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 Kotlin (1.3.71) (judge verdict: Accepted)

import java.io.BufferedReader
import java.io.InputStreamReader
import java.io.PrintWriter
import java.util.*
import kotlin.collections.ArrayList

val br = BufferedReader(InputStreamReader(System.`in`))
val out = PrintWriter(System.out)

fun main() {
    br.use { out.use {
        solve()
        out.flush()
    } }
}

// refered: https://atcoder.jp/contests/practice2/submissions/16646346
fun solve() {
    val (n, m) = readIntegerList()
    var g = Array(n) { br.readLine().toCharArray() }
    val mf = MaxFlow(n * m + 2)
    val s = n * m
    val t = s + 1
    for (i in 0 until n) {
        for (j in 0 until m) {
            if (g[i][j] == '#') continue
            if (((i xor j) and 1) == 0) {
                mf.addEdge(s, i * m + j, 1);
            } else {
                mf.addEdge(i * m + j, t, 1);
            }
        }
    }
    for (i in 0 until n) {
        for (j in (i and 1) until m step 2) {
            if (g[i][j] == '#') continue;
            if (j - 1 >= 0 && g[i][j - 1] == '.') {
                mf.addEdge(i * m + j, i * m + (j - 1), 1);
            }
            if (j + 1 < m && g[i][j + 1] == '.') {
                mf.addEdge(i * m + j, i * m + (j + 1), 1);
            }
            if (i - 1 >= 0 && g[i - 1][j] == '.') {
                mf.addEdge(i * m + j, (i - 1) * m + j, 1);
            }
            if (i + 1 < n && g[i + 1][j] == '.') {
                mf.addEdge(i * m + j, (i + 1) * m + j, 1);
            }
        }
    }
    val ans = mf.fordFulkersonMaxFlow(s, t).toInt()
    var cnt = 0;
    for (e in mf.edges) {
        if (e.flow == 0L) continue;
        val u = e.from
        val v = e.to
        if (u == s || v == t) continue;
        val ui = u / m
        val uj = u % m;
        val vi = v / m
        val vj = v % m;
        if (g[ui][uj] != '.' || g[vi][vj] != '.') {
            throw AssertionError();
        }
        if (ui == vi) {
            if (uj + 1 == vj) {
                g[ui][uj] = '>';
                g[vi][vj] = '<';
            } else if (uj == vj + 1) {
                g[ui][uj] = '<';
                g[vi][vj] = '>';
            } else {
                throw AssertionError();
            }
        } else if (uj == vj) {
            if (ui + 1 == vi) {
                g[ui][uj] = 'v';
                g[vi][vj] = '^';
            } else if (ui == vi + 1) {
                g[ui][uj] = '^';
                g[vi][vj] = 'v';
            } else {
                throw AssertionError();
            }
        } else {
            throw AssertionError();
        }
        cnt++;
    }
    if (ans != cnt) throw AssertionError();
    out.println(ans)
    for (gi in g) {
        out.println(gi)
    }
}

/**
 * https://github.com/da-louis/ac-library-kt/blob/master/src/main/kotlin/jp/atcoder/library/kotlin/maxFlow/MaxFlow.kt
 * 提出時の変更点edgesをpublicに変更
 *
 * convert from [AtCoderLibraryForJava - MaxFlow](https://github.com/NASU41/AtCoderLibraryForJava/blob/24160d880a5fc6d1caf9b95baa875e47fb568ef3/MaxFlow/MaxFlow.java)
 */
class MaxFlow(private val n: Int) {
    inner class CapEdge internal constructor(val from: Int, val to: Int, var cap: Long, val rev: Int) {
        val flow: Long
            get() = g[to][rev]!!.cap
    }

    private var m = 0
    val edges: ArrayList<CapEdge> = ArrayList()
    private val count: IntArray = IntArray(n)
    private val g: Array<Array<CapEdge?>> = Array(n) { emptyArray<CapEdge?>() }

    fun addEdge(from: Int, to: Int, cap: Long): Int {
        rangeCheck(from, 0, n)
        rangeCheck(to, 0, n)
        nonNegativeCheck(cap, "Capacity")
        val e = CapEdge(from, to, cap, count[to])
        count[from]++
        count[to]++
        edges.add(e)
        return m++
    }

    fun getEdge(i: Int): CapEdge {
        rangeCheck(i, 0, m)
        return edges[i]
    }

    fun changeEdge(i: Int, newCap: Long, newFlow: Long) {
        rangeCheck(i, 0, m)
        nonNegativeCheck(newCap, "Capacity")
        require(newFlow <= newCap) { String.format("Flow %d is greater than capacity %d.", newCap, newFlow) }
        val e = edges[i]
        val er = g[e.to][e.rev]
        e.cap = newCap - newFlow
        er!!.cap = newFlow
    }

