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: Accepted)

#![allow(unused_imports)]
#![allow(unused_macros)]
use std::cmp::{ min, max };

macro_rules! ewriteln {
    ($($args:expr),*) => { let _ = writeln!(&mut std::io::stderr(), $($args),*); };
}
macro_rules! trace {
    ($x:expr) => { ewriteln!(">>> {} = {:?}", stringify!($x), $x) };
    ($($xs:expr),*) => { trace!(($($xs),*)) }
}
macro_rules! put {
    ($x:expr) => { println!("{}", $x) };
    ($x:expr, $($xs:expr),*) => { print!("{} ", $x); put!($($xs),*) }
}

#[allow(dead_code)]
const M: i64 = 1_000_000_007;

fn eval(p: ((f64, f64), (f64, f64)), t: f64) -> (f64, f64) {
    ((p.0).0 + (p.1).0 * t, ((p.0).1 + (p.1).1 * t))
}

fn area(ps: &Vec<((f64, f64), (f64, f64))>, t: f64) -> f64 {
    let (x_, y_) = eval(ps[0], t);
    let mut x_min = x_;
    let mut y_min = y_;
    let mut x_max = x_;
    let mut y_max = y_;
    for &p in ps.iter() {
        let (x, y) = eval(p, t);
        if x_min > x { x_min = x };
        if x_max < x { x_max = x };
        if y_min > y { y_min = y };
        if y_max < y { y_max = y };
    }
    (x_max - x_min) * (y_max - y_min)
}

fn main() {
    let mut sc = Scanner::new();
    let n: usize = sc.cin();
    let ps: Vec<_> = (0..n).map(|_| {
        let x: f64 = sc.cin();
        let y: f64 = sc.cin();
        let d: char = sc.cin::<String>().chars().next().unwrap();
        match d {
            'U' => ((x, y), (0., 1.)),
            'D' => ((x, y), (0., -1.)),
            'R' => ((x, y), (1., 0.)),
            'L' => ((x, y), (-1., 0.)),
            _ => ((x,y),(0.,0.))
        }
    }).collect();

    let mut left = 0.0;
    let mut right = 2000000000.0;
    for _ in 0..1000 {
        let t1 = (left + left + right) / 3.0;
        let t2 = (left + right + right) / 3.0;
        if area(&ps, t1) < area(&ps, t2) {
            right = t2;
        } else {
            left = t1;
        }
    }
    put!(area(&ps, (left + right) / 2.0));
}

use std::io::{self, Write};
use std::str::FromStr;
use std::collections::VecDeque;

struct Scanner { stdin: io::Stdin, buffer: VecDeque<String>, }
impl Scanner {
    fn new() -> Scanner { Scanner { stdin: io::stdin(), buffer: VecDeque::new() } }
    fn cin<T: FromStr>(&mut self) -> T {
        while self.buffer.len() == 0 {
            let mut line = String::new();
            let _ = self.stdin.read_line(&mut line);
            for w in line.split_whitespace() {
                self.buffer.push_back(String::from(w));
            }
        }
        self.buffer.pop_front().unwrap().parse::<T>().ok().unwrap()
    }
}