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: Time Limit Exceeded)

def coordinate(s,x,y)
  s[y][x]
end

def can_move?(group_1, group_2)
  group_1.each do |ele|
    (-2..2).each do |i|
      (-2..2).each do |j|
        if group_2.include?([ele[0] + i, ele[1] + j])
          return true
        end
      end
    end
  end
  return false
end

def possible_move_groups(groups, now_group)
  possible_move_groups = []
  groups.each do |group|
    possible_move_groups << group if can_move?(group, now_group)
  end
  possible_move_groups
end


H, W  = gets.split(' ').map(&:to_i)
CH, CW = gets.split(' ').map(&:to_i)
DH, DW = gets.split(' ').map(&:to_i)
s = [Array.new(W, '#')]


H.times do
  s << ['#'] + gets.chomp.split('')
end

coordinates = [] # H*Wの座標が入っている [[1, 1], [2, 1], [3, 1], [4, 1], ...]
(1..H).each do |h|
  (1..W).each do |w|
    coordinates << [w, h]
  end
end

miti_masues = []

coordinates.each do |coordinate|
  if coordinate(s, coordinate[0], coordinate[1]) == '.'
    miti_masues << coordinate
  end
end

groups = []
loop do
  group = [miti_masues.first]
  miti_masues.delete(miti_masues.first)
  gensen_gourp = []

  loop do
    group_add = []

    migi = [group.last[0] + 1, group.last[1]]
    hidari = [group.last[0] - 1, group.last[1]]
    ue = [group.last[0], group.last[1] + 1]
    sita = [group.last[0], group.last[1] - 1]
    [migi, hidari, ue, sita].each do |ele|
      if miti_masues.include?(ele)
        group_add << ele
        miti_masues.delete(ele)
      end
    end

    gensen_gourp += [group.last]
    group.pop
    group += group_add
    break if group.length == 0
  end
  groups << gensen_gourp
  break if miti_masues.length == 0
end

start_group = []
goal_group = []

groups.each do |group|
  start_group = group if group.include?([CW, CH])
  goal_group = group if group.include?([DW, DH])
end

ans = 0

cost_groups_array = [[start_group]] #cost0のグループのgroupを0番目の要素に。

loop do
  cost_groups = cost_groups_array.last
  groups -= cost_groups

  next_cost_groups = []

  cost_groups.each do  |cost_group|
    next_cost_groups += possible_move_groups(groups, cost_group)
  end

  cost_groups_array << next_cost_groups

  groups -= next_cost_groups
  break if groups == [] || next_cost_groups == []
end

find_flag = false
cost_groups_array.each_with_index do |group, i|
  if group.include?(goal_group)
    puts i
    find_flag = true
  end
end
puts -1 unless find_flag