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 C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Math;
 
public class Solve{
    static public int mod = 1000000007;
    static public string al = "abcdefghijklmnopqrstuvwxyz";
    public static void Main(){
        // •ûj
        //
        var tn = inta();
        var t = tn[0];
        var y = tn[1];
        var start = inta();
        var startt = start[0]-1;
        var starty = start[1]-1;
        var goal = inta();
        var goalt = goal[0]-1;
        var goaly = goal[1]-1;
        
        var map = new char[t,y];
        for(int i=0;i<t;i++){
            var s = rstr();
            for(int j=0;j<y;j++){
                map[i,j] = s[j];
            }
        }
        
        var pq3 = new PriorityQueue3();
        pq3.Enqueue(0,startt,starty);
        var movet = new int[]{0,1,0,-1};
        var movey = new int[]{1,0,-1,0};
        var passed = new bool[t,y];
        while(pq3.Count() > 0){
            var de = pq3.Dequeue();
            var warp = de[0];
            var nowt = de[1];
            var nowy = de[2];
            if(nowt == goalt && nowy == goaly){
                WriteLine(warp);
                return;
            }
            if(passed[nowt,nowy]) continue;
            passed[nowt,nowy] = true;
            
            for(int i=0;i<4;i++){
                var nextt = nowt + movet[i];
                var nexty = nowy + movey[i];
                if(nextt < 0 || t <= nextt || nexty < 0 || y <= nexty) continue;
                if(passed[nextt,nexty] || map[nextt,nexty] == '#') continue;
                pq3.Enqueue(warp,nextt,nexty);
            }
            
            for(int i=-2;i<=2;i++){
                for(int j=-2;j<=2;j++){
                    if(i == 0 && j == 0) continue;
                    var nextt = nowt + i;
                    var nexty = nowy + j;
                    if(nextt < 0 || t <= nextt || nexty < 0 || y <= nexty) continue;
                    if(passed[nextt,nexty] || map[nextt,nexty] == '#') continue;
                    pq3.Enqueue(warp+1,nextt,nexty);
                }
            }
        }
        WriteLine(-1);
        
	
    }
    public static void swap(ref int a,ref int b){int temp = a;a= b;b = temp;}
    static void charswap(ref char a,ref char b){char temp = a;a= b;b = temp;}
    static int ncr(int n,int r){if(n<r)return 0;r = Min(r,n-r);long nn = 1;for(int i=n-r+1;i<=n;i++){nn = nn*i%mod;}long rr = 1;for(int i=1;i<=r;i++){rr = rr*i%mod;}rr = square((int)rr,mod-2);nn = nn * rr %mod;return (int)nn;}
    // a^b mod
    static int square(int a,int b){string binary = Convert.ToString(b,2);int bileng = binary.Length;long a_power = a;long value = 1;for(int i=bileng-1;i>=0;i--){if(binary[i] == '1'){value = value*a_power%mod;}a_power = a_power*a_power%mod;}return (int)value;}
    static int square2(int a,int b){long output = 1;var list = new List<long>();int sh = 1;long n = a;list.Add(a);while(sh < b){sh *= 2;n = n*n%mod;list.Add(n);}for(int i=list.Count-1;i>=0;i--){if(b > sh){b -= sh;sh /= 2;output = output*list[i]%mod;}}return (int)output;}
    //Šeí“Çæ
    static string rstr(){ return ReadLine(); }
    static int rint(){ return int.Parse(ReadLine()); }
    static long rlong(){ return long.Parse(ReadLine()); }
    static string[] stra(){ return ReadLine().Split(' '); }
    static char[] chara(){ string[] a=stra();string b="";for(int i=0;i<a.Length;i++){b+=a[i];}return b.ToCharArray();}
    static int[] inta(){ string[] read_str_array = ReadLine().Split(' '); int[] return_int_array = new int[read_str_array.Length]; for(int countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_int_array[countup_i] = int.Parse(read_str_array[countup_i]); } return return_int_array; }
    static int[,] inta2(int num_array,int in_array){ int[,] int_array2 = new int[num_array,in_array]; for(int i=0;i<num_array;i++){ int[] temp_array = inta(); for(int j=0;j<in_array;j++){ int_array2[i,j] = temp_array[j]; } } return int_array2; }
    static long[] longa(){ string[] read_str_array = ReadLine().Split(' '); long[] return_long_array = new long[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_long_array[countup_i] = long.Parse(read_str_array[countup_i]); } return return_long_array; }
    static double[] doublea(){ string[] read_str_array = ReadLine().Split(' '); double[] return_double_array = new double[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_double_array[countup_i] = double.Parse(read_str_array[countup_i]); } return return_double_array; }
    // -----------------------------
    static long GCD(long a,long b){ if(a < b){ long temp = a; a = b; b = temp; } if(a % b == 0){ return b; } else{ long temp = b; b = a%b; a = temp; return GCD(a,b); } }
    static long LCM(long a,long b){ return a * b / GCD(a,b); }
    static void WriteArray(int[,] a,int b,int c){for(int i=0;i<b;i++){for(int j=0;j<c;j++){if(j!=0) Write(" ");Write(a[i,j]);}WriteLine();}}
}



