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 PHP (5.6.30) (judge verdict: Time Limit Exceeded)

<?php

$tmp = explode(" ", trim(fgets(STDIN)));
$N = $tmp[0];
$K = $tmp[1];

$v = [];
for ($i = 0; $i < $N; $i++) {
    $t = explode(" ", trim(fgets(STDIN)));
    $v[] = [$t[0] % (2 * $K), $t[1] % (2 * $K), $t[2]];
}

$max = 0;
for ($i = 0; $i < 2 * $K; $i++) {
    for ($j = 0; $j < 2 * $K; $j++) {
        $cnt = 0;
        for ($k = 0; $k < $N; $k++) {
            if ((int)(($v[$k][0] + $i) / $K) % 2 == (int)(($v[$k][1] + $j) / $K) % 2) {
                if ($v[$k][2] == 'W') {
                    $cnt++;
                }
            } else {
                if ($v[$k][2] == 'B') {
                    $cnt++;
                }
            }
        }
        if ($cnt > $max) {
            $max = $cnt;
            if ($max == $N) {
                break 2;
            }
        }
    }
}

echo $max.PHP_EOL;