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++ (GCC 9.2.1) (judge verdict: Accepted)

///////////////////////////////////////////////////////////////////////////////

#include <bits/stdc++.h>
#include <unistd.h>
#include <time.h>
#include <sys/time.h>

using namespace std;

///////////////////////////////////////////////////////////////////////////////

#define DEBUG 0

#define pb push_back
#define V vector
#define M unordered_map
#define S static

#define rep(i,n) for(ll i=0LL; i<n; ++i)
#define srep(i,s,n) for(ll i=s; i<n; ++i)
#define rrep(i,n) for(ll i=n-1LL; i>=0LL; --i)
#define ALL(a) (a).begin(),(a).end()
#define RALL(a) (a).rbegin(),(a).rend()
#define CIN(x) do { \
        assert(!cin.eof()); \
        cin >> x; \
        assert(!cin.fail()); \
} while(0);
#if DEBUG
#define debug_print(...) _debug_print(__VA_ARGS__)
#define debug_printf(...) printf(__VA_ARGS__)
#define debug_print_time _debug_print_time
#else // DEBUG
#define debug_print(...)
#define debug_printf(...)
#define debug_print_time
#endif // DEBUG

typedef long long ll;
typedef unsigned long long ull;
typedef tuple<ll, ll> t2;
typedef tuple<ll, ll, ll> t3;
typedef tuple<ll, ll, ll, ll> t4;
typedef tuple<ll, ll, ll, ll, ll> t5;
template <typename T>
using priority_queue_incr = priority_queue<T, V<T>, greater<T> >;

///////////////////////////////////////////////////////////////////////////////

template <typename TT, typename T>
T get_m(M<TT, T> &m, TT k, T default_value)
{
    if (m.find(k) == m.end()) return m[k] = default_value;
    return m[k];
}

template <typename TT, typename T>
void incr_m(M<TT, T> &m, TT k)
{
    if (m.find(k) == m.end()) m[k] = 0;
    m[k]++;
}

template <typename TT, typename T>
void incr_m(map<TT, T> &m, TT k)
{
    if (m.find(k) == m.end()) m[k] = 0;
    m[k]++;
}

void llin(ll &a)
{
    CIN(a);
}

void llinl1(V<ll> &v, ll count)
{
    for (ll i = 0LL; i < count; ++i) {
        ll a;
        CIN(a);
        v.push_back(a);
    }
}

void llinl2(V<t2> &v, ll count)
{
    for (ll i = 0LL; i < count; ++i) {
        ll a, b;
        CIN(a >> b);
        v.push_back(t2(a, b));
    }
}

void llinl3(V<t3> &v, ll count)
{
    for (ll i = 0LL; i < count; ++i) {
        ll a, b, c;
        CIN(a >> b >> c);
        v.push_back(t3(a, b, c));
    }
}

void llinl4(V<t4> &v, ll count)
{
    for (ll i = 0LL; i < count; ++i) {
        ll a, b, c, d;
        CIN(a >> b >> c >> d);
        v.push_back(t4(a, b, c, d));
    }
}

void llina(V<ll> &v, ll count)
{
    llinl1(v, count);
}

template <typename T>
void sort(V<T> &v)
{
    sort(v.begin(), v.end());
}

template <typename T>
void sort_reverse(V<T> &v)
{
    sort(v.begin(), v.end(), greater<T>());
}

template <typename T>
void _debug_print(T x)
{
    cout << x << endl;
}

void _debug_print(const t2 &x)
{
    ll x1 = get<0>(x);
    ll x2 = get<1>(x);
    cout << "-- " << x1 << " -- " << x2 << endl;
}

void _debug_print(const t3 &x)
{
    ll x1 = get<0>(x);
    ll x2 = get<1>(x);
    ll x3 = get<2>(x);
    cout << "-- " << x1 << " -- " << x2 << " -- " << x3 << endl;
}

void _debug_print(const t4 &x)
{
    ll x1 = get<0>(x);
    ll x2 = get<1>(x);
    ll x3 = get<2>(x);
    ll x4 = get<3>(x);
    cout << "-- " << x1 << " -- " << x2
         << " -- " << x3 << " -- " << x4 << endl;
}

template <typename T>
void _debug_print(T xarray[], ll n)
{
    rep (i, n) _debug_print(xarray[i]);
}

template <typename T>
void _debug_print(const V<T> &xlist)
{
    for (auto x : xlist) {
        cout << "-- ";
        _debug_print(x);
    }
}

template <typename T>
void _debug_print(const set<T> &xset)
{
    for (auto x : xset) {
        cout << "-- ";
        _debug_print(x);
    }
}

template <typename TT, typename T>
void _debug_print(const M<TT, T> &xlist)
{
    for (auto x : xlist) {
        TT k = x.first;
        T v = x.second;
        cout << "====" << endl;
        cout << "K=";
        _debug_print(k);
        cout << "V=";
        _debug_print(v);
    }
}

template <typename TT, typename T>
void _debug_print(const map<TT, T> &xlist)
{
    for (auto x : xlist) {
        TT k = x.first;
        T v = x.second;
        cout << "====" << endl;
        cout << "K=";
        _debug_print(k);
        cout << "V=";
        _debug_print(v);
    }
}