class UnionFind{
    private int[] _unionfind;
    private int[] _unionfind_size;
    
    public UnionFind(int a){
        _unionfind = new int[a];
        _unionfind_size = new int[a];
        for(int i=0;i<a;i++){
            _unionfind[i] = i;
            _unionfind_size[i] = 1;
        }
    }
    
    public int Root(int x){
        if(_unionfind[x] == x) return x;
        return _unionfind[x] = Root(_unionfind[x]);
    }
    
    public void Unite(int a,int b){
        a = Root(a);
        b = Root(b);
        if(a == b) return;
        if(_unionfind_size[a] < _unionfind_size[b]){
            int temp = a;
            a = b;
            b = temp;
        }
        _unionfind[b] = a;
        _unionfind_size[a] += _unionfind_size[b];
    }
    
    public bool Same(int a,int b){
        return Root(a) == Root(b);
    }
    
    public int Size(int x){
        return _unionfind_size[Root(x)];
    }
}

class PriorityQueue{
    private List<int> _priorityqueue = new List<int>();
    
    public void Enqueue(int a){
        _priorityqueue.Add(a);
        int num = _priorityqueue.Count-1;
        while(true){
            if(num == 0) break;
            if(_priorityqueue[(num-1)/2] < _priorityqueue[num]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                int temp = _priorityqueue[(num-1)/2];
                _priorityqueue[(num-1)/2] = _priorityqueue[num];
                _priorityqueue[num] = temp;
                num = (num-1)/2;
            }else{
                break;
            }
        }
    }
    
    public int Dequeue(){
        int re = _priorityqueue[0];
        int temp = _priorityqueue[0];
        _priorityqueue[0] = _priorityqueue[_priorityqueue.Count-1];
        _priorityqueue[_priorityqueue.Count-1] = temp;
        _priorityqueue.RemoveAt(_priorityqueue.Count-1);
        int num = 0;
        while(true){
            int swapnum = -1;
            if((num+1)*2 < _priorityqueue.Count){
                if(_priorityqueue[num*2+1] > _priorityqueue[(num+1)*2]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                    swapnum = num*2+1;
                }else{
                    swapnum = (num+1)*2;
                }
            }else if(num*2+1 < _priorityqueue.Count){
                swapnum = num*2+1;
            }
            if(swapnum == -1) break;
            if(_priorityqueue[swapnum] > _priorityqueue[num]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                temp = _priorityqueue[swapnum];
                _priorityqueue[swapnum] = _priorityqueue[num];
                _priorityqueue[num] = temp;
                num = swapnum;
            }else{
                break;
            }
        }
        return re;
    }
    
    public int Peek(){
        return _priorityqueue[0];
    }
    
    public int Count(){
        return _priorityqueue.Count;
    }
}
class PriorityQueue2{
    private List<int> _priorityqueue = new List<int>();
    private List<int> _accompany_info = new List<int>();
    
    public void Enqueue(int a,int b){
        _priorityqueue.Add(a);
        _accompany_info.Add(b);
        int num = _priorityqueue.Count-1;
        while(true){
            if(num == 0) break;
            if(_priorityqueue[(num-1)/2] > _priorityqueue[num]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                int temp = _priorityqueue[(num-1)/2];
                  _priorityqueue[(num-1)/2] = _priorityqueue[num];
                  _priorityqueue[num] = temp;
                temp = _accompany_info[(num-1)/2];
                  _accompany_info[(num-1)/2] = _accompany_info[num];
                  _accompany_info[num] = temp;
                num = (num-1)/2;
            }else{
                break;
            }
        }
    }
    
