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Contest: Task: Related: TaskA TaskC

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

Sample Output 1

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

Sample Input 3