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 Go (1.14.1) (judge verdict: Time Limit Exceeded)

package main

import (
        "bufio"
        "fmt"
        "os"
)

// using 01-BFS
func solution(h, w int, c, d [2]int, s [][]byte) int {

        c[0], c[1] = c[0]-1, c[1]-1
        d[0], d[1] = d[0]-1, d[1]-1

        dx := [4]int{-1, 0, 1, 0}
        dy := [4]int{0, -1, 0, 1}

        const INF = 1e9
        dist := make([][]int, h)
        for i := range dist {
                dist[i] = make([]int, w)
                for j := range dist[i] {
                        dist[i][j] = INF
                }
        }
        // dequeを使う
        // cost: 0を使う辺は前からappend
        // cost: 1を使う辺は後からappend
        deque := make([][2]int, 0)
        dist[c[0]][c[1]] = 0
        deque = emplace_back(deque, [2]int{c[0], c[1]})
        for len(deque) > 0 {
                cur := deque[0]
                x := cur[0]
                y := cur[1]
                di := dist[x][y]
                deque = deque[1:]
                for i := 0; i < 4; i++ {
                        nx := x + dx[i]
                        ny := y + dy[i]
                        if nx < 0 || nx >= h || ny < 0 || ny >= w {
                                continue
                        }
                        if s[nx][ny] == '#' {
                                continue
                        }
                        if dist[nx][ny] <= di {
                                continue
                        }
                        dist[nx][ny] = di
                        deque = emplace_front(deque, [2]int{nx, ny})
                }
                // warp
                for i := -2; i <= 2; i++ {
                        for j := -2; j <= 2; j++ {
                                nx := x + i
                                ny := y + j
                                if nx < 0 || nx >= h || ny < 0 || ny >= w {
                                        continue
                                }
                                if s[nx][ny] == '#' {
                                        continue
                                }
                                if dist[nx][ny] <= di+1 {
                                        continue
                                }
                                dist[nx][ny] = di + 1
                                deque = emplace_back(deque, [2]int{nx, ny})
                        }
                }
        }
        out := dist[d[0]][d[1]]
        if out == INF {
                return -1
        }
        return out
}

func emplace_front(deque [][2]int, q [2]int) [][2]int {
        out := make([][2]int, len(deque)+1)
        out[0] = q
        copy(out[1:], deque)
        return out
}

func emplace_back(deque [][2]int, q [2]int) [][2]int {
        out := make([][2]int, len(deque)+1)
        out[len(deque)] = q
        copy(out[:len(deque)], deque)
        return out
}

func main() {
        r := bufio.NewReader(os.Stdin)
        var h, w int
        fmt.Fscan(r, &h, &w)
        var c [2]int
        fmt.Fscan(r, &c[0], &c[1])
        var d [2]int
        fmt.Fscan(r, &d[0], &d[1])
        s := make([][]byte, h)
        for i := range s {
                var t string
                s[i] = make([]byte, w)
                fmt.Fscan(r, &t)
                for j := range t {
                        s[i][j] = t[j]
                }
        }
        fmt.Println(solution(h, w, c, d, s))
}