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.Generic;
using System.Diagnostics;
using System.Linq;

namespace AtCoder.D
{
    public class Program
    {
        public static void Main() { var r = GetResult(); Debug.WriteLine(r); Console.Write(r); }

        private static object GetResult()
        {
            var (H, W) = Read2Longs();
            var (Ch, Cw) = Read2Longs();
            Ch--; Cw--; // インデックス変換
            var (Dh, Dw) = Read2Longs();
            Dh--; Dw--; // インデックス変換
            var S = new List<string>();
            for (int i = 0; i < H; i++)
            {
                S.Add(ReadText());
            }
            var table = GetTable(H, W);
            table[Ch, Cw] = 0;
            var queue = new Queue<Point>();
            queue.Enqueue(new Point(Ch, Cw));
            var warpQueue = new Queue<Point>();

            // 到達可能な範囲で埋め尽くす
            while (queue.Count > 0 || warpQueue.Count > 0)
            {
                while (queue.Count > 0)
                {
                    var p = queue.Dequeue();

                    var nexts = new Point[]
                    {
                    new Point(p.H + 1, p.W),
                    new Point(p.H, p.W + 1),
                    new Point(p.H - 1, p.W),
                    new Point(p.H, p.W - 1),
                    };

                    foreach (var next in nexts)
                    {
                        if (next.H < 0 || next.H >= H) continue;
                        if (next.W < 0 || next.W >= W) continue;
                        if (S[next.H][next.W] == '#') continue;
                        //if (table[next.H, next.W] != -1) continue;

                        // ワープ回数をカウント (ワープ不要で更新する)
                        //if (table[next.H, next.W] >= table[p.H, p.W])
                        {
                            table[next.H, next.W] = table[p.H, p.W];
                        }

                        queue.Enqueue(next);
                    }

                    var warps = new Point[]
                    {
                    new Point(p.H + 2, p.W - 2),
                    new Point(p.H + 2, p.W - 1),
                    new Point(p.H + 2, p.W),
                    new Point(p.H + 2, p.W + 1),
                    new Point(p.H + 2, p.W + 2),
                    new Point(p.H + 1, p.W - 2),
                    new Point(p.H + 1, p.W - 1),
                    //new Point(p.H + 1, p.W),
                    new Point(p.H + 1, p.W + 1),
                    new Point(p.H + 1, p.W + 2),
                    new Point(p.H, p.W - 2),
                    //new Point(p.H, p.W - 1),
                    //new Point(p.H, p.W),
                    //new Point(p.H, p.W + 1),
                    new Point(p.H, p.W + 2),
                    new Point(p.H - 1, p.W - 2),
                    new Point(p.H - 1, p.W - 1),
                    //new Point(p.H - 1, p.W),
                    new Point(p.H - 1, p.W + 1),
                    new Point(p.H - 1, p.W + 2),
                    new Point(p.H - 2, p.W - 2),
                    new Point(p.H - 2, p.W - 1),
                    new Point(p.H - 2, p.W),
                    new Point(p.H - 2, p.W + 1),
                    new Point(p.H - 2, p.W + 2),
                    };

                    foreach (var warp in warps)
                    {
                        if (warp.H < 0 || warp.H >= H) continue;
                        if (warp.W < 0 || warp.W >= W) continue;
                        if (S[warp.H][warp.W] == '#') continue;
                        if (table[warp.H, warp.W] != -1) continue;

                        // ワープ回数をカウント (ワープ不要で更新する)
                        {
                            table[warp.H, warp.W] = table[p.H, p.W] + 1;
                        }

                        warpQueue.Enqueue(warp);
                    }
                }
                if (table[Dh, Dw] != -1)
                {
                    return table[Dh, Dw];
                }
                queue = warpQueue;
                warpQueue = new Queue<Point>();
            }

            // デバッグ用
            var x = new List<string>();
            for (int i = 0; i < H; i++)
            {
                var s = "";
                for (int j = 0; j < W; j++)
                {
                    s += table[i, j].ToString();
                }
                x.Add(s);
            }

            return table[Dh, Dw];
        }

        private static int[,] GetTable(long H, long W)
        {
            var t = new int[H, W];
            for (int i = 0; i < H; i++)
            {
                for (int j = 0; j < W; j++)
                {
                    t[i, j] = -1;
                }
            }


            return t;
        }

        class Point
        {
            public int H { get; set; }
            public int W { get; set; }

            public Point(long h, long w)
            {
                H = (int)h;
                W = (int)w;
            }
        }

        #region Console.Read

        public static string ReadText() { return Console.ReadLine(); }
        public static List<string> ReadTexts() { return Console.ReadLine().Split(' ').ToList(); }
        public static long ReadLong() { return long.Parse(Console.ReadLine()); }
        public static (long, long) Read2Longs() { var ls = ReadLongs(); return (ls[0], ls[1]); }
        public static (long, long, long) Read3Longs() { var ls = ReadLongs(); return (ls[0], ls[1], ls[2]); }
        public static (long, long, long, long) Read4Longs() { var ls = ReadLongs(); return (ls[0], ls[1], ls[2], ls[3]); }
        public static (long, long, long, long, long) Read5Longs() { var ls = ReadLongs(); return (ls[0], ls[1], ls[2], ls[3], ls[4]); }
        public static List<long> ReadLongs() { return Console.ReadLine().Split(' ').Select(x => long.Parse(x)).ToList(); }

        #endregion
    }
}