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 Lua (Lua 5.3.5) (judge verdict: Time Limit Exceeded)

local List = {}
function List.new ()
    return {first = 0, last = -1}
end

function List.pushleft (list, value)
    local first = list.first - 1
    list.first = first
    list[first] = value
end

function List.pushright (list, value)
    local last = list.last + 1
    list.last = last
    list[last] = value
end

function List.popleft (list)
    local first = list.first
    local value = list[first]
    list[first] = nil
    list.first = first + 1
    return value
end

function List.popright (list)
    local last = list.last
    local value = list[last]
    list[last] = nil
    list.last = last - 1
    return value
end

function List.empty (list)
    return list.first > list.last
end

----------

local h,w=io.read("n","n")
local ch,cw=io.read("n","n")
local dh,dw=io.read("n","n","l")

local s={}
for i=1,h do
    local input=io.read()
    s[i]={}
    for j=1,w do
        s[i][j]=input:sub(j,j)
    end
end

local checkin=List.new()
local checked=List.new()
List.pushright(checkin,{ch,cw})

local cost={}
local INF=math.maxinteger
for i=1,h do
    cost[i]={}
    for j=1,w do
        cost[i][j]=INF
    end
end
cost[ch][cw]=0
local dx={1,0,-1,0}
local dy={0,1,0,-1}

while (not List.empty(checkin)) or (not List.empty(checked)) do
    if not List.empty(checkin) then
        local q=List.popleft(checkin)
        List.pushright(checked,q)
        local x,y=q[1],q[2]
        for i=1,4 do
            local nx,ny=x+dx[i],y+dy[i]
            local checker=(nx<1 or nx>h or ny<1 or ny>w)
            if not checker and s[nx][ny]~="#"then
                if cost[x][y]<cost[nx][ny] then
                    cost[nx][ny]=cost[x][y]
                    List.pushright(checkin,{nx,ny})
                end
            end
        end
    else
        local q=List.popleft(checked)
        local x,y=q[1],q[2]
        for dx=-2,2 do
            for dy=-2,2 do
                local nx,ny=x+dx,y+dy
                local checker=(nx<1 or nx>h or ny<1 or ny>w)
                if not checker and s[nx][ny]~="#"then
                    if cost[x][y]+1<cost[nx][ny] then
                        cost[nx][ny]=cost[x][y]+1
                        List.pushright(checkin,{nx,ny})
                    end
                end
            end
        end
    end
end

print(cost[dh][dw]<INF and cost[dh][dw] or -1)