///////////////////////////////////////////////////////////////////////////////

void _main();
int main()
{
    cout << setprecision(12);
#if !DEBUG
    ios::sync_with_stdio(false);
    cin.tie(0);
#endif

    _main();
    return 0;
}









struct MaxFlow {
    V<ll> *links;
    ll **capacities;
    ll *idxs;
    ll nodes;

    MaxFlow(ll nodes) {
        // i == 0       --> S
        // i == nodes+1 --> T
        links = new V<ll>[nodes+2LL];
        capacities = new ll*[nodes+2LL];
        rep (i, nodes+2LL) {
            capacities[i] = new ll[nodes+2LL];
            memset(capacities[i], 0, (nodes+2LL) * sizeof(ll));
        }
        idxs = new ll[nodes+2LL];
        memset(idxs, 0, (nodes+2LL) * sizeof(ll));
        this->nodes = nodes;
    }

    void add_link(ll a, ll b, ll capacity) {
        links[a].pb(b);
        links[b].pb(a);
        capacities[a][b] = capacity;
        capacities[b][a] = 0LL;
    }

    bool find_path(deque<ll> &path, set<ll> &checked, ll v) {
        if (checked.find(v) != checked.end()) return false;

        checked.insert(v);
        path.push_back(v);
        if (v == nodes+1LL) return true;

        rep (xxx, links[v].size()) {
            ll nowidx = idxs[v]++;
            nowidx %= links[v].size();
            ll nv = links[v][nowidx];
            if (capacities[v][nv] == 0) continue;
            if (find_path(path, checked, nv)) return true;
        }

        // checked.erase(v);
        path.pop_back();
        return false;
    }

    ll solve(void) {
        ll ret = 0LL;

        for ( ; ; ) {

            set<ll> checked;
            deque<ll> foundq;
            if (!find_path(foundq, checked, 0)) break;

            V<ll> found;
            while (foundq.size()) {
                ll v = foundq.front(); foundq.pop_front();
                found.pb(v);
            }

            ll flowcount = 1e18;
            rep (i, (ll)found.size()-1LL) {
                ll src = found[i];
                ll dst = found[i+1];
                flowcount = min(flowcount, capacities[src][dst]);
            }
            assert(flowcount > 0LL);

            rep (i, (ll)found.size()-1LL) {
                ll src = found[i];
                ll dst = found[i+1];
                capacities[src][dst] -= flowcount;
                capacities[dst][src] += flowcount;
            }
            ret += flowcount;

        }

        return ret;
    }
};













bool valid(const V<string> &grid, ll h, ll w, ll hh, ll ww)
{
    if (hh < 0) return false;
    if (hh >= h) return false;
    if (ww < 0) return false;
    if (ww >= w) return false;
    if (grid[hh][ww] == '#') return false;
    return true;
}

void _main()
{
    ll h; llin(h);
    ll w; llin(w);
    S V<string> grid;
    rep (i, h) {
        string tmp; CIN(tmp);
        grid.pb(tmp);
    }

    ll nodes = 10005;
    S MaxFlow flow(nodes);

    rep (hh, h) rep (ww, w) {
        if ((hh+ww) & 1LL) flow.add_link(hh*w + ww + 1, nodes+1, 1);
        else flow.add_link(0, hh*w + ww + 1, 1);
    }

    ll diff_h[4] = {-1, 0, 0, 1};
    ll diff_w[4] = {0, -1, 1, 0};
    V<t2> paths;

    rep (hh, h) rep (ww, w) {
        if ((hh+ww) & 1LL) continue;
        if (grid[hh][ww] == '#') continue;
        rep (i, 4) {
            ll hhh = hh + diff_h[i];
            ll www = ww + diff_w[i];
            if (valid(grid, h, w, hhh, www)) {
                flow.add_link(hh*w + ww + 1, hhh*w + www + 1, 1);
                paths.pb(t2(hh*w + ww + 1, hhh*w + www + 1));
            }
        }
    }

    ll anscnt = flow.solve();

    for (auto path : paths) {
        ll x1 = get<0>(path);
        ll x2 = get<1>(path);
        debug_printf("--------\n");
        if (flow.capacities[x1][x2]) continue;
        debug_printf("---- f=%lld\n", flow.capacities[x1][x2]);
        x1--;
        x2--;
        ll h1 = x1 / w;
        ll w1 = x1 % w;
        ll h2 = x2 / w;
        ll w2 = x2 % w;
        debug_printf("h1=%lld h2=%lld w1=%lld w2=%lld\n", h1, h2, w1, w2);
        if (h1 == h2) {
            ll wmin = min(w1, w2);
            ll wmax = max(w1, w2);
            grid[h1][wmin] = '>';
            grid[h1][wmax] = '<';
        }
        else {
            ll hmin = min(h1, h2);
            ll hmax = max(h1, h2);
            grid[hmin][w1] = 'v';
            grid[hmax][w1] = '^';
        }
    }

    debug_printf("----- ans\n");

    cout << anscnt << endl;
    rep (hh, h) {
        rep (ww, w) {
            cout << grid[hh][ww];
        }
        cout << endl;
    }
}

///////////////////////////////////////////////////////////////////////////////