Problem:
Score : 400 points

Problem Statement

A maze is composed of a grid of H \times W squares - H vertical, W horizontal.

The square at the i-th row from the top and the j-th column from the left - (i,j) - is a wall if S_{ij} is # and a road if S_{ij} is ..

There is a magician in (C_h,C_w). He can do the following two kinds of moves:

Move A: Walk to a road square that is vertically or horizontally adjacent to the square he is currently in.

Move B: Use magic to warp himself to a road square in the 5\times 5 area centered at the square he is currently in.

In either case, he cannot go out of the maze.

At least how many times does he need to use the magic to reach (D_h, D_w)?

Constraints

1 \leq H,W \leq 10^3

1 \leq C_h,D_h \leq H

1 \leq C_w,D_w \leq W

S_{ij} is # or ..

S_{C_h C_w} and S_{D_h D_w} are ..

(C_h,C_w) \neq (D_h,D_w)

Input

Input is given from Standard Input in the following format:

H W
C_h C_w
D_h D_w
S_{11}\ldots S_{1W}
\vdots
S_{H1}\ldots S_{HW}

Output

Print the minimum number of times the magician needs to use the magic. If he cannot reach (D_h,D_w), print -1 instead.

Sample Input 1

4 4
1 1
4 4
..#.
..#.
.#..
.#..

Sample Output 1

1

For example, by walking to (2,2) and then using the magic to travel to (4,4), just one use of magic is enough.

Note that he cannot walk diagonally.

Sample Input 2

4 4
1 4
4 1
.##.
####
####
.##.

Sample Output 2

-1

He cannot move from there.

Sample Input 3

4 4
2 2
3 3
....
....
....
....

Sample Output 3

0

No use of magic is needed.

Sample Input 4

4 5
1 2
2 5
#.###
####.
#..##
#..##

Sample Output 4

2

Solution in Haskell (GHC 8.8.3) (judge verdict: Wrong Answer)

{-# LANGUAGE BangPatterns        #-}
{-# LANGUAGE FlexibleContexts    #-}
{-# LANGUAGE MultiWayIf          #-}
{-# LANGUAGE RankNTypes          #-}
{-# LANGUAGE ScopedTypeVariables #-}
import           Control.Monad
import           Control.Monad.ST
import           Control.Monad.State
import           Control.Monad.Trans.State
import           Data.Array.IArray
import           Data.Array.ST
import           Data.Array.Unboxed
import           Data.Bits
import qualified Data.ByteString.Char8       as BS
import           Data.Char
import           Data.Int                    (Int64)
import           Data.List
import           Data.Maybe
import           Data.Proxy
import           Data.Sequence               (Seq, (<|), (><), (|>))
import qualified Data.Sequence               as Seq
import qualified Data.Set                    as S
import           Data.STRef
import qualified Data.Vector                 as V
import qualified Data.Vector.Mutable         as VM
import qualified Data.Vector.Unboxed         as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import           Numeric

main :: IO ()
main = do
  [h,w] <- readLnAsListWith unconsInt
  [cx,cy] <- readLnAsListWith unconsInt
  [dx,dy] <- readLnAsListWith unconsInt
  sgrid <- readLnAs2DArrayWith unconsChar h w :: IO (UArray (Int,Int) Char)

  let ggrid = gridDfs (h,w) (cx,cy) (\g (i,j) -> g ! (i,j) == '#') sgrid

  print $ ggrid ! (dx,dy)

  return ()


type Grid a = UArray (Int,Int) a
type MGrid s = STUArray s (Int,Int)

gridDfs :: (Int,Int) -> (Int,Int) -> (Grid a -> (Int,Int) -> Bool) -> Grid a -> UArray (Int,Int) Int
gridDfs (h',w') (sh,sw) f g = runSTUArray $ do
  seen <- newArray ((1,1),(h',w')) (-1) :: ST s (STUArray s (Int,Int) Int)
  go seen (sh,sw) 0
  return seen

  where go :: MGrid s Int -> (Int,Int) -> Int -> ST s ()
        go seen (h,w) color = do
          c <- readArray seen (h,w)
          if c /= -1 && c <= color
            then return ()
            else do writeArray seen (h,w) color
                    let nexts1 = filter (\(nh,nw) -> nh >= 1 && nh <= h' && nw >= 1 && nw <= w') [(h+1,w),(h-1,w),(h,w+1),(h,w-1)]
                    forM_ nexts1 $ \(nh,nw) ->
                      if g `f` (nh,nw)
                      then do return ()
                      else do go seen (nh,nw) color

                    let nexts2 = filter (\(nh,nw) -> nh >= 1 && nh <= h' && nw >= 1 && nw <= w') [(h+x,w+y) | x <- [-2..2], y <- [-2..2], (x,y) /= (0,0)]
                    forM_ nexts2 $ \(nh,nw) -> do
                      c2 <- readArray seen (nh,nw)
                      if g `f` (nh,nw) || c2 /= (-1)
                      then do return ()
                      else do go seen (nh,nw) (color+1)





