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 C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;

class F
{
	struct P
	{
		public double x, y;
		public string d;
	}

	struct V
	{
		public double x;
		public int d;
	}

	static void Main()
	{
		var n = int.Parse(Console.ReadLine());
		var ps = new int[n].Select(_ => Console.ReadLine().Split()).Select(x => new P { x = double.Parse(x[0]), y = double.Parse(x[1]), d = x[2] }).ToArray();

		var rs = ps.Where(x => x.d == "R").ToArray();
		var ls = ps.Where(x => x.d == "L").ToArray();
		var us = ps.Where(x => x.d == "U").ToArray();
		var ds = ps.Where(x => x.d == "D").ToArray();

		var xs = new List<V>();
		Edge(xs, rs, p => p.x, 1);
		Edge(xs, us.Concat(ds).ToArray(), p => p.x, 0);
		Edge(xs, ls, p => p.x, -1);

		var ys = new List<V>();
		Edge(ys, us, p => p.y, 1);
		Edge(ys, rs.Concat(ls).ToArray(), p => p.y, 0);
		Edge(ys, ds, p => p.y, -1);

		var tx = GetTurnTime(xs);
		var ty = GetTurnTime(ys);

		Func<V, double, double> f = (v, t) => v.x + v.d * t;
		Func<double, double> rect = t => (xs.Max(v => f(v, t)) - xs.Min(v => f(v, t))) * (ys.Max(v => f(v, t)) - ys.Min(v => f(v, t)));
		Console.WriteLine(Math.Min(rect(Math.Min(tx, ty)), rect(Math.Max(tx, ty))));
	}

	static void Edge(List<V> vs, P[] ps, Func<P, double> toValue, int d)
	{
		if (ps.Length == 1)
		{
			vs.Add(new V { x = toValue(ps[0]), d = d });
		}
		else if (ps.Length > 1)
		{
			vs.Add(new V { x = ps.Max(toValue), d = d });
			vs.Add(new V { x = ps.Min(toValue), d = d });
		}
	}

	static double GetTurnTime(List<V> vs)
	{
		double l = 0, r = 1000000000, t;
		while (r - l > 0.001)
		{
			t = (l + r) / 2;
			var vs2 = vs.OrderBy(v => v.x + v.d * t).ThenBy(v => v.d).ToArray();
			if (vs2[0].d == 1 && vs2.Last().d <= 0 || vs2[0].d == 0 && vs2.Last().d == -1) l = t;
			else r = t;
		}
		return Math.Round(r, 2);
	}
}