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 (3.8.2) (judge verdict: Accepted)

import numpy as np

N = int(input())

x_min = [np.inf] * 4
x_max = [-np.inf] * 4
y_min = [np.inf] * 4
y_max = [-np.inf] * 4
for _ in range(N):
    x, y, d = input().split()
    x = int(x)
    y = int(y)
    i = {'R': 0, 'L': 1, 'U': 2, 'D': 3}[d]
    x_min[i] = min(x_min[i], x)
    x_max[i] = max(x_max[i], x)
    y_min[i] = min(y_min[i], y)
    y_max[i] = max(y_max[i], y)

x_max_ = max(x_max[2], x_max[3])
x_min_ = min(x_min[2], x_min[3])
y_max_ = max(y_max[0], y_max[1])
y_min_ = min(y_min[0], y_min[1])

def W(t):
    R = max(x_max[0] + t, x_max[1] - t, x_max_)
    L = min(x_min[0] + t, x_min[1] - t, x_min_)
    return R - L

def H(t):
    U = max(y_max[2] + t, y_max[3] - t, y_max_)
    D = min(y_min[2] + t, y_min[3] - t, y_min_)
    return U - D

def WH(t):
    return W(t) * H(t)

ts = [
    0,
    (x_max[1] - x_max[0]) / 2,
    (x_min[1] - x_min[0]) / 2,
    (y_max[3] - y_max[2]) / 2,
    (y_min[3] - y_min[2]) / 2,
    x_max_ - x_max[0],
    -(x_max_ - x_max[1]),
    x_min_ - x_min[0],
    -(x_min_ - x_min[1]),
    y_max_ - y_max[2],
    -(y_max_ - y_max[3]),
    y_min_ - y_min[2],
    -(y_min_ - y_min[3]),
]

result = np.inf;
for t in ts:
    if t >= 0:
        result = min(result, WH(t))

print(result)