Problem:
Score : 700 points

Problem Statement

We have a board with an H \times W grid.
Each square in the grid is painted in black or white. The square at the i-th row from the top and j-th column from the left is black if the j-th character in S_i is #, and white if that character is ..

Snuke can perform the following operation on the grid any number of times:

Select a row or column in the grid, and invert the color of all the squares in that row or column (that is, black squares become white and vice versa).

Then, Snuke draws a rectangle along grid lines. Here, all the squares contained in the rectangle must be painted in black.

Find the maximum possible area of Snuke's rectangle when the operation is performed optimally.

Constraints

2 \leq H \leq 2000

2 \leq W \leq 2000

|S_i| = W

S_i consists of # and ..

Input

Input is given from Standard Input in the following format:

H W
S_1
S_2
:
S_H

Output

Print the maximum possible area of Snuke's rectangle.

Sample Input 1

3 3
..#
##.
.#.

Sample Output 1

6

If the first row from the top and the third column from the left are inverted, a 2 \times 3 rectangle can be drawn, as shown below:

Sample Input 2

4 4
....
....
....
....

Sample Output 2

16

Sample Input 3

10 8
##...#.#
##...#.#
..###.#.
#.##.#.#
.#..#.#.
..##.#.#
##.#.#..
...#.#..
###.#.##
###..###

Sample Output 3

27

Solution in C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

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

#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 _debug_print_time(const char *prefix)
{
    struct timeval tv;
    gettimeofday(&tv, NULL);
#if DEBUG
    struct tm *tm = localtime(&tv.tv_sec);
#endif
    debug_printf("-- %s %02d:%02d:%02d.%06ld\n",
                 prefix, tm->tm_hour, tm->tm_min, tm->tm_sec, tv.tv_usec);
}

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

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

    _main();
    return 0;
}

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

    S ll hs[4] = {0, 0, 1, 1};
    S ll ws[4] = {0, 1, 0, 1};
    S bool ngs[2010][2010];
    memset(ngs, 0, sizeof(ngs));
    rep (hh, h-1) rep (ww, w-1) {
        ll cnt = 0;
        rep (i, 4) cnt += (grid[hh+hs[i]][ww+ws[i]] == '#');
        if (cnt & 1LL) ngs[hh][ww] = true;
    }

#if DEBUG
    debug_printf("---- ngs\n");
    rep (hh, h-1) {
        rep (ww, w-1) {
            debug_printf("%d ", ngs[hh][ww]);
        }
        debug_printf("\n");
    }
#endif // DEBUG

    S V<ll> ptss_wh[2010];
    rep (hh, h) rep (ww, w) if (ngs[hh][ww]) ptss_wh[ww].pb(hh);

    S V<ll> wait_w[2010][2010];
    rep (ww, w) {
        ll idx = 0;
        rep (hs, h) {
            if (idx >= ptss_wh[ww].size()) break;
            ll hh = ptss_wh[ww][idx];
            if (hs > hh) idx++;
            if (idx >= ptss_wh[ww].size()) break;
            hh = ptss_wh[ww][idx];
            wait_w[hs][hh].pb(ww);
        }
    }

    rep (hs, h) srep (hh, hs, h) {
        debug_printf("---- wait_w %lld %lld\n", hs, hh);
        debug_print(wait_w[hs][hh]);
    }

#if 0
    rep (hs, 0) {
        ll idx = 0;
        srep (hh, hs, h) {
            ll h0 = ptss_wh[hh][idx];
            wait_w[h0][hh].pb(ww);
            for (hh : ptss_wh[ww]) {
                ll h0 = ptss_wh[ww][i];
                ll h1 = ptss_wh[ww][i+1];
                srep (hh, h0, h1) pts_hw_wait[hh][].pb(ww);
            }
#if 0
            auto iter = lower_bound(ALL(ptss_wh[ww]), hs);
            if (iter == ptss_wh[ww].end()) continue;
            pts_hw_wait[*iter].pb(ww);
#endif
        }
    }
#endif

    ll ans = 0;
    ll id = 0;
    set<ll> ptsset_orig; ptsset_orig.insert(-1); ptsset_orig.insert(w-1);
    set<t2> m_lenset_orig; m_lenset_orig.insert(t2(-w, id++));




    cout << 0 << endl;
    return;




    rep (hs, h) {
        set<ll> ptsset = ptsset_orig;
        set<t2> m_lenset = m_lenset_orig;

        srep (he, hs+1, h+1) {
            auto ml = *(m_lenset.begin());
            ll len = -get<0>(ml);
            ans = max(ans, (he-hs) * len);

            for (auto ww : wait_w[hs][he-1]) {
                auto iter = ptsset.lower_bound(ww);
                ll w2 = *iter;
                iter--;
                ll w1 = *iter;
                auto to_delete = m_lenset.lower_bound(t2(-(w2-w1), -1));
                m_lenset.erase(to_delete);
                ptsset.insert(ww);
                m_lenset.insert(t2(-(ww-w1), id++));
                m_lenset.insert(t2(-(w2-ww), id++));
            }
        }
    }

    debug_printf("---- ans\n");
    cout << ans << endl;
}

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