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)

def sum(dp,x1,y1,x2,y2)
  dp[x2][y2]+dp[x1-1][y1-1]-dp[x2][y1-1]-dp[x1-1][y2]
end

n,k=gets.split.map(&:to_i)
cond={}
cond[:B]=Array.new(4*k){Array.new(4*k,0)}
cond[:W]=Array.new(4*k){Array.new(4*k,0)}
n.times{
  x,y,c=gets.split
  x=x.to_i%(2*k)
  y=y.to_i%(2*k)
  c=c.to_sym
  2.times{|dx|2.times{|dy|cond[c][x+dx*2*k][y+dy*2*k]+=1}}
}
dp={}
dp[:B]=Array.new(4*k+1){Array.new(4*k+1,0)}
dp[:W]=Array.new(4*k+1){Array.new(4*k+1,0)}

[:B,:W].each{|c|
  (4*k).times{|x|
    row_sum=0
    (4*k).times{|y|
      row_sum+=cond[c][x][y]
      dp[c][x][y]=dp[c][x-1][y]+row_sum
  }}
}

max=0
(2*k).times{|x|(2*k).times{|y|
  temp=0
  temp+=sum(dp[:B],x+k,y+k,x+2*k-1,y+2*k-1)
  temp+=sum(dp[:B],x  ,y  ,x+  k-1,y+  k-1)
  temp+=sum(dp[:W],x+k,y  ,x+2*k-1,y+  k-1)
  temp+=sum(dp[:W],x  ,y+k,x+  k-1,y+2*k-1)
  max=temp if max<temp
}}

puts max