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: Runtime Error)

N, K = gets.split.map &:to_i

XY = 2*K+1
$imos = XY.times.map{[0] * XY}

q = ->x_from, x_to, y_from, y_to{
  $imos[x_from][y_from] += 1
  $imos[x_from][y_to] -= 1
  $imos[x_to][y_from] -= 1
  $imos[x_to][y_to] += 1
}

xyc = $<.map{|s|
  x, y, c = s.split
  x_add = c == ?W ? 0 : K
  x_from = (x.to_i + x_add).%  2*K
  y_from = y.to_i.% 2*K
  # p [x_from, y_from]

  x_to = x_from + K
  y_to = y_from + K
  q[x_from, [x_to, 2*K].min, y_from, [y_to, 2*K].min]
  if x_to > 2*K && y_to > 2*K
    q[0, x_to-2*K, 0, y_to-2*K]
    q[0, x_to-2*K, y_from, 2*K]
    q[x_from, 2*K, 0, x_to-2*K]
  elsif x_to > 2*K
    q[0, x_to-2*K, y_from, y_to]
  elsif y_to > 2*K
    q[x_from, y_to, 0, x_to-2*K]
  end
}
XY.times{|x|
  (1...XY).each{|y|
    $imos[x][y] += $imos[x][y-1]
  }
}
(1...XY).each{|x|
  XY.times{|y|
    $imos[x][y] += $imos[x-1][y]
  }
}
# p$imos
p$imos.map(&:max).max