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 JavaScript (Node.js 12.16.1) (judge verdict: Wrong Answer)

//Don't have to see. start------------------------------------------
var read = require('readline').createInterface({
  input: process.stdin, output: process.stdout
});
var obj; var inLine = [];
read.on('line', function(input){inLine.push(input);});
read.on('close', function(){
  obj = init(inLine);
  console.error('\n↑入力 ↓出力');
  Main();
});
function makeClone(obj){return (obj instanceof Set) ? new Set(Array.from(obj)) : JSON.parse(JSON.stringify(obj));}
function nextInt(){return myconv(next(),1);} function nextStrArray(){return myconv(next(),2);}
function nextIntArray(){return myconv(next(),4);} function nextCharArray(){return myconv(next(),6);}
function next(){return obj.next();} function hasNext(){return obj.hasNext();}
function init(input){  
  return {
    list : input, index : 0, max : input.length,
    hasNext : function(){return (this.index < this.max);},
    next : function(){if(this.hasNext()){return this.list[this.index++];}else{throw 'ArrayIndexOutOfBoundsException ‚There is no more input';}}
  };
}
function myout(s){console.log(s);}
function myerr(s){console.error('debug:' + require('util').inspect(s,false,null));}
//param "no" is
//unknown or outlier : return i. 1: parseInt.
//2: split space. 4: split space and parseInt.
//6: split 1 character. 7: split 1 character and parseInt.
//8: join space. 9: join nextline. 0: join no character.
function myconv(i,no){try{switch(no){case 1:return parseInt(i);case 2:return i.split(' ');case 4:return i.split(' ').map(Number);case 6:return i.split('');case 7:return i.split('').map(Number);case 8:return i.join(' ');case 9:return i.join('\n');case 0:return i.join('');default:return i;}}catch(e){return i;}}

//Don't have to see. end------------------------------------------
function Main(){
  var one = nextIntArray();
  var H = one[0];
  var W = one[1];
  one = nextIntArray();
  var sy = one[0] - 1;
  var sx = one[1] - 1;
  one = nextIntArray();
  var ey = one[0] - 1;
  var ex = one[1] - 1;
  var dy = [-1,0,1,0];
  var dx = [0,-1,0,1];
  var list = new Array(H);
  for(var i = 0; i < H; i++){
    list[i] = new Array(W);
    var tmp = nextCharArray();
    for(var j = 0; j < W; j++){
      list[i][j] = {
        s : tmp[j],
        access : false,
        count : -1
      };
    }
  }
  var queue = [[sy, sx]];
  list[sy][sx].access = true;
  list[sy][sx].count = 0;
  while(queue.length > 0){
    var now = queue.shift();
    var nowY = now[0];
    var nowX = now[1];
    for(var i = 0; i < dy.length; i++){
      var nextY = nowY + dy[i];
      var nextX = nowX + dx[i];
      if(nextY >= 0 && nextY < H && nextX >= 0 && nextX < W){
        if(!list[nextY][nextX].access && list[nextY][nextX].s == "."){
          queue.unshift([nextY, nextX]);
          list[nextY][nextX].count = list[nowY][nowX].count;
          list[nextY][nextX].access = true;
        }
      }
    }
    for(var k = nowY - 2; k <= nowY + 2; k++){
      for(var l = nowX - 2; l <= nowX + 2; l++){
        if(k >= 0 && k < H && l >= 0 && l < W){
          if(list[k][l].s == "." && !list[k][l].access){
            list[k][l].count = list[nowY][nowX].count + 1;
            list[k][l].access = true;
            queue.push([k, l]);
          }
        }
      }
    }
  }
  
  myout((list[ey][ex].count != 0) ? list[ey][ex].count : -1);
}