Problem: 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:

cba927b2484fad94fb5ff7473e9aadef.png

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.

Examples

Input

4 3
0 1 W
1 2 W
5 3 B
5 4 B


Output

4


Input

2 1000
0 0 B
0 1 W


Output

2


Input

6 2
1 2 B
2 1 W
2 2 B
1 0 B
0 6 W
4 5 W


Output

4

Solution (in PYTHON3):

Solution:
import sys
input = sys.stdin.readline

def cumsum_2d(inlist,h,w):
    outlist = [ [ inlist[i-1][j-1] if i != 0 and j != 0 else 0 for i in range(k+1) ] for j in range(k+1) ]
    for i in range(1,h+1):
        tmp = 0
        for j in range(1,w+1):
            tmp += outlist[i][j]
            outlist[i][j] = tmp
    for i in range(1,w+1):
        tmp = 0
        for j in range(1,h+1):
            tmp += outlist[j][i]
            outlist[j][i] = tmp

    return outlist
    
n, k = [ int(v) for v in input().split() ]
hope = [ [ a for a in input().split() ] for i in range(n) ]

b_board = [ [ 0 for i in range(k)] for j in range(k) ]
w_board = [ [ 0 for i in range(k)] for j in range(k) ]



for h, w, c in hope:
    a1, a2 = divmod(int(w),k)
    b1, b2 = divmod(int(h),k)
    if (a1+b1) % 2 == 1:
        if c == "B":
            c = "W"
        else:
            c = "B"

    if c == "B":
        b_board[int(h)%k][int(w)%k] += 1
    else:
        w_board[int(h)%k][int(w)%k] += 1
    
b_board_3 = cumsum_2d(b_board,k,k)
w_board_3 = cumsum_2d(w_board,k,k)


total_b, total_w = b_board_3[k][k], w_board_3[k][k]
ans = 0

for i in range(1,k+1):
    for j in range(1,k+1):
        b1 = b_board_3[i][j] - b_board_3[0][j] - b_board_3[i][0]
        b2 = total_b - b_board_3[i][k] - b_board_3[k][j] + b_board_3[i][j]
        w1 = w_board_3[i][j] - w_board_3[0][j] - w_board_3[i][0]
        w2 = total_w - w_board_3[i][k] - w_board_3[k][j] + w_board_3[i][j]
        ans = max(b1+b2+(total_w-w1-w2),w1+w2+(total_b-b1-b2),ans)

print(ans)