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 Kotlin (1.0.0) (judge verdict: Wrong Answer)

fun main(args: Array<String>) {
    val n = readLine()!!.toInt()
    val points = (1..n).map {
        val line = readLine()!!.split(" ")
        Point(line[0].toInt()*2, line[1].toInt()*2, line[2])
    }
    
    var tl = 0
    var tr = 500000000

    var dmid: Long
    var tm = 0
    while (tl<tr-1) {
        tm = (tl+tr)/2
        dmid = points.time(tm).ds()
        if (dmid<0) tl = tm else if (dmid>0) tr = tm else break
    }
    (Math.max(0, tm-3)..(tm+3)).map { points.time(it).area() }.min()!!
        .let { println(it / 4.0) }

}

data class Point(var x: Int, var y: Int, val d: String)

fun List<Point>.time(t: Int): List<Point> {
    return this.map {
        when (it.d){
            "R" -> Point(it.x + t, it.y, it.d)
            "L" -> Point(it.x - t, it.y, it.d)
            "U" -> Point(it.x, it.y + t, it.d)
            "D" -> Point(it.x, it.y - t, it.d)
            else -> null
        }!!
    }
}

fun List<Point>.area(): Long {
    val w = this.maxBy { it.x }!!.x - this.minBy { it.x }!!.x
    val h = this.maxBy { it.y }!!.y - this.minBy { it.y }!!.y
    return 1L * w * h
}

fun List<Point>.ds() = this.time(1).area() - this.area()