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: Accepted)

{-# LANGUAGE BangPatterns #-}
import           Control.Exception           (assert)
import           Control.Monad
import           Control.Monad.Primitive
import qualified Control.Monad.ST            as ST
import qualified Data.Array.IO               as IO
import qualified Data.Array.ST               as ST
import qualified Data.Array.Unboxed          as A
import           Data.Bits
import qualified Data.ByteString.Char8       as BS
import           Data.Char
import           Data.Foldable
import           Data.List
import qualified Data.Map.Strict             as Map
import           Data.Maybe
import qualified Data.Sequence               as Seq
import qualified Data.Set                    as Set
import qualified Data.Vector                 as VB
import qualified Data.Vector.Mutable         as VBM
import qualified Data.Vector.Unboxed         as V
import qualified Data.Vector.Unboxed.Mutable as VM
import           Debug.Trace

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine
getIntVec n = V.unfoldrN n (BS.readInt . BS.dropWhile isSpace) <$> BS.getLine

readInteger = fst . fromJust . BS.readInteger
readIntegerList = map readInteger . BS.words
getInteger = readInteger <$> BS.getLine
getIntegerList = readIntegerList <$> BS.getLine

inf :: Int
inf = 10^18

main = do
  [h, w] <- getIntList
  [ch, cw] <- getIntList
  [dh, dw] <- getIntList

  s <- replicateM h $ BS.getLine
  let s' = VB.fromList s

  arr <- IO.newArray ((1,1),(h,w)) inf :: IO (IO.IOUArray (Int,Int) Int)

  let getC y x | y < 1 || y > h || x < 1 || x > w = '#'
               | otherwise = (s' VB.! (y-1)) `BS.index` (x-1)

  let bfs :: Seq.Seq (Int, Int, Int) -> IO ()
      bfs q
        | Seq.null q = return ()
        | otherwise = do
            let (ct,y,x) Seq.:< q' = Seq.viewl q
                c = getC y x
            if c == '#'
              then bfs q'
              else do v <- IO.readArray arr (y, x)
                      if v <= ct
                        then bfs q'
                        else do IO.writeArray arr (y, x) ct
                                if (y == dh && x == dw)
                                  then bfs q'
                                  else do
                                  let walkable = [(ct, y',x') |
                                                  (y',x') <- [(y-1, x),(y+1, x), (y, x-1), (y, x+1)],
                                                  getC y' x' /= '#']
                                      warpable = [(ct+1, y+dy,x+dx) |
                                                  dy <- [(-2)..2], dx <- [(-2)..2],
                                                  getC (y+dy) (x+dx) /= '#']
                                      q1 = (Seq.fromList walkable) Seq.>< q'
                                      q2 = q1 Seq.>< (Seq.fromList warpable)
                                  bfs $ q2

  bfs $ Seq.fromList [(0,ch,cw)]
  ans <- IO.readArray arr (dh, dw)
  print $ if ans >= inf then (-1) else ans