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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using static System.Console;
using System.Text;
using System.IO;
using static System.Math;
using System.Numerics;

namespace AtCoder
{
    class Program
    {
        static public long[] Sarray() { return ReadLine().Split().Select(long.Parse).ToArray(); }
        static public List<long> Slist() { return ReadLine().Split().Select(long.Parse).ToList(); }
        static long Mod = (long)1e9 + 7;
        static void Main(string[] args)
        {
            SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
            Solve();
            Out.Flush();
            Read();
        }
        static void Solve()
        {
            var NK = Sarray();
            var N = NK[0];
            var K = NK[1];
            var K2 = 2 * K;
            var black = new long[K2, K2];
            var white = new long[K2, K2];
            for (var i = 0; i < N; ++i)
            {
                var xyc = ReadLine().Split();
                var x = long.Parse(xyc[0]) % K2;
                var y = long.Parse(xyc[1]) % K2;
                if (xyc[2] == "B")
                    ++black[x, y];
                else
                    ++white[x, y];
            }
            var bRui = new long[K2 + 1, K2 + 1];
            var wRui = new long[K2 + 1, K2 + 1];
            for (var x = 0; x < K2; ++x)
                for (var y = 0; y < K2; ++y)
                {
                    bRui[x + 1, y + 1] = bRui[x, y + 1] + bRui[x + 1, y] - bRui[x, y] + black[x, y];
                    wRui[x + 1, y + 1] = wRui[x, y + 1] + wRui[x + 1, y] - wRui[x, y] + white[x, y];
                }
            var max = 0L;
            for (var x = 0; x < K2; ++x)
                for (var y = 0; y < K2; ++y)
                {
                    var sum = 0L;
                    sum += Get2DRui(bRui, x, y, x + K, y + K);
                    sum += Get2DRui(bRui, x + K, y + K, x + K2, y + K2);
                    sum += Get2DRui(wRui, x + K, y, x + K2, y + K);
                    sum += Get2DRui(wRui, x, y + K, x + K, y + K2);
                    max = Max(max, sum);
                }
            WriteLine(max);
        }
        //[xl,xr)x[yd,yu)の範囲の累積和
        static long Get2DRui(long[,] rui,long xl,long yd, long xr, long yu)
        {
            var xLim = rui.GetLength(0) - 1;
            var yLim = rui.GetLength(1) - 1;
            long xpr = 0, xpl = 0, ypu = 0, ypd = 0;
            if (xLim <= xl)
            {
                xl %= xLim;
                xr %= xLim;
            }
            if (yLim <= yd)
            {
                yd %= yLim;
                yu %= yLim;
            }
            if (xLim < xr)
            {
                xpr = xr % xLim;
                xr = xLim;
            }
            if (yLim < yu)
            {
                ypu = yu % yLim;
                yu = yLim;
            }

            var men = 0L;
            men += rui[xr,   yu] - rui[xl,   yu] - rui[xr,   yd] + rui[xl,   yd];
            men += rui[xr,  ypu] - rui[xl,  ypu] - rui[xr,  ypd] + rui[xl,  ypd];
            men += rui[xpr,  yu] - rui[xpl,  yu] - rui[xpr,  yd] + rui[xpl,  yd];
            men += rui[xpr, ypu] - rui[xpl, ypu] - rui[xpr, ypd] + rui[xpl, ypd];
            return men;
        }
    }
}