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 Ruby (2.3.3) (judge verdict: Accepted)

start = Time.now
N = gets.to_i
XYD = $<.map{|s|
  x, y, d = s.split
  [x.to_i, y.to_i, d.intern]
}
f = ->n{
  tbl = {
    R: [1, 0],
    L: [-1, 0],
    U: [0, 1],
    D: [0, -1],
  }
  min_x = min_y = 1.0/0
  max_x = max_y = -1.0/0
  XYD.each{|x, y, d|
    x *= 2
    y *= 2
    mul_x, mul_y = tbl[d]
    x += n * mul_x
    y += n * mul_y
    min_x = x if x < min_x
    max_x = x if x > max_x
    min_y = y if y < min_y
    max_y = y if y > max_y
  }
  (max_x - min_x) * (max_y - min_y)
}

l = 0
r = 10**9
# p f[0]
50.times{|i|
  if f[(l * 2 + r) / 3] < f[(l + r * 2) / 3]
    r = (l + r * 2) / 3
  else
    l = (l * 2 + r) / 3
  end
  break if Time.now - start > 4.9
}
p f[(l + r) / 2].fdiv 4