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;

namespace AtCoderABC130F
{
    class Program
    {
        static void Main(string[] args)
        {
            float ans = float.MaxValue;
            int N = int.Parse(Console.ReadLine());

            int MinXR = int.MaxValue /10;
            int MaxXR = int.MinValue/10;
            int MinXL = int.MaxValue/10;
            int MaxXL = int.MinValue/10;
            int MinYU = int.MaxValue/10;
            int MaxYU = int.MinValue/10;
            int MinYD = int.MaxValue/10;
            int MaxYD = int.MinValue/10;

            int MinX = int.MaxValue/10;
            int MaxX = int.MinValue/10;
            int MinY = int.MaxValue/10;
            int MaxY = int.MinValue/10;

            for (int i = 0; i < N; i++)
            {
                string[] s = Console.ReadLine().Split(' ');
                int x = int.Parse(s[0]);
                int y = int.Parse(s[1]);

                switch(s[2])
                {
                    case "R":
                        if (MinXR > x)
                            MinXR = x;
                        if (MaxXR < x)
                            MaxXR = x;
                        if (MinY > y)
                            MinY = y;
                        if (MaxY < y)
                            MaxY = y;
                        break;
                    case "L":
                        if (MinXL > x)
                            MinXL = x;
                        if (MaxXL < x)
                            MaxXL = x;
                        if (MinY > y)
                            MinY = y;
                        if (MaxY < y)
                            MaxY = y;
                        break;
                    case "U":
                        if (MinYU > y)
                            MinYU = y;
                        if (MaxYU < y)
                            MaxYU = y;
                        if (MinX > x)
                            MinX = x;
                        if (MaxX < x)
                            MaxX = x;
                        break;
                    case "D":
                        if (MinYD > y)
                            MinYD = y;
                        if (MaxYD < y)
                            MaxYD = y;
                        if (MinX > x)
                            MinX = x;
                        if (MaxX < x)
                            MaxX = x;
                        break;
                }
            }

            int pow = 10000000;
            long MinSS = long.MaxValue;
            int NextRange = 0;

            int from = 0;
            int to = 10;

            int offset = 0;

            for (int p = 0; p <8; p++)
            {
                for (int i = from; i < to; i++)
                {
                    int time = i * pow + offset;

                    int MaxXv = Math.Max(Math.Max(MaxXR + time, MaxXL - time), MaxX);
                    int MinXv = Math.Min(Math.Min(MinXR + time, MinXL - time), MinX);
                    int MaxYv = Math.Max(Math.Max(MaxYU + time, MaxYD - time), MaxY);
                    int MinYv = Math.Min(Math.Min(MinYU + time, MinYD - time), MinY);

                    long SS = (long)(MaxXv - MinXv) * (MaxYv - MinYv);
                    if (SS < MinSS)
                    {
                        MinSS = SS;
                        NextRange = i;
                    }
                }

                if (NextRange == 0)
                {
                    from = 0;
                    to = 20;
                    offset = 0;
                }
                else
                {
                    from = 0;
                    to = 30;
                    offset = (NextRange - 1) * pow;
                }

                pow = pow / 10;
                MinSS = long.MaxValue;

            }

            for (int i = 0; i < 60; i++)
            {
                float time = (float)offset + (float)i / 2;
                float MaxXv = Math.Max(Math.Max((float)MaxXR + time, (float)MaxXL - time), (float)MaxX);
                float MinXv = Math.Min(Math.Min((float)MinXR + time, (float)MinXL - time), (float)MinX);
                float MaxYv = Math.Max(Math.Max((float)MaxYU + time, (float)MaxYD - time), (float)MaxY);
                float MinYv = Math.Min(Math.Min((float)MinYU + time, (float)MinYD - time), (float)MinY);

                float SS = (MaxXv - MinXv) * (MaxYv - MinYv);
                if (SS < ans)
                    ans = SS;
            }


            Console.WriteLine(ans.ToString());
        }
    }
}