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 PyPy3 (2.4.0) (judge verdict: Accepted)

from bisect import *
from collections import *
from fractions import gcd
from math import factorial
from itertools import *
from heapq import *
import copy
#input()
N,K=map(int,input().split())
XYC=[input().split() for i in range(N)]
for i in range(N):
    x,y,c=XYC[i]
    XYC[i]=[int(x),int(y),c]

lstB=[[0 for i in range(K)] for m in range(K)]
lstW=[[0 for i in range(K)] for m in range(K)]
for x,y,c in XYC:

    if ((x//K + y//K) + (c=="W") * 1)%2==0:
        lstB[x%K][y%K]+=1
    else:
        lstW[x%K][y%K]+=1

#print(lstB)
#print(lstW)
sumB=[sum(lstB[x]) for x in range(K)]
sumW=[sum(lstW[x]) for x in range(K)]
ruisumB=[0]
ruisumW=[0]
for i in range(0,K):
    ruisumB.append(ruisumB[i]+sumB[i])
    ruisumW.append(ruisumW[i]+sumW[i])


ruiBy=[[0] for k in range(K)]
ruiWy=[[0] for k in range(K)]
for y in range(K):
    for x in range(K):
        ruiBy[y].append(lstB[x][y]+ruiBy[y][x])
        ruiWy[y].append(lstW[x][y]+ruiWy[y][x])
Revalue=0
for x in range(K+1):
    value=0
    value+=ruisumW[x]
    value+=ruisumB[K]-ruisumB[x]
    Revalue=max(Revalue,value)
    for y in range(K):
        value+=ruiWy[y][K]-2*ruiWy[y][x]
        value-=ruiBy[y][K]-2*ruiBy[y][x]
        Revalue=max(Revalue,value)

for x in range(K+1):
    value=0
    value+=ruisumB[x]
    value+=ruisumW[K]-ruisumW[x]
    Revalue=max(Revalue,value)
    for y in range(K):
        value+=ruiBy[y][K]-2*ruiBy[y][x]
        value-=ruiWy[y][K]-2*ruiWy[y][x]
        Revalue=max(Revalue,value)

print(Revalue)