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 $K$ $\times$ $K$ 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

• $1$ $\leq$ $N$ $\leq$ $10^5$
• $1$ $\leq$ $K$ $\leq$ $1000$
• $0$ $\leq$ $x_i$ $\leq$ $10^9$
• $0$ $\leq$ $y_i$ $\leq$ $10^9$
• If $i$ $\neq$ $j$, then $(x_i,y_i)$ $\neq$ $(x_j,y_j)$.
• $c_i$ 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.

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

4

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

2 1000
0 0 B
0 1 W

2

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

4