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.map(&:to_i)
K2 = 2 * K
m = Array.new(K2 + 1){Array.new(K2 + 1, 0)}
readlines.map(&:split).each do |x, y, c|
  x = x.to_i % K2
  y = (y.to_i + (c == 'W' ? K : 0)) % K2
  m[x + 1][y + 1] += 1
  m[(x + K) % K2 + 1][(y + K) % K2 + 1] += 1
end
[*1 .. K2].product([*1 .. K2]).each{|i, j| m[i][j] += m[i - 1][j] + m[i][j - 1] - m[i - 1][j - 1]}
puts [*1 .. K].repeated_permutation(2).flat_map{|i, j| c = m[i + K][j + K] - m[i][j + K] - m[i + K][j] + m[i][j]; [c, N - c]}.max