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_macros)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

#[derive(Default, Debug)]
struct Border {
    pl: Option<i64>,
    zr: Option<i64>,
    mi: Option<i64>,
}

use std::cmp::{max, min};

impl Border {
    fn update_max(&mut self, v: i64, sig: i64) {
        if sig > 0 {
            if self.pl.is_none() || self.pl < Some(v) {
                self.pl = Some(v);
            }
        } else if sig == 0 {
            if self.zr.is_none() || self.zr < Some(v) {
                self.zr = Some(v);
            }
        } else if sig < 0 {
            if self.mi.is_none() || self.mi < Some(v) {
                self.mi = Some(v);
            }
        }
    }

    fn update_min(&mut self, v: i64, sig: i64) {
        if sig > 0 {
            if self.pl.is_none() || self.pl > Some(v) {
                self.pl = Some(v);
            }
        } else if sig == 0 {
            if self.zr.is_none() || self.zr > Some(v) {
                self.zr = Some(v);
            }
        } else if sig < 0 {
            if self.mi.is_none() || self.mi > Some(v) {
                self.mi = Some(v);
            }
        }
    }

    fn cur_max(&self, t: i64) -> i64 {
        let mut ret = i64::min_value();

        if let Some(p) = self.pl {
            ret = max(ret, p * 2 + t);
        }
        if let Some(p) = self.zr {
            ret = max(ret, p *2);
        }
        if let Some(p) = self.mi {
            ret = max(ret, p * 2 - t);
        }

        ret
    }

    fn cur_min(&self, t: i64) -> i64 {
        let mut ret = i64::max_value();

        if let Some(p) = self.pl {
            ret = min(ret, p * 2 + t);
        }
        if let Some(p) = self.zr {
            ret = min(ret, p *2);
        }
        if let Some(p) = self.mi {
            ret = min(ret, p * 2 - t);
        }

        ret
    }
}

fn main() {
    input! {
        n: usize,
        xyd: [(i64, i64, chars); n],
    }

    let mut xmax = Border::default();
    let mut xmin = Border::default();
    let mut ymax = Border::default();
    let mut ymin = Border::default();

    for (x, y, d) in xyd {
        let xsig = if d[0] == 'R' {1} else if d[0] == 'L' {-1} else {0};
        let ysig = if d[0] == 'U' {1} else if d[0] == 'D' {-1} else {0};

        xmax.update_max(x, xsig);
        xmin.update_min(x, xsig);
        ymax.update_max(y, ysig);
        ymin.update_min(y, ysig);
    }

    // dbg!(&xmax);
    // dbg!(&xmin);
    // dbg!(&ymax);
    // dbg!(&ymin);

    let mut ans = i64::max_value();

    for t in 0..2 * 1_0000_0000 {
        let cur = (xmax.cur_max(t) - xmin.cur_min(t)) * (ymax.cur_max(t) - ymin.cur_min(t));

        // dbg!(cur);
        ans = min(ans, cur);
    }

    println!("{}", ans as f64 / 4.0);
}