Problem:
Score : 600 points

Problem Statement

Snuke, a water strider, lives in a rectangular pond that can be seen as a grid with H east-west rows and W north-south columns. Let (i,j) be the square at the i-th row from the north and j-th column from the west.

Some of the squares have a lotus leaf on it and cannot be entered.
The square (i,j) has a lotus leaf on it if c_{ij} is @, and it does not if c_{ij} is ..

In one stroke, Snuke can move between 1 and K squares (inclusive) toward one of the four directions: north, east, south, and west.
The move may not pass through a square with a lotus leaf. Moving to such a square or out of the pond is also forbidden.

Find the minimum number of strokes Snuke takes to travel from the square (x_1,y_1) to (x_2,y_2).
If the travel from (x_1,y_1) to (x_2,y_2) is impossible, point out that fact.

Constraints

1 \leq H,W,K \leq 10^6

H \times W \leq 10^6

1 \leq x_1,x_2 \leq H

1 \leq y_1,y_2 \leq W

x_1 \neq x_2 or y_1 \neq y_2.

c_{i,j} is . or @.

c_{x_1,y_1} = .

c_{x_2,y_2} = .

All numbers in input are integers.

Input

Input is given from Standard Input in the following format:

H W K
x_1 y_1 x_2 y_2
c_{1,1}c_{1,2} .. c_{1,W}
c_{2,1}c_{2,2} .. c_{2,W}
:
c_{H,1}c_{H,2} .. c_{H,W}

Output

Print the minimum number of strokes Snuke takes to travel from the square (x_1,y_1) to (x_2,y_2), or print -1 if the travel is impossible.

Sample Input 1

3 5 2
3 2 3 4
.....
.@..@
..@..

Sample Output 1

5

Initially, Snuke is at the square (3,2).
He can reach the square (3, 4) by making five strokes as follows:

From (3, 2), go west one square to (3, 1).

From (3, 1), go north two squares to (1, 1).

From (1, 1), go east two squares to (1, 3).

From (1, 3), go east one square to (1, 4).

From (1, 4), go south two squares to (3, 4).

Sample Input 2

1 6 4
1 1 1 6
......

Sample Output 2

2

Sample Input 3

3 3 1
2 1 2 3
.@.
.@.
.@.

Sample Output 3

-1

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

using System;
using System.Collections.Generic;
using System.Collections;
using static Assistant.Input;
using static Assistant.Debug;
using System.Linq;
using Assistant;

namespace ABC170F
{
    class Program
    {
        static void Main(string[] args)
        {
            var H = RInt;
            var W = RInt;
            var K = RInt;

            int xs = RInt;
            int ys = RInt;
            int xt = RInt;
            int yt = RInt;

            char[,] map = new char[H + 2, W + 2];
            for (int i = 0; i < H + 2; i++)
            {
                string s = "";
                if (i != 0 && i != H + 1) s = Console.ReadLine();
                for (int j = 0; j < W + 2; j++)
                {
                    if (i == 0 || i == H + 1 || j == 0 || j == W + 1) map[i, j] = '@';
                    else map[i, j] = s[j - 1];
                }
            }

            Draw2D(map);

            int[,] dist = new int[H + 2, W + 2];
            for (int i = 0; i < H + 2; i++)
            {
                for (int j = 0; j < W + 2; j++)
                {
                    dist[i, j] = int.MaxValue / 2;
                }
            }
            dist[xs, ys] = 0;
            PriorityQueue<int, Data> que = new PriorityQueue<int, Data>(d => d.d, false);
            que.Enqueue(new Data() { x = xs, y = ys, d = 0 });
            map[xs, ys] = '@';

            int[] dx = new int[4] { 0, 1, 0, -1 };
            int[] dy = new int[4] { 1, 0, -1, 0 };
            while (que.Count > 0)
            {
                var q = que.Dequeue();
                int x = q.x;
                int y = q.y;

                for (int i = 0; i < 4; i++)
                {
                    for (int k = 1; k <= K; k++)
                    {
                        int nx = x + k * dx[i];
                        int ny = y + k * dy[i];
                        if (map[nx, ny] != '@')
                        {
                            if (dist[nx, ny] < dist[x, y] + 1) break;
                            dist[nx, ny] = dist[x, y] + 1;

                            que.Enqueue(new Data() { x = nx, y = ny, d = q.d + 1 });
                        }
                        else break;
                    }
                }

                if (dist[xt, yt] != int.MaxValue / 2) break;
            }

            Draw2D(dist);

            int ans = dist[xt, yt];
            if (ans == int.MaxValue / 2) ans = -1;
            Console.WriteLine(ans);
        }

        struct Data
        {
            public int x;
            public int y;
            public int d;
        }
    }

    public class PriorityQueue<TKey, TValue> : IEnumerable<TValue>
    {
        private readonly List<KeyValuePair<TKey, TValue>> _data = new List<KeyValuePair<TKey, TValue>>();
        private readonly bool _isDescending;
        private readonly Func<TValue, TKey> _keySelector;
        private readonly IComparer<TKey> _keyComparer;

        public PriorityQueue(Func<TValue, TKey> keySelector, bool isDescending = true)
            : this(keySelector, Comparer<TKey>.Default, isDescending)
        {
        }

        public PriorityQueue(Func<TValue, TKey> keySelector, IComparer<TKey> keyComparer, bool isDescending = true)
        {
            _keySelector = keySelector;
            _keyComparer = keyComparer;
            _isDescending = isDescending;
        }

