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: Accepted)

N,K = gets.split.map(&:to_i)
KK = 2*K

$tot = [0,0]
$elvis = (0..1).map{ (0..K).map{ Array.new(K+1,0) } }
N.times{  
  x,y,c = gets.split
  x,y = [x,y].map{ |r| r.to_i % KK }
  c = c == "B" ? 0 : 1
  if x >= K
    x %= K; c ^= 1
  end
  if y >= K
    y %= K; c ^= 1
  end
  $elvis[c][x][y] += 1
  $tot[c] += 1
}

def C(color, x, y)
  return 0 if x < 0 || y < 0
  $elvis[color][x][y]
end

$sum = (0..1).map{ |color|
  lastline = Array.new(K+1, 0)
  (0..K).map{ |y|
    linesum = 0
    lastline = (0..K).map{ |x|
      linesum += C(color, x, y)
      lastline[x] + linesum
    }
    lastline
  }
}

#p $elvis[0], $sum[0]
#p $elvis[1], $sum[1]

def S(color, x, y)
  return 0 if x < 0 || y < 0
  $sum[color][x][y]
end
#exit
ans = 0
(0..K).each{ |x|
  (0..K).each{ |y|
	bb = 2*S(0, x-1, y-1) - S(0, K, y-1) - S(0, x-1, K)
	ww = 2*S(1, x-1, y-1) - S(1, K, y-1) - S(1, x-1, K)
    ans = [ans, bb+$tot[0]-ww, ww+$tot[1]-bb].max
    #p ans
  }
}
puts ans