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: Accepted)

using System.Linq;
using static System.Console;
using static System.Math;
class Z { static void Main() => new _K(); }
class _K
{
	int[] G => Strs.Select(int.Parse).ToArray();
	string[] Strs => ReadLine().Split();
	int K, L;
	readonly int[,,] sum;
	public _K()
	{
		var I = G;
		K = I[1];
		L = 4 * K;
		sum = new int[L + 1, L + 1, 2];
		var tot = new int[2];
		for (var i = 0; i < I[0]; i++)
		{
			var S = Strs;
			var c = S[2] == "B" ? 0 : 1;
			Add(int.Parse(S[0]) % (2 * K) + 1, int.Parse(S[1]) % (2 * K) + 1, c);
			tot[c]++;
		}
		for (var c = 0; c < 2; c++)
		{
			for (var x = 0; x < L; x++) for (var y = 0; y <= L; y++) sum[x + 1, y, c] += sum[x, y, c];
			for (var x = 0; x <= L; x++) for (var y = 0; y < L; y++) sum[x, y + 1, c] += sum[x, y, c];
		}
		var max = 0;
		for (var i = 0; i < K; i++) for (var j = 0; j < K; j++)
			{
				var s = Sum(i, j);
				max = Max(max, Max(s, I[0] - s));
			}
		WriteLine(max);
	}
	void Add(int x, int y, int c)
	{
		sum[x, y, c]++;

		{
			sum[x + 2 * K, y, c]++;
			sum[x + 2 * K, y + 2 * K, c]++;
		}
		sum[x, y + 2 * K, c]++;
	}
	int Sum(int i, int j) => Sum(i, j, 0) + Sum(i + K, j + K, 0) + Sum(i, j + K, 1) + Sum(i + K, j, 1);
	int Sum(int i, int j, int c) => sum[i + K, j + K, c] - sum[i + K, j, c] - sum[i, j + K, c] + sum[i, j, c];
}