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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

{-# OPTIONS_GHC -O2 #-}
{-# LANGUAGE BangPatterns, BinaryLiterals, CPP, DerivingStrategies  #-}
{-# LANGUAGE DerivingVia, FlexibleContexts, FlexibleInstances       #-}
{-# LANGUAGE GeneralizedNewtypeDeriving, KindSignatures, LambdaCase #-}
{-# LANGUAGE MagicHash, MultiParamTypeClasses, MultiWayIf           #-}
{-# LANGUAGE NumericUnderscores, OverloadedStrings, PatternSynonyms #-}
{-# LANGUAGE RankNTypes, RecordWildCards, ScopedTypeVariables       #-}
{-# LANGUAGE StandaloneDeriving, TupleSections, TypeApplications    #-}
{-# LANGUAGE TypeFamilies, TypeInType, UnboxedTuples, ViewPatterns  #-}

import           Control.Applicative
import           Control.Exception
import           Control.Monad
import           Control.Monad.Primitive
import           Control.Monad.Reader
import           Control.Monad.ST
import           Control.Monad.State.Strict
import           Data.Bifunctor
import           Data.Bool
import qualified Data.ByteString                   as B
import qualified Data.ByteString.Builder           as B
import qualified Data.ByteString.Char8             as C
import qualified Data.ByteString.Internal          as B
import qualified Data.ByteString.Unsafe            as B
import           Data.Char
import qualified Data.Foldable                     as F
import           Data.Function
import           Data.Functor.Identity
import qualified Data.IntMap.Strict                as IM
import qualified Data.IntSet                       as IS
import qualified Data.List                         as L
import qualified Data.Map.Strict                   as M
import           Data.Monoid
import           Data.Ord
import           Data.Primitive
import           Data.Proxy
import           Data.Ratio
import           Data.Semigroup
import qualified Data.Set                          as S
import           Data.Tuple
import qualified Data.Vector                       as V
import qualified Data.Vector.Algorithms.Intro      as Intro
import qualified Data.Vector.Fusion.Stream.Monadic as MS
import qualified Data.Vector.Fusion.Bundle.Monadic as MB
import           Data.Vector.Fusion.Util
import qualified Data.Vector.Generic               as G
import qualified Data.Vector.Generic.Mutable       as GM
import qualified Data.Vector.Mutable               as VM
import qualified Data.Vector.Primitive             as P
import qualified Data.Vector.Primitive.Mutable     as PM
import qualified Data.Vector.Unboxed               as U
import qualified Data.Vector.Unboxed.Mutable       as UM
import           Debug.Trace
import           Foreign                           hiding (void)
import           GHC.Exts
import           GHC.TypeLits
import           System.IO
import           Unsafe.Coerce

#define MOD 1000000007

main :: IO ()
main = do
    [n, k] <- map read.words <$> getLine
    qs <- U.unfoldrN n (runParser $ (,,) <$> int <*> int <*> (skipSpaces *> char)) <$> C.getContents
    print $ solve n k qs

solve :: Int -> Int -> U.Vector (Int, Int, Char) -> Int
solve n k qs = runST $ do
    mat0 <- UM.replicate (h * w) (0 :: Int)
    mat1 <- UM.replicate (h * w) (0 :: Int)
    U.forM_ qs $ \(x,y,c) -> do
        let x' = rem x (2 * k) + 1
            y' = rem y (2 * k) + 1
        if c == 'W'
        then UM.modify mat0 (+1) (ix x' y')
        else UM.modify mat1 (+1) (ix x' y')

    rep h $ \i -> do
        rep (w - 1) $ \j -> do
            mij <- UM.unsafeRead mat0 (ix i j)
            UM.unsafeModify mat0 (+mij) (ix i (j + 1))
    rep w $ \j -> do
        rep (h - 1) $ \i -> do
            mij <- UM.unsafeRead mat0 (ix i j)
            UM.unsafeModify mat0 (+mij) (ix (i + 1) j)
    rep h $ \i -> do
        rep (w - 1) $ \j -> do
            mij <- UM.unsafeRead mat1 (ix i j)
            UM.unsafeModify mat1 (+mij) (ix i (j + 1))
    rep w $ \j -> do
        rep (h - 1) $ \i -> do
            mij <- UM.unsafeRead mat1 (ix i j)
            UM.unsafeModify mat1 (+mij) (ix (i + 1) j)

    let query mat i0 j0 i1 j1 = do
            a <- UM.unsafeRead mat (ix i1 j1)
            b <- UM.unsafeRead mat (ix i0 j1)
            c <- UM.unsafeRead mat (ix i1 j0)
            d <- UM.unsafeRead mat (ix i0 j0)
            return $! a - b - c + d

