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)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use ordered_float::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
#[allow(unused_imports)]
use std::f64::consts::*;

#[allow(unused)]
const INF: usize = std::usize::MAX / 4;
#[allow(unused)]
const M: usize = 1000000007;

fn calc_area(xyd: &Vec<(f64, f64, f64, f64)>, t: f64) -> f64 {
    let x_min = xyd
        .iter()
        .map(|item| OrderedFloat::from(item.0 + t * item.2))
        .min()
        .unwrap()
        .into_inner();
    let x_max = xyd
        .iter()
        .map(|(xi, _, dxi, _)| OrderedFloat::from(xi + t * dxi))
        .max()
        .unwrap()
        .into_inner();
    let y_min = xyd
        .iter()
        .map(|(_, yi, _, dyi)| OrderedFloat::from(yi + t * dyi))
        .min()
        .unwrap()
        .into_inner();
    let y_max = xyd
        .iter()
        .map(|(_, yi, _, dyi)| OrderedFloat::from(yi + t * dyi))
        .max()
        .unwrap()
        .into_inner();
    (x_max - x_min) * (y_max - y_min)
}

fn main() {
    input! {
        n: usize,
        xyd: [(f64, f64, char); n],
    }
    let xyd = xyd
        .into_iter()
        .map(|(xi, yi, di)| {
            let (dx, dy) = match di {
                'R' => (1., 0.),
                'L' => (-1., 0.),
                'U' => (0., 1.),
                'D' => (0., -1.),
                _ => (0., 0.),
            };
            (xi, yi, dx, dy)
        })
        .collect::<Vec<_>>();
    let mut l = 0.;
    let mut r = 100000000000.;
    for _ in 0..1000 {
        let m = (l + r) / 2.;
        if calc_area(&xyd, (l + m) / 2.) < calc_area(&xyd, (m + r) / 2.) {
            r = m;
        } else {
            l = m;
        }
    }
    eprintln!("{} {}", l, r);
    println!("{}", calc_area(&xyd, l));
}
*/

fn main() {
    let exe = "/tmp/bin7F1DE299";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";