Problem:
Score : 600 points

Problem Statement

There are N points in a two-dimensional plane. The initial coordinates of the i-th point are (x_i, y_i). Now, each point starts moving at a speed of 1 per second, in a direction parallel to the x- or y- axis. You are given a character d_i that represents the specific direction in which the i-th point moves, as follows:

If d_i = R, the i-th point moves in the positive x direction;

If d_i = L, the i-th point moves in the negative x direction;

If d_i = U, the i-th point moves in the positive y direction;

If d_i = D, the i-th point moves in the negative y direction.

You can stop all the points at some moment of your choice after they start moving (including the moment they start moving).
Then, let x_{max} and x_{min} be the maximum and minimum among the x-coordinates of the N points, respectively. Similarly, let y_{max} and y_{min} be the maximum and minimum among the y-coordinates of the N points, respectively.

Find the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}) and print it.

Constraints

1 \leq N \leq 10^5

-10^8 \leq x_i,\ y_i \leq 10^8

x_i and y_i are integers.

d_i is R, L, U, or D.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 d_1
x_2 y_2 d_2
.
.
.
x_N y_N d_N

Output

Print the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}).

The output will be considered correct when its absolute or relative error from the judge's output is at most 10^{-9}.

Sample Input 1

2
0 3 D
3 0 L

Sample Output 1

0

After three seconds, the two points will meet at the origin. The value in question will be 0 at that moment.

Sample Input 2

5
-7 -10 U
7 -6 U
-8 7 D
-3 3 D
0 -6 R

Sample Output 2

97.5

The answer may not be an integer.

Sample Input 3

20
6 -10 R
-4 -9 U
9 6 D
-3 -2 R
0 7 D
4 5 D
10 -10 U
-1 -8 U
10 -6 D
8 -5 U
6 4 D
0 3 D
7 9 R
9 -4 R
3 10 D
1 9 U
1 -6 U
9 -8 R
6 7 D
7 -3 D

Sample Output 3

273

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <algorithm>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <limits>
#include <map>
#include <queue>
#include <set>
#include <tuple>
#include <vector>

using namespace std;

#define rep(i, n) for (int64_t i = 0; i < (int64_t)(n); i++)
#define irep(i, n) for (int64_t i = 0; i <= (int64_t)(n); i++)
#define rrep(i, n) for (int64_t i = (n)-1; i >= 0; i--)
#define rirep(i, n) for (int64_t i = n; i >= 0; i--)

using Event = tuple<int64_t, int64_t>;
using AEvent = tuple<int64_t, int64_t, char>;

bool byTime(const AEvent& lh, const AEvent& rh)
{
    return get<0>(lh) < get<0>(rh);
}

vector<Event> solve(int64_t x0, int64_t x1, int64_t x2)
{
    int64_t t01 = 2 * (x0 - x1);
    int64_t t12 = 2 * (x1 - x2);
    int64_t t20 = x0 - x2;

    vector<Event> events;
    events.push_back(make_pair(0, -1));

    if (x0 + x2 >= 2 * x1) {
        events.push_back(make_pair(max(0L, t20), 1));
    } else {
        events.push_back(make_pair(max(0L, t01), 0));
        events.push_back(make_pair(max(0L, t12), 1));
    }

    return events;
}

vector<Event> mergeEvents(const vector<Event>& ub, const vector<Event>& lb)
{
    vector<AEvent> xev;

    for (auto& p : ub) {
        int64_t time;
        int64_t slope;
        tie(time, slope) = p;
        xev.push_back(make_tuple(time, slope, '+'));
    }

    for (auto& p : lb) {
        int64_t time;
        int64_t slope;
        tie(time, slope) = p;
        xev.push_back(make_tuple(time, -slope, '-'));
    }

    stable_sort(xev.begin(), xev.end(), byTime);

    vector<Event> result;
    int64_t smax = 0, smin = 0;
    for (int i = 0; i < xev.size(); i++) {
        int64_t time;
        int64_t slope;
        char axis;
        tie(time, slope, axis) = xev[i];

        if (axis == '+') {
            smax = slope;
        } else {
            smin = slope;
        }
        result.push_back(make_tuple(time, smax - smin));
    }

    return result;
}

int main()
{
    int n;
    cin >> n;

    int64_t xumax, xumin, xcmax, xcmin, xdmax, xdmin;
    int64_t yumax, yumin, ycmax, ycmin, ydmax, ydmin;
    xumax = xcmax = xdmax = yumax = ycmax = ydmax = -2e8;
    xumin = xcmin = xdmin = yumin = ycmin = ydmin = 2e8;

    rep(i, n)
    {
        int64_t x, y;
        char d;
        cin >> x >> y >> d;
        if (d == 'R' || d == 'L') {
            if (d == 'R') {
                xumax = max(xumax, x);
                xumin = min(xumin, x);
            }
            if (d == 'L') {
                xdmax = max(xdmax, x);
                xdmin = min(xdmin, x);
            }
            ycmax = max(ycmax, y);
            ycmin = min(ycmin, y);
        }
        if (d == 'U' || d == 'D') {
            if (d == 'U') {
                yumax = max(yumax, y);
                yumin = min(yumin, y);
            }
            if (d == 'D') {
                ydmax = max(ydmax, y);
                ydmin = min(ydmin, y);
            }
            xcmax = max(xcmax, x);
            xcmin = min(xcmin, x);
        }
    }

    auto xevmax = solve(xdmax, xcmax, xumax);
    auto xevmin = solve(-xumin, -xcmin, -xdmin);
    auto xev = mergeEvents(xevmax, xevmin);
    auto yevmax = solve(ydmax, ycmax, yumax);
    auto yevmin = solve(-yumin, -ycmin, -ydmin);
    auto yev = mergeEvents(yevmax, yevmin);

    rep(i, xev.size())
    {
        int64_t time, slope;
        tie(time, slope) = xev[i];
    }
    rep(i, yev.size())
    {
        int64_t time, slope;
        tie(time, slope) = yev[i];
    }

    vector<tuple<int64_t, int64_t, char>> ev;
    rep(i, xev.size())
    {
        int64_t time, slope;
        tie(time, slope) = xev[i];
        ev.push_back(make_tuple(time, slope, 'x'));
    }
    rep(i, yev.size())
    {
        int64_t time, slope;
        tie(time, slope) = yev[i];
        ev.push_back(make_tuple(time, slope, 'y'));
    }

    stable_sort(ev.begin(), ev.end(), byTime);

    double result = 1e18;
    int64_t ptime = 0;
    double xw = max(xumax, max(xdmax, xcmax)) - min(xumin, min(xcmin, xdmin));
    double yw = max(yumax, max(ydmax, ycmax)) - min(yumin, min(ycmin, ydmin));
    int64_t xslope = 0, yslope = 0;
    rep(i, ev.size())
    {
        int64_t time, slope;
        char axis;
        tie(time, slope, axis) = ev[i];

        if (ptime != time) {
            double ab = xslope * yslope;
            double t = (time - ptime) / 2.0;
            double nx = xw + xslope * t, ny = yw + yslope * t;
            result = min(result, min(xw * yw, nx * ny));
            xw = nx;
            yw = ny;
        }

        ptime = time;
        if (axis == 'x') {
            xslope = slope;
        } else {
            yslope = slope;
        }

        //cout << "(" << time << ", " << slope << ", " << axis << ") " << xw << " " << yw << endl;
    }

    cout << result << endl;
    return 0;
}