    private fun buildGraph() {
        for (i in 0 until n) {
            g[i] = arrayOfNulls(count[i])
        }
        val idx = IntArray(n)
        for (e in edges) {
            g[e.to][idx[e.to]++] = CapEdge(e.to, e.from, 0, idx[e.from])
            g[e.from][idx[e.from]++] = e
        }
    }

    fun maxFlow(s: Int, t: Int): Long {
        return flow(s, t, INF)
    }

    fun flow(s: Int, t: Int, flowLimit: Long): Long {
        rangeCheck(s, 0, n)
        rangeCheck(t, 0, n)
        buildGraph()
        var flow: Long = 0
        val level = IntArray(n)
        val que = IntArray(n)
        val iter = IntArray(n)
        while (true) {
            java.util.Arrays.fill(level, -1)
            dinicBFS(s, t, level, que)
            if (level[t] < 0) return flow
            java.util.Arrays.fill(iter, 0)
            while (true) {
                val d = dinicDFS(t, s, flowLimit - flow, iter, level)
                if (d <= 0) break
                flow += d
            }
        }
    }

    private fun dinicBFS(s: Int, t: Int, level: IntArray, que: IntArray) {
        var hd = 0
        var tl = 0
        que[tl++] = s
        level[s] = 0
        while (tl > hd) {
            val u = que[hd++]
            for (e in g[u]) {
                val v = e!!.to
                if (e.cap <= 0 || level[v] >= 0) continue
                level[v] = level[u] + 1
                if (v == t) return
                que[tl++] = v
            }
        }
    }

    private fun dinicDFS(cur: Int, s: Int, f: Long, iter: IntArray, level: IntArray): Long {
        if (cur == s) return f
        var res: Long = 0
        while (iter[cur] < count[cur]) {
            val er = g[cur][iter[cur]++]
            val u = er!!.to
            val e = g[u][er.rev]
            if (level[u] >= level[cur] || e!!.cap <= 0) continue
            val d = dinicDFS(u, s, Math.min(f - res, e.cap), iter, level)
            if (d <= 0) continue
            e.cap -= d
            er.cap += d
            res += d
            if (res == f) break
        }
        return res
    }

    fun fordFulkersonMaxFlow(s: Int, t: Int): Long {
        return fordFulkersonFlow(s, t, INF)
    }

    fun fordFulkersonFlow(s: Int, t: Int, flowLimit: Long): Long {
        rangeCheck(s, 0, n)
        rangeCheck(t, 0, n)
        buildGraph()
        val used = BooleanArray(n)
        var flow: Long = 0
        while (true) {
            java.util.Arrays.fill(used, false)
            val f = fordFulkersonDFS(s, t, flowLimit - flow, used)
            if (f <= 0) return flow
            flow += f
        }
    }

    private fun fordFulkersonDFS(cur: Int, t: Int, f: Long, used: BooleanArray): Long {
        if (cur == t) return f
        used[cur] = true
        for (e in g[cur]) {
            if (used[e!!.to] || e.cap <= 0) continue
            val d = fordFulkersonDFS(e.to, t, Math.min(f, e.cap), used)
            if (d <= 0) continue
            e.cap -= d
            g[e.to][e.rev]!!.cap += d
            return d
        }
        return 0
    }

    fun minCut(s: Int): BooleanArray {
        rangeCheck(s, 0, n)
        val reachable = BooleanArray(n)
        val stack = IntArray(n)
        var ptr = 0
        stack[ptr++] = s
        reachable[s] = true
        while (ptr > 0) {
            val u = stack[--ptr]
            for (e in g[u]) {
                val v = e!!.to
                if (reachable[v] || e.cap <= 0) continue
                reachable[v] = true
                stack[ptr++] = v
            }
        }
        return reachable
    }

    private fun rangeCheck(i: Int, minInclusive: Int, maxExclusive: Int) {
        if (i < minInclusive || i >= maxExclusive) {
            throw IndexOutOfBoundsException(String.format("Index %d out of bounds for length %d", i, maxExclusive))
        }
    }

    private fun nonNegativeCheck(cap: Long, attribute: String) {
        require(cap >= 0) { String.format("%s %d is negative.", attribute, cap) }
    }

    companion object {
        private const val INF = Long.MAX_VALUE
    }
}

fun readInteger() = Integer.parseInt(br.readLine())
fun readLong() = java.lang.Long.parseLong(br.readLine())
fun readStringList() = br.readLine()!!.split(' ')
fun readIntegerList() = readStringList().map(Integer::parseInt)
fun readLongList() = readStringList().map(java.lang.Long::parseLong)
fun readDoubleList() = readStringList().map(java.lang.Double::parseDouble)