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 Pascal (FPC 2.6.2) (judge verdict: Memory Limit Exceeded)

uses math;
var n,k,i,j,sumw,sumb,m,ans,x,y:longint;
var bs,ws,b1,w1:array[-3020..3020,-3020..3020] of longint;
var c,c1:char;
function gsw(x1,y1,x2,y2:longint):longint;
begin
  exit(ws[x2,y2]-ws[x2,y1-1]-ws[x1-1,y2]+ws[x1-1,y1-1]);
end;
function gsb(x1,y1,x2,y2:longint):longint;
begin
  exit(bs[x2,y2]-bs[x2,y1-1]-bs[x1-1,y2]+bs[x1-1,y1-1]);
end;
begin
  fillchar(bs,sizeof(bs),0);fillchar(ws,sizeof(ws),0);
  readln(n,k);
  for i:=1 to n do
  begin
    readln(x,y,c,c1);x:=x mod 2*k;y:=y mod 2*k;
    if c1='B' then inc(b1[x,y]) else inc(w1[x,y]);
  end;
  for i:=1 to 2*k-1 do
    for j:=1 to 2*k-1 do
    begin
      bs[i,j]:=b1[i,j]+bs[i-1,j]+bs[i,j-1]-bs[i-1,j-1];
      ws[i,j]:=w1[i,j]+ws[i-1,j]+ws[i,j-1]-ws[i-1,j-1];
    end;
  ans:=0;
  for i:=1 to 2*k do
    for j:=1 to 2*k do
    begin
      {sumw:=ws[i-k+1,k-j]-ws[i-k+1-k,k+j]-ws[i-k+1,k-j-k]+ws[i-k+1-k,k-j-k]+
            //ws[i-k+1,2*k-j+1+k-1]-ws[i-k+1,2*k-j]-ws[i-k+1-k,3*k-j]+ws[i+1,2*k-j-1]+
            ws[i-k+1,3*k-i-1]-ws[i-k+1-k,3*k-i-1]-ws[i-k+1,2*k-i-1]+ws[i-2*k+1,2*k-i-1]+
            ws[i+1,2*k-j]-ws[i+1-k,2*k-j]-ws[i+1,2*k-j]+ws[i+1-k,2*k-j-j]+
            ws[k+i+1,2*k-i+1]-ws[i+1,2*k-i+1]-ws[k+i+1,k-i+1]+ws[i+1,k-i+1]+
            ws[k+i+1,k-i]-ws[i+1,k-i]-ws[k+i+1,-i]+ws[i+1,-i];
      sumb:=bs[  }
      ans:=max(ans,gsw(1,1,i-1,j-1));
      ans:=max(ans,gsw(1,j+k,i-1,2*k));
      ans:=max(ans,gsw(i+k,1,2*k,j-1));
      ans:=max(ans,gsw(i,j,i+k-1,j+k-1));
      ans:=max(ans,gsw(i+k,j+k,2*k,2*k));
      ans:=max(ans,gsb(1,j,i-1,j+k-1));
      ans:=max(ans,gsb(i,1,i+k-1,j-1));
      ans:=max(ans,gsb(i,j+k,i+k-1,2*k));
      ans:=max(ans,gsb(i+k,j,2*k,j+k-1));

      ans:=max(ans,gsb(1,1,i-1,j-1));
      ans:=max(ans,gsb(1,j+k,i-1,2*k));
      ans:=max(ans,gsb(i+k,1,2*k,j-1));
      ans:=max(ans,gsb(i,j,i+k-1,j+k-1));
      ans:=max(ans,gsb(i+k,j+k,2*k,2*k));
      ans:=max(ans,gsw(1,j,i-1,j+k-1));
      ans:=max(ans,gsw(i,1,i+k-1,j-1));
      ans:=max(ans,gsw(i,j+k,i+k-1,2*k));
      ans:=max(ans,gsw(i+k,j,2*k,j+k-1));
    end;
  writeln(ans);
end.