Problem:
Score : 500 points

Problem Statement

AtCoDeer is thinking of painting an infinite two-dimensional grid in a checked pattern of side K.
Here, a checked pattern of side K is a pattern where each square is painted black or white so that each connected component of each color is a K × K square.
Below is an example of a checked pattern of side 3:

AtCoDeer has N desires.
The i-th desire is represented by x_i, y_i and c_i.
If c_i is B, it means that he wants to paint the square (x_i,y_i) black; if c_i is W, he wants to paint the square (x_i,y_i) white.
At most how many desires can he satisfy at the same time?

Constraints

1 ≤ N ≤ 10^5

1 ≤ K ≤ 1000

0 ≤ x_i ≤ 10^9

0 ≤ y_i ≤ 10^9

If i ≠ j, then (x_i,y_i) ≠ (x_j,y_j).

c_i is B or W.

N, K, x_i and y_i are integers.

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1 c_1
x_2 y_2 c_2
:
x_N y_N c_N

Output

Print the maximum number of desires that can be satisfied at the same time.

Sample Input 1

4 3
0 1 W
1 2 W
5 3 B
5 4 B

Sample Output 1

4

He can satisfy all his desires by painting as shown in the example above.

Sample Input 2

2 1000
0 0 B
0 1 W

Sample Output 2

2

Sample Input 3

6 2
1 2 B
2 1 W
2 2 B
1 0 B
0 6 W
4 5 W

Sample Output 3

4

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Linq;

namespace arc089_b
{
    class Program
    {
        static void Main(string[] args)
        {
            int[] x = Console.ReadLine().Split().Select(int.Parse).ToArray();
            int[,] grid = new int[x[0],2]; bool[] ban = new bool[x[0]];
            for (int i = 0; i < x[0]; i++)
            {
                string[] s = Console.ReadLine().Split();
                grid[i,0] = int.Parse(s[0]);
                grid[i,1] = int.Parse(s[1]);
                ban[i] = s[2] == "B";
            }
            int ret = 0;
            for (int i = 0; i < x[1] * 2; i++)
            {
                for (int j = 0; j < x[1] * 2; j++)
                {
                    int now = 0;
                    for (int k = 0; k < x[0]; k++)
                    {
                        if ((((grid[k,0] + i) / x[1] + (grid[k,1] + j) / x[1]) % 2 == 0) == ban[k])
                        {
                            now++;
                        }
                    }
                    if (ret < now)
                    {
                        ret = now;
                    }
                }
            }
            Console.WriteLine(ret);
        }
    }
}