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 Python (3.4.3) (judge verdict: Wrong Answer)

import numpy as np
from scipy.optimize import fmin

mapping = {
    "R": (1, 0),
    "L": (-1, 0),
    "U": (0, 1),
    "D": (0, -1)
}

def f(x, vx, t):
    nx = x + vx * t
    return nx.max() - nx.min()

def solve(n, x, y, d):
    x = np.asarray(x, dtype=np.float)
    y = np.asarray(y, dtype=np.float)
    vx = np.zeros(n, dtype=np.float)
    vy = np.zeros(n, dtype=np.float)
    for i in range(n):
        vx[i], vy[i] = mapping[d[i]]
    h = lambda t: f(y, vy, t) * f(x, vx, t)
    t = fmin(h, x0=100, ftol=1e-9, disp=False)[0]
    return h(t)

n = int(input())
x, y, d = [-1]*n, [-1]*n, [-1]*n
for i in range(n):
    x[i], y[i], d[i] = input().split()
    x[i] = int(x[i])
    y[i] = int(y[i])
print(solve(n, x, y, d))