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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
//#include <boost/multiprecision/cpp_ll.hpp>
//typedef boost::multiprecision::cpp_ll ll;
typedef long double dd;
#define i_7 (ll)(1E9+7)
//#define i_7 998244353
#define i_5 i_7-2
ll mod(ll a){
    ll c=a%i_7;
    if(c>=0)return c;
    return c+i_7;
}
typedef pair<ll,ll> l_l;
ll inf=(ll)1E18;
#define rep(i,l,r) for(ll i=l;i<=r;i++)
#define pb push_back
ll max(ll a,ll b){if(a<b)return b;else return a;}
ll min(ll a,ll b){if(a>b)return b;else return a;}
void Max(ll &pos,ll val){pos=max(pos,val);}//Max(dp[n],dp[n-1]);
void Min(ll &pos,ll val){pos=min(pos,val);}
void Add(ll &pos,ll val){pos=mod(pos+val);}
dd EPS=1E-9;
#define endl "\n"
#define fastio ios::sync_with_stdio(false); cin.tie(0); cout.tie(0);



int main(){fastio
    ll n,k;cin>>n>>k;
    ll x[n],y[n];
    string c[n];
    rep(i,0,n-1){
        cin>>x[i]>>y[i]>>c[i];
        x[i]=x[i]%(2*k);
        y[i]=y[i]%(2*k);
    }
    ll desk[4*k+1][4*k+1];
    rep(i,0,4*k){
        rep(j,0,4*k){
            desk[i][j]=0;
        }
    }
    rep(i,0,n-1){
        if(c[i]=="B"){
            desk[x[i]+k+1][y[i]+k+1]++;
            desk[x[i]+2*k+1][y[i]+2*k+1]++;
            desk[x[i]+2*k+1][y[i]+k+1]--;
            desk[x[i]+k+1][y[i]+2*k+1]--;
            
            desk[x[i]+1][y[i]+1]++;
            desk[x[i]+k+1][y[i]+k+1]++;
            desk[x[i]+k+1][y[i]+1]--;
            desk[x[i]+1][y[i]+k+1]--;
        }else{
            desk[x[i]+k+1][y[i]+1]++;
            desk[x[i]+2*k+1][y[i]+k+1]++;
            desk[x[i]+k+1][y[i]+k+1]--;
            desk[x[i]+2*k+1][y[i]+1]--;
            
            desk[x[i]+1][y[i]+k+1]++;
            desk[x[i]+k+1][y[i]+2*k+1]++;
            desk[x[i]+1][y[i]+2*k+1]--;
            desk[x[i]+k+1][y[i]+k+1]--;
        }
    }
    
    
    rep(i,0,4*k){
        rep(j,1,4*k){
            desk[i][j]=desk[i][j-1]+desk[i][j];
        }
    }
    
    
    rep(j,0,4*k){
        rep(i,1,4*k){
            desk[i][j]=desk[i-1][j]+desk[i][j];
        }
    }
    
    ll a_d[2*k][2*k];
    ll ans=0;
    rep(i,0,2*k-1){
        rep(j,0,2*k-1){
            a_d[i][j]=0;
            for(ll ii=i;ii<=4*k;ii+=2*k){
                for(ll jj=j;jj<=4*k;jj+=2*k){
                    a_d[i][j]+=desk[ii][jj];
                }
            }
            ans=max(ans,a_d[i][j]);
        }
    }
    cout<<ans<<endl;
    
    return 0;
}