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)

INF = 1 << 30
def min(a,b); a < b ? a : b; end
def max(a,b); a > b ? a : b; end

N = gets.to_i
P = Hash.new{|h,k| h[k] = {x: {max: -INF, min: INF}, y: {max: -INF, min: INF}}}

N.times do
  x,y,d = gets.split
  x = x.to_i; y = y.to_i; d = d.to_sym
  P[d][:x][:max] = max(P[d][:x][:max], x) 
  P[d][:x][:min] = min(P[d][:x][:min], x) 
  P[d][:y][:max] = max(P[d][:y][:max], y) 
  P[d][:y][:min] = min(P[d][:y][:min], y)
end

def bounding_box(t)
  x_min = min(min(P[:U][:x][:min], P[:D][:x][:min]), min(P[:L][:x][:min] - t, P[:R][:x][:min] + t))
  x_max = max(max(P[:U][:x][:max], P[:D][:x][:max]), max(P[:L][:x][:max] - t, P[:R][:x][:max] + t))
  y_min = min(min(P[:L][:y][:min], P[:R][:y][:min]), min(P[:D][:y][:min] - t, P[:U][:y][:min] + t))
  y_max = max(max(P[:L][:y][:max], P[:R][:y][:max]), max(P[:D][:y][:max] - t, P[:U][:y][:max] + t))
  (x_max - x_min) * (y_max - y_min)
end

T = [ P[:L][:x][:min] - min(P[:U][:x][:min], P[:D][:x][:min]), min(P[:U][:x][:min], P[:D][:x][:min]) - P[:R][:x][:min],  (P[:L][:x][:min] - P[:R][:x][:min]) / 2.0,
      P[:L][:x][:max] - max(P[:U][:x][:max], P[:D][:x][:max]), max(P[:U][:x][:max], P[:D][:x][:max]) - P[:R][:x][:max],  (P[:L][:x][:max] - P[:R][:x][:max]) / 2.0,
      P[:D][:y][:min] - min(P[:L][:y][:min], P[:R][:y][:min]), min(P[:L][:y][:min], P[:R][:y][:min]) - P[:U][:y][:min],  (P[:D][:y][:min] - P[:U][:y][:min]) / 2.0,
      P[:D][:y][:max] - max(P[:L][:y][:max], P[:R][:y][:max]), max(P[:L][:y][:max], P[:R][:y][:max]) - P[:U][:y][:max],  (P[:D][:y][:max] - P[:U][:y][:max]) / 2.0,
      0, INF ].map{|t| max(t,0) }

puts T.map{|t| bounding_box(t) }.min