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 Ruby (2.7.1) (judge verdict: Wrong Answer)

class Problem
  attr_accessor *(?a..?z).to_a.map(&:to_sym)
  attr_accessor :map, :history
  def initialize
    @h,@w = gets.split.map(&:to_i)
    @c = gets.split.map(&:to_i).map{ _1 - 1 }
    @d = gets.split.map(&:to_i).map{ _1 - 1 }
    @map = Array.new(h){ gets.chomp }
    @history = Array.new(h){ Array.new(w,0) }
  end

  def eachgrid(y, x)
    yield y + 1, x if y < h - 1
    yield y - 1, x if y > 0
    yield y , x + 1 if x < w - 1
    yield y , x - 1 if x > 0
  end

  def each_warp(y,x)
    (y-2).upto(y+2) do |ny|
      (x-2).upto(x+2) do |nx|
        next if ny == y && nx == x
        next if nx < 0
        next if ny < 0
        next if h <= ny
        next if w <= nx
        next if map[ny][nx] == "#"
        yield ny,nx
      end
    end
  end

  def set_area_number
    seen = Array.new(h){ Array.new(w,false) }
    id = 0
    h.times do |y|
      w.times do |x|
        next if seen[y][x]
        seen[y][x] = true
        next if map[y][x] == "#"
        id += 1
        history[y][x] = id
        que = [[y,x]]
        while que.size > 0
          y,x = que.shift
          eachgrid(y,x) do |ny,nx|
            next if seen[ny][nx]
            seen[ny][nx] = true
            next if map[ny][nx] == "#"
            history[ny][nx] = id
            que << [ny,nx]
          end
        end
      end
    end
    @m = id
  end

  def make_graph
    @r = Array.new(m+1){ [] }
    h.times do |y|
      w.times do |x|
        next if map[y][x] == "#"
        sid = history[y][x]
        each_warp(y,x) do |ny,nx|
          next if map[ny][nx] == "#"
          did = history[ny][nx]
          next if sid == did
          r[sid] << did unless r[sid].include?(did)
          r[did] << sid unless r[did].include?(sid)
        end
      end
    end
  end

  def solve
    cy,cx = c
    dy,dx = d
    ch = history[cy][cx]
    dh = history[dy][dx]
    return 0 if ch == dh
    seen = Array.new(m+1, false)
    seen[ch] = true
    que = [[ch,0]]
    while que.size > 0
      v,cnt = que.shift
      return cnt if v == dh
      r[v].each do |nv|
        next if seen[nv]
        seen[nv] = true
        que << [nv, cnt + 1]
      end
    end
    return -1
  end

  def show(ans)
    puts ans
  end
end

Problem.new.instance_eval do
  set_area_number
  make_graph
  show(solve)
  # pp self
end