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)

require 'pp'
require 'set'

h, w = gets.split.map(&:to_i)
ch, cw = gets.split.map(&:to_i)
dh, dw = gets.split.map(&:to_i)
s = []
h.times do |i|
    t = gets.chomp!.chars
    s[i] = t
end

# grouping
added = Array.new(h)
h.times do |i|
    added[i] = Array.new(w, 0)
end
groups = []
g_num = 1
h.times do |i2|
    w.times do |j2|
        next if s[i2][j2] != "."
        
        #added[i2][j2] = g_num
        queue = []
        queue << [i2, j2, g_num]

        found = 0

        while queue.empty?.!
            i, j, num = queue.pop
            if added[i][j] == 0
                added[i][j] = num
                found = 1
                groups[num] ||= []
                groups[num] << [i, j]
            end
            if j > 0 && s[i][j-1] && s[i][j-1] == '.'
                if added[i][j-1] == 0
                    added[i][j-1] = num
                    groups[num] ||= []
                    groups[num] << [i, j-1]
                    found = 1
                    queue << [i, j-1, num]
                end
            end
            if j < w - 1 && s[i][j+1] && s[i][j+1] == "."
                if added[i][j+1] == 0
                    added[i][j+1] = num
                    groups[num] ||= []
                    groups[num] << [i, j+1]
                    found = 1
                    queue << [i, j+1, num]
                end
            end
            if i > 0 && s[i-1][j] && s[i-1][j] == "."
                if added[i-1][j] == 0
                    added[i-1][j] = num
                    groups[num] ||= []
                    groups[num] << [i-1, j]
                    found = 1
                    queue << [i-1, j, num]
                end
            end
            if i < h - 1 && s[i+1][j] && s[i+1][j] == "."
                if added[i+1][j] == 0
                    added[i+1][j] = num
                    groups[num] ||= []
                    groups[num] << [i+1, j]
                    found = 1
                    queue << [i+1, j, num]
                end
            end
        end

        if found == 1
            g_num += 1
        end
    end
end
=begin
pp added
pp groups
g2 = Array.new(g_num)
g_num.times do |x|
    g2[x] = Array.new(g_num, 0)
end
h.times do |i2|
    w.times do |j2|
        next if added[i2][j2] == 0

        g = added[i2][j2]
    
        (-2).upto(2) do |i|
            (-2).upto(2) do |j|
                next if i2 + i < 0 || i2 + i >= h || j2 + j < 0 || j2 + j >= w
                next if added[i2+i][j2+j] == 0
                
                if g != added[i2+i][j2+j]
                    g2[g][added[i2+i][j2+j]] = 1
                end
            end
        end
    end
end

#pp g2


start = added[ch-1][cw-1]
ed = added[dh-1][dw-1]

#pp start
#pp ed
v = Array.new(g_num, 10**10)
q2 = []
q2 << [start, 0]

while q2.empty?.!
    t, d = q2.pop

    next if d >= v[t] 

    v[t] = d

    g2[t].each_with_index do |k, i|
        next if k == 0
        q2.push([i, d+1])
    end
end

if v[ed] == 10 ** 10
    puts "-1"
else
    puts v[ed]
end
=end