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 Rust (1.15.1) (judge verdict: Wrong Answer)

// Try AtCoder
// author: Leonardone @ NEETSDKASU

fn fmin(a: f64, b: f64) -> f64 {
    if a < b { a } else { b }
}
fn fmax(a: f64, b: f64) -> f64 {
    if a > b { a } else { b }
}
fn fabs(a: f64) -> f64 {
    if a < 0.0 { -a } else { a }
}

fn main() {
    let mut stdin = String::new();
    std::io::Read::read_to_string(
        &mut std::io::stdin(),
        &mut stdin).unwrap();
    let mut stdin = stdin.split_whitespace();
    let mut get = || stdin.next().unwrap();
    macro_rules! get {
        ($t:ty) => (get().parse::<$t>().unwrap());
        () => (get!(i64));
    }
    
    let n = get!();
    let mut vs = vec![];
    let mut y_min = 1e19;
    let mut y_max = -1e19;
    let mut x_min = 1e19;
    let mut x_max = 1e19;
    
    for _ in 0 .. n {
        let x = get!(f64);
        let y = get!(f64);
        let c = get().as_bytes()[0];
        vs.push((x, y, c));
        y_min = fmin(y_min, y);
        y_max = fmax(y_max, y);
        x_min = fmin(x_min, x);
        x_max = fmax(x_max, x);
    }
    let mut ans = (x_max-x_min) * (y_max-y_min);
    let mut lp = 0.0;
    let mut rp = 3e8;
    
    while fabs(rp - lp) > 1e-18 {
        let hp = (lp + rp) / 2.0;
        let mut y_min = 1e19;
        let mut y_max = -1e19;
        let mut x_min = 1e19;
        let mut x_max = -1e19;
        macro_rules! f {
            ($x:expr, $y:expr) => {
                y_min = fmin(y_min, $y);
                y_max = fmax(y_max, $y);
                x_min = fmin(x_min, $x);
                x_max = fmax(x_max, $x);
            }
        }
        for &(x, y, c) in vs.iter() {
            match c {
            b'U' => {
                f!(x, y + hp);
            },
            b'D' => {
                f!(x, y - hp);
            },
            b'R' => {
                f!(x + hp, y);
            },
            _ => {
                f!(x - hp, y);
            },
            }
        }
        let tmp = (x_max-x_min)*(y_max-y_min);
        if tmp < ans {
            rp = hp;
            ans = tmp;
        } else {
        	lp = hp-1.0;
        }
        rp = hp;
    }
    println!("{}", ans);
}