    public int[] Dequeue(){
        var re = new int[]{_priorityqueue[0],_accompany_info[0]};
        var qcount = _priorityqueue.Count;
        var temp = _priorityqueue[0]; _priorityqueue[0] = _priorityqueue[qcount-1]; _priorityqueue[qcount-1] = temp;
        temp = _accompany_info[0]; _accompany_info[0] = _accompany_info[qcount-1]; _accompany_info[qcount-1] = temp;
        _priorityqueue.RemoveAt(qcount-1);
        _accompany_info.RemoveAt(qcount-1);
        int num = 0;
        while(true){
            int swapnum = -1;
            if((num+1)*2 < _priorityqueue.Count){
                if(_priorityqueue[num*2+1] < _priorityqueue[(num+1)*2]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                    swapnum = num*2+1;
                }else{
                    swapnum = (num+1)*2;
                }
            }else if(num*2+1 < _priorityqueue.Count){
                swapnum = num*2+1;
            }
            if(swapnum == -1) break;
            if(_priorityqueue[swapnum] < _priorityqueue[num]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                temp = _priorityqueue[swapnum];_priorityqueue[swapnum] = _priorityqueue[num];_priorityqueue[num] = temp;
                temp = _accompany_info[swapnum];_accompany_info[swapnum] = _accompany_info[num];_accompany_info[num] = temp;
                num = swapnum;
            }else{
                break;
            }
        }
        return re;
    }
    
    public int[] Peek(){
        var temp = new int[]{_priorityqueue[0],_accompany_info[0]};
        return temp;
    }
    
    public int Count(){
        return _priorityqueue.Count;
    }
}
class PriorityQueue3{
    private List<int> _priorityqueue = new List<int>();
    private List<int> _accompany_info = new List<int>();
    private List<int> _accompany_info2 = new List<int>();
    
    public void Enqueue(int a,int b,int c){
        _priorityqueue.Add(a);
        _accompany_info.Add(b);
        _accompany_info2.Add(c);
        int num = _priorityqueue.Count-1;
        while(true){
            if(num == 0) break;
            if(_priorityqueue[(num-1)/2] > _priorityqueue[num]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                int temp = _priorityqueue[(num-1)/2];
                  _priorityqueue[(num-1)/2] = _priorityqueue[num];
                  _priorityqueue[num] = temp;
                temp = _accompany_info[(num-1)/2];
                  _accompany_info[(num-1)/2] = _accompany_info[num];
                  _accompany_info[num] = temp;
                  temp = _accompany_info2[(num-1)/2];
                  _accompany_info2[(num-1)/2] = _accompany_info2[num];
                  _accompany_info2[num] = temp;
                num = (num-1)/2;
            }else{
                break;
            }
        }
    }
    
    public int[] Dequeue(){
        var re = new int[]{_priorityqueue[0],_accompany_info[0],_accompany_info2[0]};
        var qcount = _priorityqueue.Count;
        var temp = _priorityqueue[0]; _priorityqueue[0] = _priorityqueue[qcount-1]; _priorityqueue[qcount-1] = temp;
        temp = _accompany_info[0]; _accompany_info[0] = _accompany_info[qcount-1]; _accompany_info[qcount-1] = temp;
        temp = _accompany_info2[0]; _accompany_info2[0] = _accompany_info2[qcount-1]; _accompany_info2[qcount-1] = temp;
        _priorityqueue.RemoveAt(qcount-1);
        _accompany_info.RemoveAt(qcount-1);
        _accompany_info2.RemoveAt(qcount-1);
        int num = 0;
        while(true){
            int swapnum = -1;
            if((num+1)*2 < _priorityqueue.Count){
                if(_priorityqueue[num*2+1] < _priorityqueue[(num+1)*2]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                    swapnum = num*2+1;
                }else{
                    swapnum = (num+1)*2;
                }
            }else if(num*2+1 < _priorityqueue.Count){
                swapnum = num*2+1;
            }
            if(swapnum == -1) break;
            if(_priorityqueue[swapnum] < _priorityqueue[num]){ //‘å¬‹t‚Ì˜M‚é‚Æ‚±
                temp = _priorityqueue[swapnum];_priorityqueue[swapnum] = _priorityqueue[num];_priorityqueue[num] = temp;
                temp = _accompany_info[swapnum];_accompany_info[swapnum] = _accompany_info[num];_accompany_info[num] = temp;
                temp = _accompany_info2[swapnum];_accompany_info2[swapnum] = _accompany_info2[num];_accompany_info2[num] = temp;
                num = swapnum;
            }else{
                break;
            }
        }
        return re;
    }
    
    public int[] Peek(){
        var temp = new int[]{_priorityqueue[0],_accompany_info[0],_accompany_info2[0]};
        return temp;
    }
    
    public int Count(){
        return _priorityqueue.Count;
    }
}