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: Accepted)

DEBUG = 0

INF = 10**9
WALK = [[0, 1], [0, -1], [1, 0], [-1, 0]]
WARP = [*[-2, -1, 1, 2].product([-2, -1, 1, 2]), [2, 0], [-2, 0], [0, 2], [0, -2]]

h, w = gets.split.map &:to_i
ch, cw = gets.split.map &:to_i
dh, dw = gets.split.map &:to_i
# 番兵をつけておく
v_margin = [[-1] * (w + 4)] * 2
dists = v_margin + h.times.map { [-1, -1] + gets.chomp.each_char.map { _1 == '.' ? INF : -2 } + [-1, -1] } + v_margin

puts maze if DEBUG >= 1

# 01-BFS
start = [ch - 1 + 2, cw - 1 + 2, 0]
goal_x, goal_y = dh - 1 + 2, dw - 1 + 2
dists[start[0]][start[1]] = 0
queue = [start]
until queue.empty?
  p queue if DEBUG >= 1
  puts maze if DEBUG >= 2
  i, j, cost = *queue.shift
  
  # ゴールなら
  if i == goal_x and j == goal_y
    p cost
    exit
  end

  is_edge = false
  WALK.each do |(di, dj)|
    _i, _j = i + di, j + dj
    if dists[_i][_j] <= cost
      # 枝刈り1用
      is_edge = true if dists[_i][_j] == -2

      next
    end
    dists[_i][_j] = cost
    queue.unshift [_i, _j, cost]
  end

  # 枝刈り1: 徒歩範囲の内部ではワープしない
  if is_edge
    WARP.each do |(di, dj)|
      _i, _j, _cost = i + di, j + dj, cost + 1

      # 枝刈り2: 更新できないときはキューに追加しない
      next if _cost >= dists[_i][_j]

      dists[_i][_j] = _cost
      queue << [_i, _j, _cost]
    end
  end
end

# 到達できない場合
p -1