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 (Clang 3.8.0) (judge verdict: Wrong Answer)

#include <algorithm>
#include <iostream>
#include <iomanip>
#include <cstring>
#include <string>
#include <vector>
#include <random>
#include <bitset>
#include <queue>
#include <cmath>
#include <stack>
#include <set>
#include <map>
typedef long long ll;
using namespace std;
const ll MOD = 1000000007LL;
const double INF = 1000000000;

int main() {
    cin.sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);

    int n;
    cin >> n;

    double UpMin = -INF, UpMax = -INF, UpStay = -INF;
    double DownMin = INF, DownMax = INF, DownStay = INF;
    double LeftMin = INF, LeftMax = INF, LeftStay = INF;
    double RightMin = -INF, RightMax = -INF, RightStay = -INF;

    for (int i = 0; i < n; i++) {
        double x, y;
        char d;
        cin >> x >> y >> d;

        if (d == 'D') {
            UpMin = max(UpMin, y);
            DownMax = min(DownMax, y);
            LeftStay = min(LeftStay, x);
            RightStay = max(RightStay, x);
        } else if (d == 'U') {
            UpMax = max(UpMax, y);
            DownMin = min(DownMin, y);
            LeftStay = min(LeftStay, x);
            RightStay = max(RightStay, x);
        } else if (d == 'R') {
            UpStay = max(UpStay, y);
            DownStay = min(DownStay, y);
            LeftMin = min(LeftMin, x);
            RightMax = max(RightMax, x);
        } else if (d == 'L') {
            UpStay = max(UpStay, y);
            DownStay = min(DownStay, y);
            LeftMax = min(LeftMax, x);
            RightMin = max(RightMin, x);
        }
    }

    // ぶつかる時間のvector
    vector <double> vec(1, 0.0);

    // Up
    if (UpMin > -INF && UpStay > -INF) vec.push_back(UpMin - UpStay);
    if (UpMax > -INF && UpStay > -INF) vec.push_back(UpStay - UpMax);
    if (UpMin > -INF && UpMax > -INF)  vec.push_back((UpMin - UpMax) / 2);

    // Down
    if (DownMax < INF && DownStay < INF) vec.push_back(DownMax - DownStay);
    if (DownMin < INF && DownStay < INF) vec.push_back(DownStay - DownMin);
    if (DownMax < INF && DownMin < INF)  vec.push_back((DownMax - DownMin) / 2);

    // Right
    if (RightMin > -INF && RightStay > -INF) vec.push_back(RightMin - RightStay);
    if (RightMax > -INF && RightStay > -INF) vec.push_back(RightStay - RightMax);
    if (RightMin > -INF && RightMax > -INF)  vec.push_back((RightMin - RightMax) / 2);

    // Left
    if (LeftMax < INF && LeftStay < INF) vec.push_back(LeftMax - LeftStay);
    if (LeftMin < INF && LeftStay < INF) vec.push_back(LeftStay - LeftMin);
    if (LeftMax < INF && LeftMin < INF)  vec.push_back((LeftMax - LeftMin) / 2);

    vector <double> vec2;
    for (int i = 0; i < vec.size(); i++) {
        vec2.push_back(vec[i]);
        for (int j = 0; j < i; j++) {
            vec2.push_back((vec[i] + vec[j]) / 2.0);
        }
    }

    double ans = INF;
    for (double d : vec2) {
        double u = max({UpStay, UpMin - d, UpMax + d});
        double f = min({DownStay, DownMin + d, DownMax - d});
        double l = min({LeftStay, LeftMin + d, LeftMax - d});
        double r = max({RightStay, RightMin - d, RightMax + d});

        ans = min(ans, (u - f) * (r - l));
    }

    cout << ans << "\n";
    return 0;
}