-- gridBfs :: (Int,Int) -> (Int,Int) -> (Grid a -> (Int,Int) -> Bool) -> Grid a -> ()
-- gridBfs (h',w') (cx,cy) f g = runST $ do
--   seen <- newArray ((1,1),(h',w')) (-1)
--   let queue = pushBack empty (cx,cy)



--   -- go seen (sh,sw)

--   where go :: MGrid s Bool -> Queue (Int,Int) -> (Int,Int) -> Int -> ST s ()
--         go seen q (h,w) color = do
--           writeArray seen (h,w) color
--           let cand = filter (\(nh,nw) -> not (nh < 1 || nh > h' || nw < 1 || nw > w' || g `f` (nh,nw)) ) [(h+1,w),(h-1,w),(h,w+1),(h,w-1)]
--               q2 = foldl pushBack rest cand







--           forM_ nexts $ \(nh,nw) -> do
--             s <- readArray seen (nh,nw)
--             if s || nh < 1 || nh > h' || nw < 1 || nw > w' || g `f` (nh,nw)
--               then do return ()
--               else do go seen (nh,nw)




-- data Queue a = Q { front :: [a]
--                  , rear  :: [a]
--                  , thunk :: [a]
--                  } deriving (Eq)

-- empty :: Queue a
-- empty = Q [] [] []

-- pushBack :: Queue a -> a -> Queue a
-- pushBack e (Q f r t) = check $! Q f (e:r) t

-- popFront :: Queue a -> (a, Queue a)
-- popFront (Q (e:f) r t) = (e, check $! Q f r t)
-- popFront (Q [] _ _)    = error "empty queue"

-- check :: Queue a -> Queue a
-- check (Q f r (_:t)) = Q f r t
-- check (Q f r []) = let ff = rotate f r []
--                    in Q ff [] ff
--   where rotate :: [a] -> [a] -> [a] -> [a]
--         rotate [] (y:_) zs      = y:zs
--         rotate (x:xs) (y:ys) zs = x : rotate xs ys (y:zs)

-- instance Show a => Show (Queue a) where
--   show (Q f r t) = "Q { " <> show f <> "\n" <>
--                    "    " <> show r <> "\n" <>

--                    "    " <> show t <> "\n" <>
--                    "  }"




-- converter
unconsChar :: StateT BS.ByteString Maybe Char
unconsChar = StateT BS.uncons

unconsInt :: StateT BS.ByteString Maybe Int
unconsInt = StateT $ BS.readInt . BS.dropWhile isSpace

unconsInteger :: StateT BS.ByteString Maybe Integer
unconsInteger = StateT $ BS.readInteger . BS.dropWhile isSpace

-- 1D Data
-- for list
readLnAsListWith :: StateT BS.ByteString Maybe a -> IO [a]
readLnAsListWith !st = unfoldr (runStateT st) <$> BS.getLine

-- for array (boxed or unboxed)
readLnAsArrayWith :: IArray a e => StateT BS.ByteString Maybe e -> Int -> IO (a Int e)
readLnAsArrayWith !st !n = listArray (1,n) <$> readLnAsListWith st

-- for boxed vector
readLnAsVecWith :: StateT BS.ByteString Maybe a -> Int -> IO (V.Vector a)
readLnAsVecWith !st !n = V.unfoldrN n (runStateT st) <$> BS.getLine

-- for unboxed vector
readLnAsUVecWith :: VU.Unbox a => StateT BS.ByteString Maybe a -> Int -> IO (VU.Vector a)
readLnAsUVecWith !st !n = VU.unfoldrN n (runStateT st) <$> BS.getLine

-- example: 1D tuple vector
readLnAsUVecWith2Tuple :: VU.Unbox a => StateT BS.ByteString Maybe a -> Int -> IO (VU.Vector (a,a))
readLnAsUVecWith2Tuple !st !n = VU.replicateM n $ (\vec -> (vec VU.! 0, vec VU.! 1)) <$> readLnAsUVecWith st 2


-- 2D Data
-- n: number of lines
-- m: number of values per a line

-- for boxed array
readLnAs2DArrayWith :: IArray a e => StateT BS.ByteString Maybe e -> Int -> Int -> IO (a (Int,Int) e)
readLnAs2DArrayWith !st !n !m = listArray ((1,1),(n,m)) . unfoldr (runStateT st) . BS.concat <$> replicateM n BS.getLine

-- for boxed vector
readLnAs2DVecWith :: StateT BS.ByteString Maybe a -> Int -> Int -> IO (V.Vector (V.Vector a))
readLnAs2DVecWith !st !n !m = V.replicateM n $ V.unfoldrN m (runStateT st) <$> BS.getLine

-- for unboxed vector
readLnAs2DUVecWith :: VU.Unbox a => StateT BS.ByteString Maybe a -> Int -> Int -> IO (V.Vector (VU.Vector a))
readLnAs2DUVecWith !st !n !m = V.replicateM n $ VU.unfoldrN m (runStateT st) <$> BS.getLine