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 (1.42.0) (judge verdict: Accepted)

#![allow(unused, non_snake_case, dead_code, non_upper_case_globals)]
use {
    proconio::{marker::*, *},
    std::{cmp::*, collections::*, mem::*},
};
// from:
// https://atcoder.jp/contests/abc130/submissions/5990273

#[fastout]
fn main() {
    input! {
        n:usize,
        xyd:[(f64,f64,char);n]
    }
    let mut xd = D::new();
    let mut yd = D::new();
    for &(x, y, d) in xyd.iter() {
        // x,y どちらも速度を +1 する
        match d {
            'L' => {
                xd.add(x, 0);
                yd.add(y, 1);
            }
            'R' => {
                xd.add(x, 2);
                yd.add(y, 1);
            }
            'U' => {
                xd.add(x, 1);
                yd.add(y, 2);
            }
            _ => {
                xd.add(x, 1);
                yd.add(y, 0);
            }
        }
    }
    // 交差する時間リスト
    let mut ts = vec![0f64];
    for &t in xd.events().iter().chain(yd.events().iter()) {
        ts.push(t);
    }
    let mut ans = 1e18;
    for t in ts {
        let now = xd.get(t) * yd.get(t);
        if ans > now {
            ans = now;
        }
    }
    println!("{}", ans);
}

struct D {
    l: Vec<f64>,
    r: Vec<f64>,
}
impl D {
    fn new() -> D {
        D {
            l: vec![INF; 3], //速度が3通り(0,1,2)
            r: vec![-INF; 3],
        }
    }
    fn add(&mut self, x: f64, v: usize) {
        // 向きごとに最大・最小を格納
        if self.l[v] > x {
            self.l[v] = x;
        }
        if self.r[v] < x {
            self.r[v] = x;
        }
    }
    fn get(&mut self, t: f64) -> f64 {
        // 時刻が与えられたときに幅を出力
        let mut nl = INF;
        let mut nr = -INF;
        for i in 0..3 {
            if nl > self.l[i] + i as f64 * t {
                nl = self.l[i] + i as f64 * t;
            }
            if nr < self.r[i] + i as f64 * t {
                nr = self.r[i] + i as f64 * t;
            }
        }
        nr - nl
    }
    fn events(&mut self) -> Vec<f64> {
        // 交差する時間リストを出力
        let mut ts = Vec::new();
        for i in 0..3 {
            for j in 0..i {
                // 座標の差/相対速度
                let mut t = (self.l[j] - self.l[i]) / (i - j) as f64;
                if t > 0f64 {
                    ts.push(t);
                }
                let mut t = (self.r[j] - self.r[i]) / (i - j) as f64;
                if t > 0f64 {
                    ts.push(t);
                }
            }
        }
        ts
    }
}
const INF: f64 = 1e11;