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 C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace Abc086D
{
    class Program
    {
        static int N, K;
        static List<Desire> Desires = new List<Desire>();
        struct Desire
        {
            public int X, Y;
            public char Color;
            public Desire(int x, int y, char color) {
                X = x <= (K + K) ? x : x - (x / (K + K)) * (K + K);
                Y = y <= (K + K) ? y : y - (y / (K + K)) * (K + K);
                Color = color;
            }
        }
        static void InitializeProblem(System.IO.TextReader reader) {
            var buffer = reader.ReadLine().Split();
            N = int.Parse(buffer[0]);
            K = int.Parse(buffer[1]);
            for(var i = 0; i != N; ++i) {
                buffer = reader.ReadLine().Split();
                Desires.Add(new Desire(int.Parse(buffer[0]), int.Parse(buffer[1]), buffer[2][0]));
            }
        }

        static int GetNumberOfSatisfaction(int bottom, int left, int width, int height, char color) {
            return Desires.Where(d => (left <= d.X && d.X <= left + width - 1
            && bottom <= d.Y && d.Y <= bottom + height - 1
            && d.Color == color)).Count();
        }
        internal static int SolveProblem(System.IO.TextReader reader) {
            InitializeProblem(reader);
            var satisfactionMax = 0;
            for(var w = K; w >= 1; ++w) {
                for(var h = K; h >= 1; ++h) {
                    var numberOfSatisfactions = GetNumberOfSatisfaction(0, 0, w, h, 'W')
                        + GetNumberOfSatisfaction(w, 0, K, h, 'B')
                        + GetNumberOfSatisfaction(w + K, 0, K - w, h, 'W')
                        + GetNumberOfSatisfaction(0, h, w, K, 'B')
                        + GetNumberOfSatisfaction(w, h, K, K, 'W')
                        + GetNumberOfSatisfaction(w + K, h, K - w, K, 'B')
                        + GetNumberOfSatisfaction(0, h + K, w, K - h, 'W')
                        + GetNumberOfSatisfaction(w, h + K, K, K - h, 'B')
                        + GetNumberOfSatisfaction(w + K, h + K, K - w, K - h, 'W');
                    satisfactionMax = Math.Max(numberOfSatisfactions, N - numberOfSatisfactions);
                    if(satisfactionMax == N) {
                        return N;
                    }
                }
            }
            return satisfactionMax;
        }

        static void Main(string[] args) {
            Console.WriteLine(SolveProblem(Console.In));
        }
    }
}