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.IO;
using System.Linq;
using System.Net;

namespace Tasks
{
    public class D
    {
        static void Main(string[] args)
        {
            var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            Console.SetOut(sw);
            Solve();
            Console.Out.Flush();
        }

        public static void Solve()
        {
            var (N, M) = Scanner.Scan<int, int>();
            var G = new char[N][].Select(x => Scanner.Scan<string>().ToCharArray()).ToArray();
            var mfg = new MaxFlowGraph(N * M + 2);
            var s = N * M;
            var t = N * M + 1;

            for (var i = 0; i < N; i++)
            {
                for (var j = 0; j < M; j++)
                {
                    if (G[i][j] == '#') continue;
                    var v = i * M + j;
                    if ((i + j) % 2 == 0) mfg.AddEdge(s, v, 1);
                    else mfg.AddEdge(v, t, 1);
                }
            }

            var di = new[] { -1, 0, 1, 0 };
            var dj = new[] { 0, -1, 0, 1 };
            for (var i = 0; i < N; i++)
            {
                for (var j = 0; j < M; j++)
                {
                    if ((i + j) % 2 == 1 || G[i][j] == '#') continue;
                    var v0 = i * M + j;
                    for (var k = 0; k < 4; k++)
                    {
                        var ci = i + di[k];
                        var cj = j + dj[k];
                        if (ci < 0 || N <= ci) continue;
                        if (cj < 0 || M <= cj) continue;
                        if (G[ci][cj] != '.') continue;
                        var v1 = ci * M + cj;
                        mfg.AddEdge(v0, v1, 1);
                    }
                }
            }

            Console.WriteLine(mfg.FlowOf(s, t));

            foreach (var edge in mfg.GetEdges().Where(x => x.From != s && x.To != t && x.Flow != 0))
            {
                var i0 = edge.From / M;
                var j0 = edge.From % M;
                var i1 = edge.To / M;
                var j1 = edge.To % M;

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

            Console.WriteLine(string.Join("\n", G.Select(x => new string(x))));
        }

        public class MaxFlowGraph
        {
            public readonly struct Edge
            {
                public readonly int From;
                public readonly int To;
                public readonly long Cap;
                public readonly long Flow;

                public Edge(int from, int to, long cap, long flow)
                {
                    From = from;
                    To = to;
                    Cap = cap;
                    Flow = flow;
                }
            }

            private readonly struct InternalEdge
            {
                public readonly int To;
                public readonly int Rev;
                public readonly long Cap;

                public InternalEdge(int to, int rev, long cap)
                {
                    To = to;
                    Rev = rev;
                    Cap = cap;
                }
            }

            private readonly int _n;
            private readonly List<InternalEdge>[] _edges;
            private readonly List<(int X, int Y)> _pos;

            public MaxFlowGraph(int n = 0)
            {
                _n = n;
                _edges = new List<InternalEdge>[n].Select(x => new List<InternalEdge>()).ToArray();
                _pos = new List<(int X, int Y)>();
            }

            public int AddEdge(int from, int to, long cap)
            {
                if (from < 0 || _n <= from) throw new IndexOutOfRangeException(nameof(from));
                if (to < 0 || _n <= to) throw new IndexOutOfRangeException(nameof(to));
                if (cap < 0) throw new ArgumentException(nameof(cap));
                var m = _pos.Count;
                _pos.Add((from, _edges[from].Count));
                _edges[from].Add(new InternalEdge(to, _edges[to].Count, cap));
                _edges[to].Add(new InternalEdge(from, _edges[from].Count - 1, 0));
                return m;
            }

            public Edge GetEdge(int i)
            {
                var m = _pos.Count;
                if (i < 0 || m <= i) throw new IndexOutOfRangeException(nameof(i));
                var e = _edges[_pos[i].X][_pos[i].Y];
                var re = _edges[e.To][e.Rev];
                return new Edge(_pos[i].X, e.To, e.Cap + re.Cap, re.Cap);
            }

            public IEnumerable<Edge> GetEdges()
            {
                for (var i = 0; i < _pos.Count; i++) yield return GetEdge(i);
            }

