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


N, K = map(int, input().split())
cnt_B = 0
cnt_W = 0
black = [[0] * (2 * K) for _ in range(2 * K)]
white = [[0] * (2 * K) for _ in range(2 * K)]
for s in sys.stdin.readlines():
    x, y, c = s.split()
    x = int(x); y = int(y)
    if c == 'B':
        black[x % (2 * K)][y % (2 * K)] += 1
        cnt_B += 1
    else:
        white[x % (2 * K)][y % (2 * K)] += 1
        cnt_W += 1

S_black = [[0] * (2 * K + 1) for _ in range(2 * K)]
S_white = [[0] * (2 * K + 1) for _ in range(2 * K)]
for i in range(2 * K):
    for j in range(2 * K):
        S_black[i][j + 1] = S_black[i][j] + black[i][j]
        S_white[i][j + 1] = S_white[i][j] + white[i][j]

ans = 0
for j in range(K):
    var_B, var_W = 0, 0
    for i in range(K):
        var_B += S_black[i][j + K] - S_black[i][j]
        var_W += S_white[i][j + K] - S_white[i][j]
        
        var_B += S_black[i + K][2 * K] - (S_black[i + K][j + K] - S_black[i + K][j])
        var_W += S_white[i + K][2 * K] - (S_white[i + K][j + K] - S_white[i + K][j])

    ans = max(ans, max(var_B + (cnt_W - var_W), var_W + (cnt_B - var_B)))
    for i in range(K):
        var_B += (S_black[i + K][j + K] - S_black[i + K][j]) - (S_black[i][j + K] - S_black[i][j])
        var_W += (S_white[i + K][j + K] - S_white[i + K][j]) - (S_white[i][j + K] - S_white[i][j])

        var_B -= S_black[i + K][2 * K] - (S_black[i + K][j + K] - S_black[i + K][j])
        var_B += S_black[i][2 * K] - (S_black[i][j + K] - S_black[i][j])
        var_W -= S_white[i + K][2 * K] - (S_white[i + K][j + K] - S_white[i + K][j])
        var_W += S_white[i][2 * K] - (S_white[i][j + K] - S_white[i][j])

        ans = max(ans, max(var_B + (cnt_W - var_W), var_W + (cnt_B - var_B)))

print(ans)