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: Wrong Answer)

N, K = gets.split.map(&:to_i)
K2 = K*2
PTS = []
while line = gets
  x_, y_, c = line.split
  x = (x_.to_i + (c == 'W' ? K : 0)) % K2
  y =  y_.to_i % K2
  PTS << [x, y]
end
#p PTS: PTS

def dump(m)
  return
  puts
  m.reverse_each do |row|
    puts row.map{|v| format("%3d", v)}.join
  end
end

MAP = Array.new(K2){ Array.new(K2, 0) }
def add(x, y, v)
  return if x >= K2 || y >= K2
  x = 0 if x < 0
  y = 0 if y < 0
  MAP[y][x] += v
end

def add_sq(x, y)
  return if (x+K-1)<0 && (y+K-1)<0
  add(x,   y,   +1)
  add(x+K, y,   -1)
  add(x  , y+K, -1)
  add(x+K, y+K, +1)
end

for x, y in PTS
  add_sq(x-K+1, y-K+1)
  add_sq(x-K2+1, y-K2+1)
  add_sq(x   +1, y-K2+1)
  add_sq(x-K2+1, y   +1)
  add_sq(x   +1, y   +1)
end

dump(MAP)

for y in 0...K2
  for x in 1...K2
    MAP[y][x] += MAP[y][x-1]
  end
end

dump(MAP)

max = MAP[0].max
for x in 0...K2
  for y in 1...K2
    v = MAP[y][x] + MAP[y-1][x]
    max = v if v > max
    MAP[y][x] = v
  end
end

dump(MAP)

p max