            public void ChangeEdge(int i, long newCap, long newFlow)
            {
                var m = _pos.Count;
                if (i < 0 || m <= i) throw new IndexOutOfRangeException(nameof(i));
                if (newFlow < 0 || newCap < newFlow) throw new ArgumentException();
                var e = _edges[_pos[i].X][_pos[i].Y];
                var re = _edges[e.To][e.Rev];
                _edges[_pos[i].X][_pos[i].Y] = new InternalEdge(e.To, e.Rev, newCap - newFlow);
                _edges[e.To][e.Rev] = new InternalEdge(re.To, re.Rev, newFlow);
            }

            public long FlowOf(int s, int t) => FlowOf(s, t, long.MaxValue);

            public long FlowOf(int s, int t, long flowLimit)
            {
                if (s < 0 || _n <= s) throw new IndexOutOfRangeException(nameof(s));
                if (t < 0 || _n <= t) throw new IndexOutOfRangeException(nameof(t));
                var queue = new Queue<int>();
                int[] depth;
                int[] iter;

                void BFS()
                {
                    depth = Enumerable.Repeat(-1, _n).ToArray();
                    depth[s] = 0;
                    queue.Clear();
                    queue.Enqueue(s);
                    while (queue.Any())
                    {
                        var v = queue.Dequeue();
                        foreach (var edge in _edges[v].Where(edge => edge.Cap != 0 && depth[edge.To] < 0))
                        {
                            depth[edge.To] = depth[v] + 1;
                            if (edge.To == t) return;
                            queue.Enqueue(edge.To);
                        }
                    }
                }

                long DFS(int v, long up)
                {
                    if (v == s) return up;
                    var ret = 0L;
                    var dv = depth[v];
                    for (var i = iter[v]; i < _edges[v].Count; i++)
                    {
                        var e = _edges[v][i];
                        if (dv <= depth[e.To] || _edges[e.To][e.Rev].Cap == 0) continue;
                        var d = DFS(e.To, System.Math.Min(up - ret, _edges[e.To][e.Rev].Cap));
                        if (d <= 0) continue;
                        e = _edges[v][i];
                        _edges[v][i] = new InternalEdge(e.To, e.Rev, e.Cap + d);
                        var re = _edges[e.To][e.Rev];
                        _edges[e.To][e.Rev] = new InternalEdge(re.To, re.Rev, re.Cap - d);
                        ret += d;
                        if (ret == up) break;
                    }

                    return ret;
                }

                var flow = 0L;
                while (flow < flowLimit)
                {
                    BFS();
                    if (depth[t] == -1) break;
                    iter = new int[_n];
                    while (flow < flowLimit)
                    {
                        var f = DFS(t, flowLimit - flow);
                        if (f == 0) break;
                        flow += f;
                    }
                }

                return flow;
            }

            public IEnumerable<bool> MinCut(int s)
            {
                var visited = new bool[_n];
                var queue = new Queue<int>();
                queue.Enqueue(s);
                while (queue.Any())
                {
                    var p = queue.Dequeue();
                    visited[p] = true;
                    foreach (var edge in _edges[p].Where(edge => edge.Cap > 0 && !visited[edge.To]))
                    {
                        visited[edge.To] = true;
                        queue.Enqueue(edge.To);
                    }
                }

                return visited;
            }
        }

        public static class Scanner
        {
            private static Queue<string> queue = new Queue<string>();
            public static T Next<T>()
            {
                if (!queue.Any()) foreach (var item in Console.ReadLine().Trim().Split(" ")) queue.Enqueue(item);
                return (T)Convert.ChangeType(queue.Dequeue(), typeof(T));
            }
            public static T Scan<T>() => Next<T>();
            public static (T1, T2) Scan<T1, T2>() => (Next<T1>(), Next<T2>());
            public static (T1, T2, T3) Scan<T1, T2, T3>() => (Next<T1>(), Next<T2>(), Next<T3>());
            public static (T1, T2, T3, T4) Scan<T1, T2, T3, T4>() => (Next<T1>(), Next<T2>(), Next<T3>(), Next<T4>());
            public static (T1, T2, T3, T4, T5) Scan<T1, T2, T3, T4, T5>() => (Next<T1>(), Next<T2>(), Next<T3>(), Next<T4>(), Next<T5>());
            public static IEnumerable<T> ScanEnumerable<T>() => Console.ReadLine().Trim().Split(" ").Select(x => (T)Convert.ChangeType(x, typeof(T)));
        }
    }
}