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)

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();

    let mut static_x = ValRange::new();
    let mut static_y = ValRange::new();

    let mut y_u = ValRange::new();
    let mut y_d = ValRange::new();

    let mut x_l = ValRange::new();
    let mut x_r = ValRange::new();

    for _ in 0..n {
        let x: i64 = sc.read();
        let y: i64 = sc.read();
        let d = sc.chars()[0];
        match d {
            'R' => {
                static_y.set(y);
                x_r.set(x);
            }
            'L' => {
                static_y.set(y);
                x_l.set(x);
            }
            'U' => {
                static_x.set(x);
                y_u.set(y);
            }
            'D' => {
                static_x.set(x);
                y_d.set(y);
            }
            _ => unreachable!(),
        }
    }
    let xd = |t: f64| d(t, &static_x, &x_r, &x_l);
    let yd = |t: f64| d(t, &static_y, &y_u, &y_d);

    let mut ts = vec![0.0];
    cross_t3(&static_y, &y_u, &y_d, &mut ts);
    cross_t3(&static_x, &x_r, &x_l, &mut ts);

    let ans = ts.iter().map(|&t| OrdF64(xd(t) * yd(t))).min().unwrap().0;
    println!("{}", ans);
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}

#[derive(Copy, Clone)]
struct ValRange {
    min: f64,
    max: f64,
}
impl ValRange {
    fn new() -> Self {
        ValRange {
            min: std::f64::INFINITY,
            max: std::f64::NEG_INFINITY,
        }
    }
    fn set(&mut self, value: i64) {
        self.min = self.min.min(value as f64);
        self.max = self.max.max(value as f64);
    }
    #[inline]
    fn at(&self, t: f64) -> Self {
        ValRange {
            min: self.min + t,
            max: self.max + t,
        }
    }
    #[inline]
    fn union(&self, other: Self) -> Self {
        ValRange {
            min: self.min.min(other.min),
            max: self.max.max(other.max),
        }
    }
    #[inline]
    fn d(&self) -> f64 {
        self.max - self.min
    }
}

fn d(t: f64, val_static: &ValRange, val_up: &ValRange, val_down: &ValRange) -> f64 {
    val_static.union(val_down.at(-t)).union(val_up.at(t)).d()
}
fn cross_t3(val_static: &ValRange, val_up: &ValRange, val_down: &ValRange, ts: &mut Vec<f64>) {
    cross_t2(val_up, val_down, 2.0, ts);
    cross_t2(val_up, val_static, 1.0, ts);
    cross_t2(val_static, val_down, 1.0, ts);
}
fn cross_t2(val_up: &ValRange, val_down: &ValRange, step: f64, ts: &mut Vec<f64>) {
    cross_t(val_up.max, val_down.max, step, ts);
    cross_t(val_up.max, val_down.min, step, ts);
    cross_t(val_up.min, val_down.max, step, ts);
    cross_t(val_up.min, val_down.min, step, ts);
}
fn cross_t(val_up: f64, val_down: f64, step: f64, ts: &mut Vec<f64>) {
    if val_up.is_finite() && val_down.is_finite() && val_up < val_down {
        ts.push((val_down - val_up) / step);
    }
}

#[derive(PartialEq, PartialOrd)]
pub struct OrdF64(pub f64);

impl OrdF64 {
    pub fn new(value: f64) -> Self {
        assert!(!value.is_nan());
        OrdF64(value)
    }
}

impl Eq for OrdF64 {}
impl Ord for OrdF64 {
    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
        self.partial_cmp(&other).unwrap()
    }
}