        public void Enqueue(TValue item)
        {
            _data.Add(new KeyValuePair<TKey, TValue>(_keySelector(item), item));
            var childIndex = _data.Count - 1;
            while (childIndex > 0)
            {
                var parentIndex = (childIndex - 1) / 2;
                if (Compare(_data[childIndex].Key, _data[parentIndex].Key) >= 0)
                    break;
                Swap(childIndex, parentIndex);
                childIndex = parentIndex;
            }
        }

        public TValue Dequeue()
        {
            var lastIndex = _data.Count - 1;
            var firstItem = _data[0];
            _data[0] = _data[lastIndex];
            _data.RemoveAt(lastIndex--);
            var parentIndex = 0;
            while (true)
            {
                var childIndex = parentIndex * 2 + 1;
                if (childIndex > lastIndex)
                    break;
                var rightChild = childIndex + 1;
                if (rightChild <= lastIndex && Compare(_data[rightChild].Key, _data[childIndex].Key) < 0)
                    childIndex = rightChild;
                if (Compare(_data[parentIndex].Key, _data[childIndex].Key) <= 0)
                    break;
                Swap(parentIndex, childIndex);
                parentIndex = childIndex;
            }
            return firstItem.Value;
        }

        public TValue Peek()
        {
            return _data[0].Value;
        }

        private void Swap(int a, int b)
        {
            var tmp = _data[a];
            _data[a] = _data[b];
            _data[b] = tmp;
        }

        private int Compare(TKey a, TKey b)
        {
            return _isDescending ? _keyComparer.Compare(b, a) : _keyComparer.Compare(a, b);
        }

        public int Count => _data.Count;

        public IEnumerator<TValue> GetEnumerator()
        {
            return _data.Select(r => r.Value).GetEnumerator();
        }

        IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
    }
}

namespace Assistant
{
    static class Input
    {
        static List<string> line = new List<string>();
        static int index = 0;
        static String RNext()
        {
            if (line.Count <= index) line.AddRange(Console.ReadLine().Split());
            return line[index++];
        }
        public static int RInt => int.Parse(RNext());
        public static long RLong => long.Parse(RNext());
        public static int[] RInts => Console.ReadLine().Split().Select(int.Parse).ToArray();
        public static long[] RLongs => Console.ReadLine().Split().Select(long.Parse).ToArray();
        public static string RString => RNext();
        //以下未テスト
        public static int[] RIntsC(int c) => Enumerable.Repeat(0, c).Select(x => int.Parse(RNext())).ToArray();
        public static long[] RLongsC(int c) => Enumerable.Repeat(0, c).Select(x => long.Parse(RNext())).ToArray();
        public static char[][] RMap(int h) => Enumerable.Repeat(0, h).Select(x => Console.ReadLine().ToCharArray()).ToArray();
    }

    public struct Mlong
    {
        long _v;
        const long mod = 1000000007;
        public Mlong(long n = 0) : this() { _v = n >= mod ? n % mod : n; }

        public static implicit operator Mlong(long _x) => new Mlong(_x);
        public static implicit operator long(Mlong _x) => _x._v;
        public static Mlong operator +(Mlong m1, Mlong m2) { long m = m1._v + m2._v; return m >= mod ? m - mod : m; }
        public static Mlong operator -(Mlong m1, Mlong m2) { long m = m1._v - m2._v; return m >= 0 ? m : m + mod; }
        public static Mlong operator *(Mlong m1, Mlong m2) => m1._v * m2._v % mod;
        public static Mlong operator /(Mlong m1, Mlong m2) => m1._v * ModPow(m2._v, mod - 2) % mod;

        public static long ModPow(long a, long n)
        {
            if (n == 0) return 1;
            else if (n % 2 == 1) return a * ModPow(a, n - 1) % mod;
            else return ModPow(a * a, n / 2);
        }

        static Mlong[] fac, finv, inv;
        public static void nCkInit(int max)
        {
            fac = new Mlong[max]; finv = new Mlong[max]; inv = new Mlong[max];
            fac[0] = fac[1] = 1;
            finv[0] = finv[1] = 1;
            inv[1] = 1;
            for (int i = 2; i < max; i++)
            {
                fac[i] = fac[i - 1] * i;
                inv[i] = mod - inv[mod % i] * (mod / i);
                finv[i] = finv[i - 1] * inv[i];
            }
        }

        public static Mlong nCk(int n, int k)
        {
            if (n < k) return 0;
            if (n < 0 || k < 0) return 0;
            return fac[n] * finv[k] * finv[n - k];
        }
    }

    static class Debug
    {
        static public void Draw2D<T>(T[,] map, int mode = 1)
        {
#if DEBUG
            int W = map.GetLength(0);
            int H = map.GetLength(1);

            string[,] map2 = new string[W + 1, H + 1];
            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (i == 0 && j == 0) map2[i, j] = 0.ToString();
                    else if (i == 0) map2[i, j] = (j - 1).ToString();
                    else if (j == 0) map2[i, j] = (i - 1).ToString();
                    else map2[i, j] = map[i - 1, j - 1].ToString();
                }
            }

            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (mode == 0) Console.Write(map2[i, j].Last());
                    if (mode == 1) Console.Write(map2[i, j] + " ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
#endif
        }

        public static void Draw1D<T>(T[] array, int mode = 0)
        {
#if DEBUG
            Console.WriteLine(string.Join(" ", array));
#endif
        }
    }
}