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 <bits/stdc++.h>
using namespace std;
using ll = long long;

int gcd(int a, int b) { return (b == 0) ? a : gcd(b, a % b); }

#define CIN(T, x) T x; cin >> x;
#define CINS(T, x, N) vector<T> x(N); for(int i = 0; i < (N); i++) { cin >> x[i]; }
#define SHOW(x) for(int i = 0; i < x.size(); i++) { cout << x[i] << " "; } cout << "\n";
#define SHOW2(x) for(int j = 0; j < x.size(); j++) { SHOW(x[j]); } cout << "\n";

ll Lnon = 1e9, Rnon = -1e9, LtoL = 1e9, RtoL = -1e9, LtoR = 1e9, RtoR = -1e9;
ll Dnon = 1e9, Unon = -1e9, DtoD = 1e9, UtoD = -1e9, DtoU = 1e9, UtoU = -1e9;

double getL(double t) {
    double temp = 1e9;
    if (abs(Lnon) < 1e9 ) temp = Lnon;
    if (abs(LtoL) < 1e9 ) temp = min(temp, LtoL - t);
    if (abs(LtoR) < 1e9 ) temp = min(temp, LtoR + t);
    return temp;
}
double getR(double t) {
    double temp = -1e9;
    if (abs(Rnon) < 1e9 ) temp = Rnon;
    if (abs(RtoL) < 1e9 ) temp = max(temp, RtoL - t);
    if (abs(RtoR) < 1e9 ) temp = max(temp, RtoR + t);
    return temp;
}
double getD(double t) {
    double temp = 1e9;
    if (abs(Dnon) < 1e9 ) temp = Dnon;
    if (abs(DtoD) < 1e9 ) temp = min(temp, DtoD - t);
    if (abs(DtoU) < 1e9 ) temp = min(temp, DtoU + t);
    return temp;
}
double getU(double t) {
    double temp = -1e9;
    if (abs(Unon) < 1e9 ) temp = Unon;
    if (abs(UtoD) < 1e9 ) temp = max(temp, UtoD - t);
    if (abs(UtoU) < 1e9 ) temp = max(temp, UtoU + t);
    return temp;
}


int main() {

    CIN(int, N);

    for (int i = 0; i < N; i++) {
        ll x, y;
        char d;
        cin >> x >> y >> d;
        if ( d == 'R' ) {
            LtoR = min(LtoR, x);
            RtoR = max(RtoR, x);
            Dnon = min(Dnon, y);
            Unon = max(Unon, y);
        } else if ( d == 'L' ) {
            LtoL = min(LtoL, x);
            RtoL = max(RtoL, x);
            Dnon = min(Dnon, y);
            Unon = max(Unon, y);
        } else if ( d == 'U' ) {
            DtoU = min(DtoU, y);
            UtoU = max(UtoU, y);
            Lnon = min(Lnon, x);
            Rnon = max(Rnon, x);
        } else {
            DtoD = min(DtoD, y);
            UtoD = max(UtoD, y);
            Lnon = min(Lnon, x);
            Rnon = max(Rnon, x);
        }
    }

    // cout << Lnon << " " << Rnon << " " << LtoL << " " << RtoL << " " << LtoR << " " << RtoR << "\n";
    // cout << Dnon << " " << Unon << " " << DtoD << " " << UtoD << " " << DtoU << " " << UtoU << "\n";

    double ans;
    double dx1, dy1, dx2, dy2;
    dx2 = getR(0) - getL(0);
    dy2 = getU(0) - getD(0);
    ans = dx2 * dy2;
    double di = 1e4;

//    for (double i = 0; i <= 4 * 1e8 + 1e5; i = i + di) {
    for (double i = 0; i <= 4 * 1e8 + 1e5; i++) {
        double l, r, u, d;
        double temp;
        l = getL( i / 2 );
        r = getR( i / 2 );
        u = getU( i / 2 );
        d = getD( i / 2 );
        dx1 = r - l;
        dy1 = u - d;
        // std::cout << l << " " << r << " " << d << " " << u << "\n";

        temp = (r - l) * (u - d);
        ans = min(ans, temp);
        if (ans == 0) break;
        if ( (dx1 >= dx2 && dy1 > dy2) || (dx1 > dx2 && dy1 >= dy2) ) {
            if ( di == 1e4 ) {
                i = i - di;
                di = 1;
                continue;
            } else {
                break;
            }
        }
        dx2 = dx1;
        dy2 = dy1;
    }

    printf("%.16f", ans);
    // std::cout << ans << "\n";
    return 0;
}