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 PyPy3 (2.4.0) (judge verdict: Wrong Answer)

from itertools import chain
n = int(input())

p = [[] for _ in range(4)]
for _ in range(n):
    a, b, d = input().split()
    a, b = map(int, (a, b))
    p[0 if d == 'R' else 1 if d == 'L' else 2 if d == 'U' else 3].append((a, b))
to_min = [None] * 4
to_max = [None] * 4
for i in range(4):
    if p[i]:
        to_min[i] = (max if i & 1 else min)(a if i < 2 else b for a, b in p[i])
        to_max[i] = (min if i & 1 else max)(a if i < 2 else b for a, b in p[i])
still = [None] * 2
for i in range(2):
    j = (1-i)*2
    if p[j] or p[j + 1]:
        still[i] = min(b if i else a for a, b in chain(p[j], p[j+1])), max(b if i else a for a, b in chain(p[j], p[j+1])) 

def f_width(i, t):
    if still[i]:
        leftmost, rightmost = still[i]
    else:
        leftmost, rightmost = 10 ** 18, - 10 ** 18
    i *= 2
    if to_min[i] != None:
        leftmost = min(leftmost, to_min[i] + t)
    if to_max[i+1] != None:
        leftmost = min(leftmost, to_max[i+1] - t)
    if to_min[i+1] != None:
        rightmost = max(rightmost, to_min[i+1] - t)
    if to_max[i] != None:
        rightmost = max(rightmost, to_max[i] + t)
    return rightmost - leftmost

t = []
for i in range(2):
    l, r = 0, 10 ** 18
    cnt = 0
    while cnt < 100000:
        cnt += 1
        c1, c2 = l + (r - l) / 3, l + (r - l) * 2 / 3
        if f_width(i, c1) > f_width(i, c2):
            l = c1
        else:
            r = c2
    t.append(r)
l, r = min(t), max(t)
cnt = 0
while cnt < 100000:
    cnt += 1
    c1, c2 = l + (r - l) / 3, l + (r - l) * 2 / 3
    if f_width(0, c1) * f_width(1, c1) > f_width(0, c2) * f_width(1, c2):
        l = c1
    else:
        r = c2
print('{0:.9f}'.format(f_width(0, r) * f_width(1, r)))