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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

N,K = gets.split(/\s/).map(&:to_i)
hope = Array.new(N)
for i in 0..N-1
	hope[i] = gets.split(/\s/)
	if hope[i][2] == "B"
		hope[i][0] = hope[i][0].to_i % (2*K)
		hope[i][1] = hope[i][1].to_i % (2*K)
	else
		hope[i][0] = (hope[i][0].to_i-K) % (2*K)
		hope[i][1] = hope[i][1].to_i % (2*K)
	end
	
end

# hope[x,y,color]

y_ex = Array.new(2*K)

for i in 0..2*K-1
	y_ex[i] = 0
end

s_max = 0

for i in 0..2*K-1
	
	score = 0
	
	for j in 0..2*K-1
		y_ex[j] = 0
	end
	
	for j in 0..N-1
		if (hope[j][0]-i)%(2*K)<K
			if hope[j][1] < K
				y_ex[hope[j][1]] += 1
				score += 1
			else y_ex[hope[j][1]] -= 1
		end
		else
			if hope[j][1] < K
				y_ex[hope[j][1]] -= 1
			else
				y_ex[hope[j][1]] += 1
				score += 1
			end
		end
	end

	if s_max < score
		s_max = score
	end
	if s_max < N - score
		s_max = N - score
	end
	
	
	for j in 0..K-1
		score = score - y_ex[(K+j)%(2*K)] - y_ex[j]
		if s_max < score
			s_max = score
		end
		if s_max < N - score
			s_max = N - score
		end
	end
end
puts s_max