Long() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long) Long2(long offset = 0)
			=> (Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long, long) Long3(long offset = 0)
			=> (Long(offset), Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long, long, long) Long4(long offset = 0)
			=> (Long(offset), Long(offset), Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long[] ArrayLong(int length, long offset = 0)
		{
			long[] Array = new long[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Long(offset);
			}
			return Array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double() => double.Parse(Next(), CultureInfo.InvariantCulture);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double(double offset) => Double() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double) Double2(double offset = 0)
			=> (Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double, double) Double3(double offset = 0)
			=> (Double(offset), Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double, double, double) Double4(double offset = 0)
			=> (Double(offset), Double(offset), Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double[] ArrayDouble(int length, double offset = 0)
		{
			double[] Array = new double[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Double(offset);
			}
			return Array;
		}

		private byte Read()
		{
			if (isEof_) {
				throw new EndOfStreamException();
			}

			if (index_ >= length_) {
				index_ = 0;
				if ((length_ = stream_.Read(buf_, 0, BUFFER_SIZE)) <= 0) {
					isEof_ = true;
					return 0;
				}
			}

			return buf_[index_++];
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}