    res0 <- MS.foldM'
        (\oacc i -> MS.foldM'
            (\iacc j -> do
                cnts <- sequence
                    [ query mat0 0 0       i j
                    , query mat1 0 j       i (j + k)
                    , query mat0 0 (j + k) i (2 * k)
                    , query mat1 i 0       (i + k) j
                    , query mat0 i j       (i + k) (j + k)
                    , query mat1 i (j + k) (i + k) (2 * k)
                    , query mat0 (i + k) 0       (2 * k) j
                    , query mat1 (i + k) j       (2 * k) (j + k)
                    , query mat0 (i + k) (j + k) (2 * k) (2 * k)
                    ]
                return $! max iacc (sum cnts)
            ) oacc (stream 0 k)
        ) 0 (stream 0 k)

    res1 <- MS.foldM'
        (\oacc i -> MS.foldM'
            (\iacc j -> do
                cnts <- sequence
                    [ query mat1 0 0       i j
                    , query mat0 0 j       i (j + k)
                    , query mat1 0 (j + k) i (2 * k)
                    , query mat0 i 0       (i + k) j
                    , query mat1 i j       (i + k) (j + k)
                    , query mat0 i (j + k) (i + k) (2 * k)
                    , query mat1 (i + k) 0       (2 * k) j
                    , query mat0 (i + k) j       (2 * k) (j + k)
                    , query mat1 (i + k) (j + k) (2 * k) (2 * k)
                    ]
                return $! max iacc (sum cnts)
            ) oacc (stream 0 k)
        ) 0 (stream 0 k)

    return $ max res0 res1
  where
    w = 2 * k + 1
    h = 2 * k + 1
    ix :: Int -> Int -> Int
    ix i j = i * w + j
    {-# INLINE ix #-}

-------------------------------------------------------------------------------
-- Utils
-------------------------------------------------------------------------------
rep :: (Monad m) => Int -> (Int -> m ()) -> m ()
rep n = flip MS.mapM_ (stream 0 n)
{-# INLINE rep #-}
rev :: (Monad m) => Int -> (Int -> m ()) -> m ()
rev !n = flip MS.mapM_ (streamR 0 n)
{-# INLINE rev #-}
stream :: (Monad m) => Int -> Int -> MS.Stream m Int
stream !l !r = MS.Stream step l where { step x | x < r = return $ MS.Yield x (x + 1) | otherwise = return MS.Done; {-# INLINE [0] step #-}}
{-# INLINE [1] stream #-}
streamR :: (Monad m) => Int -> Int -> MS.Stream m Int
streamR !l !r = MS.Stream step (r - 1) where { step x | x >= l = return $ MS.Yield x (x - 1) | otherwise = return MS.Done; {-# INLINE [0] step #-}}
{-# INLINE [1] streamR #-}
stream' :: (Monad m) => Int -> Int -> Int -> MS.Stream m Int
stream' !l !r !d = MS.Stream step l where { step x | x < r = return $ MS.Yield x (x + d) | otherwise = return MS.Done; {-# INLINE [0] step #-}}
{-# INLINE [1] stream' #-}
infixl 8 `shiftRL`, `unsafeShiftRL`
shiftRL :: Int -> Int -> Int
shiftRL = unsafeShiftRL
{-# INLINE shiftRL #-}
unsafeShiftRL :: Int -> Int -> Int
unsafeShiftRL (I# x#) (I# i#) = I# (uncheckedIShiftRL# x# i#)
{-# INLINE unsafeShiftRL #-}
type Parser a = StateT C.ByteString Maybe a
runParser :: Parser a -> C.ByteString -> Maybe (a, C.ByteString)
runParser = runStateT
{-# INLINE runParser #-}
int :: Parser Int
int = coerce $ C.readInt . C.dropWhile isSpace
{-# INLINE int #-}
int1 :: Parser Int
int1 = fmap (subtract 1) int
{-# INLINE int1 #-}
char :: Parser Char
char = coerce C.uncons
{-# INLINE char #-}
byte :: Parser Word8
byte = coerce B.uncons
{-# INLINE byte #-}
skipSpaces :: Parser ()
skipSpaces = modify' (C.dropWhile isSpace)
{-# INLINE skipSpaces #-}
lowerBoundM :: (Monad m) => Int -> Int -> (Int -> m Bool) -> m Int
lowerBoundM low high p = go low high where { go !low !high | high <= low = return high | otherwise = p mid >>= bool (go (mid + 1) high) (go low mid) where { mid = low + unsafeShiftRL (high - low) 1}}
{-# INLINE lowerBoundM #-}
upperBoundM :: (Monad m) => Int -> Int -> (Int -> m Bool) -> m Int
upperBoundM low high p = do { flg <- p high; if flg then return high else subtract 1 <$!> lowerBoundM low high (fmap not . p)}
{-# INLINE upperBoundM #-}
lowerBound :: Int -> Int -> (Int -> Bool) -> Int
lowerBound low high p = runIdentity (lowerBoundM low high (return . p))
{-# INLINE lowerBound #-}
upperBound :: Int -> Int -> (Int -> Bool) -> Int
upperBound low high p = runIdentity (upperBoundM low high (return . p))
{-# INLINE upperBound #-}