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: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;

class Program
{
    static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        Solve();
        Console.Out.Flush();
    }

    static void Solve()
    {
        var H = sc.ReadInt();
        var W = sc.ReadInt();
        var K = sc.ReadInt();
        var h1 = sc.ReadInt() - 1;
        var w1 = sc.ReadInt() - 1;
        var h2 = sc.ReadInt() - 1;
        var w2 = sc.ReadInt() - 1;
        var C = new string[H];
        for (int i = 0; i < H; i++)
        {
            C[i] = sc.ReadStr();
        }

        var pq = new PriorityQueue<(int, int, int, int, int)>();
        var dp = new int[H, W];
        var inf = int.MaxValue / 3;
        for (int i = 0; i < H; i++)
        {
            for (int j = 0; j < W; j++)
            {
                dp[i, j] = inf;
            }
        }
        dp[h1, w1] = 0;
        pq.Enqueue((0, 0, h1, w1, -1));
        while (pq.Count > 0)
        {
            var (cost, k, h, w, dir) = pq.Dequeue();
            if (dp[h, w] < cost) continue;

            var dh = new int[] { -1, 0, 1, 0 };
            var dw = new int[] { 0, -1, 0, 1 };
            for (int d = 0; d < 4; d++)
            {
                var nh = h + dh[d];
                var nw = w + dw[d];
                if (0 <= nh && nh < H && 0 <= nw && nw < W)
                {
                    if (C[nh][nw] == '@') continue;
                    if (dir == -1)
                    {
                        dp[nh, nw] = 1;
                        pq.Enqueue((1, 1, nh, nw, d));
                    }

                    var ncost = cost + 1;
                    var nk = 1;
                    if ((d == dir || dir == -1) && k < K)
                    {
                        // 同じ方向
                        ncost--;
                        nk = k + 1;
                    }
                    if (d == dir && k == K)
                    {
                        // 同じ方向だけど限界まで進んだ
                        nk = 1;
                        ncost++;
                    }
                    if (dp[nh, nw] > ncost)
                    {
                        dp[nh, nw] = ncost;
                        pq.Enqueue((ncost, nk, nh, nw, d));
                    }
                }
            }
        }
        var ans = dp[h2, w2];
        if (ans == inf) ans = -1;
        Console.WriteLine(ans);
    }
    class PriorityQueue<T>
    {
        public int Count;
        private int Capacity;
        private bool IsMaxHeap;
        private T[] Buffer;
        private IComparer<T> Cmp;
        public PriorityQueue(bool isMaxHeap = true, IComparer<T> cmp = null)
        {
            Count = 0;
            Capacity = 1 << 10;
            IsMaxHeap = isMaxHeap;
            Buffer = new T[Capacity];
            if (cmp != null) Cmp = cmp;
            else Cmp = Comparer<T>.Default;
        }
        public PriorityQueue(int capacity, bool isMaxHeap = true, IComparer<T> cmp = null)
        {
            Count = 0;
            Capacity = Math.Max(16, capacity);
            IsMaxHeap = isMaxHeap;
            Buffer = new T[Capacity];
            if (cmp != null) Cmp = cmp;
            else Cmp = Comparer<T>.Default;
        }
        private void Resize()
        {
            Capacity <<= 1;
            Array.Resize(ref Buffer, Capacity);
        }
        public void Enqueue(T value)
        {
            if (Count == Capacity) Resize();
            Buffer[Count] = value;
            Count++;
            UpHeap();
        }
        public T Dequeue()
        {
            var res = Buffer[0];
            Swap(0, Count - 1);
            Count--;
            DownHeap();
            return res;
        }
        public T Peek() { return Buffer[0]; }
        private void Swap(int i, int j)
        {
            var tmp = Buffer[i];
            Buffer[i] = Buffer[j];
            Buffer[j] = tmp;
        }
        private void UpHeap()
        {
            var n = this.Count - 1;
            while (n != 0)
            {
                var parent = (n - 1) / 2;
                if (Compare(Buffer[n], Buffer[parent]) > 0)
                {
                    Swap(n, parent);
                    n = parent;
                }
                else break;
            }
        }
        private void DownHeap()
        {
            var parent = 0;
            while (true)
            {
                var child = 2 * parent + 1;
                if (child > this.Count - 1) break;
                if (child < Count - 1 && Compare(Buffer[child], Buffer[child + 1]) < 0)
                {
                    child += 1;
                }
                if (Compare(Buffer[parent], Buffer[child]) < 0)
                {
                    Swap(parent, child);
                    parent = child;
                }
                else break;
            }
        }
        private int Compare(T x, T y)
        {
            var res = Cmp.Compare(x, y);
            if (!IsMaxHeap) res *= -1;
            return res;
        }
    }
    static Scanner sc = new Scanner();
}
class Scanner
{
    string[] S = new string[0];
    int Index = 0;
    char[] Separators = new char[] { ' ' };

    public string Next()
    {
        if (this.Index < this.S.Length) return this.S[this.Index++];
        var line = "";
        while (line == "") line = Console.ReadLine();
        this.S = line.Split(this.Separators, StringSplitOptions.RemoveEmptyEntries);
        if (this.S.Length == 0) return this.Next();
        this.Index = 0;
        return this.S[this.Index++];
    }
    public string ReadStr()
    {
        return this.Next();
    }
    public char ReadChar()
    {
        return this.Next()[0];
    }
    public int ReadInt()
    {
        return int.Parse(this.Next());
    }
    public uint ReadUInt()
    {
        return uint.Parse(this.Next());
    }
    public long ReadLong()
    {
        return long.Parse(this.Next());
    }
    public double ReadDouble()
    {
        return double.Parse(this.Next());
    }
    public Tuple<int, int> ReadTup(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add);
    }
    public Tuple<long, long> ReadTupLong(int add = 0)
    {
        return Tuple.Create(this.ReadLong() + add, this.ReadLong() + add);
    }
    public Tuple<int, int, int> ReadTup3(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add);
    }
    public Tuple<int, int, int, int> ReadTup4(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add);
    }
    public int[] ReadIntArray(int n)
    {
        var array = new int[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadInt();
        }
        return array;
    }
    public long[] ReadLongArray(int n)
    {
        var array = new long[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadLong();
        }
        return array;
    }
    public double[] ReadDoubleArray(int n)
    {
        var array = new double[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadDouble();
        }
        return array;
    }
    public char[] ReadCharArray(int n)
    {
        var array = new char[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadChar();
        }
        return array;
    }
    public string[] ReadStrArray(int n)
    {
        var array = new string[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadStr();
        }
        return array;
    }
    public Tuple<long, long>[] ReadTupLongArray(int n, int add = 0)
    {
        var array = new Tuple<long, long>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTupLong(add);
        }
        return array;
    }
    public Tuple<int, int>[] ReadTupArray(int n, int add = 0)
    {
        var array = new Tuple<int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup(add);
        }
        return array;
    }
    public Tuple<int, int, int>[] ReadTup3Array(int n, int add = 0)
    {
        var array = new Tuple<int, int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup3(add);
        }
        return array;
    }
    public Tuple<int, int, int, int>[] ReadTup4Array(int n, int add = 0)
    {
        var array = new Tuple<int, int, int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup4(add);
        }
        return array;
    }
}