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 Perl (v5.18.2) (judge verdict: Accepted)

($N,$K)=split " ",<STDIN>;
$K2=$K*2;
for($i=0;$i<$N;$i++){
	($x,$y,$c)=split " ",<STDIN>;
	$x=$x%$K2;
	$y=$y%$K2;
	$v=($c eq "B")?1:-1;
	if ($x>=$K) { $x=$x-$K; $v=-$v; }
	if ($y>=$K) { $y=$y-$K; $v=-$v; }
	$ddv[$x][$y] +=$v*4;
}

$v=0;
for($y=0;$y<$K;$y++){ $sdv[$y]=0;$vmp[$y]=0;$vmm[$y]=0;$v[$y]=0; }

for($x=0;$x<$K;$x++){
	$dv=0;for($y=0;$y<$K;$y++){ $dv+=$ddv[$x][$y]; } $dv=-$dv/2;
	for($y=0;$y<$K;$y++){
		$dv+=$ddv[$x][$y];
		$v=$v[$y]+$dv;$v[$y]=$v;
		$sdv[$y]+=$dv;
		if ($vmp[$y]<$v) { $vmp[$y]=$v; }
		if ($vmm[$y]>$v) { $vmm[$y]=$v; }
	}
}
$s=0;
for($y=0;$y<$K;$y++){
	$vv=-$sdv[$y]/2;
	$vmp=$vmp[$y]+$vv;
	$vmm=$vmm[$y]+$vv;
	if($s< $vmp){$s= $vmp;}
	if($s<-$vmm){$s=-$vmm;}
}

$ss=($s+$N)/2;
print $ss;