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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.*;
import java.io.*;

public class Main {
	static BufferedReader br;
	static PrintWriter pw;
	static int inf = (int) 1e9;
	static long mod = (long) 1e9 + 7;
	static long[] memo;
	static int neg_inf = Integer.MIN_VALUE;

	public static void main(String[] args) throws NumberFormatException, IOException {
		br = new BufferedReader(new InputStreamReader(System.in));
		PrintWriter pw = new PrintWriter(System.out);
		StringTokenizer st = new StringTokenizer(br.readLine());
		n = Integer.parseInt(st.nextToken());
		m = Integer.parseInt(st.nextToken());
		k = Integer.parseInt(st.nextToken());
		st = new StringTokenizer(br.readLine());
		int x1 = Integer.parseInt(st.nextToken()) - 1;
		int y1 = Integer.parseInt(st.nextToken()) - 1;
		int x2 = Integer.parseInt(st.nextToken()) - 1;
		int y2 = Integer.parseInt(st.nextToken()) - 1;
		a = new char[n][m];
		for (int i = 0; i < a.length; i++) {
			a[i] = br.readLine().toCharArray();
		}
		pw.println(dijkstra(new pair(x1, y1), new pair(x2, y2)));

		pw.flush();
	}

	static final int INF = (int) 1e9; // don't increase, avoid overflow
	static ArrayList<Edge>[] adjList;
	static int V, n, m, k;

	/*
	 * 1. Dijkstra's Algorithm for SSSP on weighted graphs
	 */
	static int dijkstra(pair S, pair T) // O(E log E)
	{
		int[][][] dist = new int[4][n][m];
		for (int i = 0; i < n; i++) {
			Arrays.fill(dist[0][i], INF);
			Arrays.fill(dist[1][i], INF);
			Arrays.fill(dist[2][i], INF);
			Arrays.fill(dist[3][i], INF);
		}
		PriorityQueue<Edge> pq = new PriorityQueue<Edge>();
		dist[0][S.x][S.y] = 0;
		dist[1][S.x][S.y] = 0;
		dist[2][S.x][S.y] = 0;
		dist[3][S.x][S.y] = 0;
		pq.add(new Edge(S, 0, 0, -1)); // may add more in case of MSSP (Mult-Source)
		while (!pq.isEmpty()) {
			Edge cur = pq.remove();
			if (cur.node.equals(T)) { // remove if all computations are needed
//				for (int i = 0; i < n; i++) {
//					System.out.print(Arrays.toString(dist[0][i]));
//					System.out.print(Arrays.toString(dist[1][i]));
//					System.out.print(Arrays.toString(dist[2][i]));
//					System.out.println(Arrays.toString(dist[3][i]));
//				}
				return cur.cost;

			}
			for (int i = 0; i < 4; i++) {
				int x = cur.node.x + dx[i];
				int y = cur.node.y + dy[i];
				if (valid(x, y)) {
					if (cur.w == i && cur.c < k) {
						if (cur.cost < dist[i][x][y])
							pq.add(new Edge(new pair(x, y), dist[i][x][y] = cur.cost, cur.c + 1, i));
					} else {
						if (cur.cost + 1 < dist[i][x][y])
							pq.add(new Edge(new pair(x, y), dist[i][x][y] = cur.cost + 1, 1, i));
					}
				}
			}

		}
		return -1;
	}

	static char[][] a;

	static int[] dx = new int[] { -1, 0, 1, 0 };
	static int[] dy = new int[] { 0, 1, 0, -1 };

	static boolean valid(int x, int y) {
		return x > -1 && x < n && y > -1 && y < m && a[x][y] != '@';
	}

	static class Edge implements Comparable<Edge> {
		pair node;
		int cost;
		int c;
		int w;

		Edge(pair a, int b, int c, int d) {
			node = a;
			cost = b;
			this.c = c;
			this.w = d;
		}

		public int compareTo(Edge e) {
			return cost - e.cost;
		}
	}

	static class pair implements Comparable<pair> {
		int x;
		int y;

		public pair(int d, int u) {
			x = d;
			y = u;
		}

		public int compareTo(pair o) {
			return o.y - y;
		}

		public boolean equals(Object o) {
			// TODO Auto-generated method stub
			pair obj = (pair) o;
			return obj.x == x && obj.y == y;
		}

	}

	static class triple implements Comparable<triple> {
		int x;
		int y;
		int z;
		int w;

		public triple(int a, int b, int c) {
			x = a;
			y = b;
			z = c;
		}

		public triple(int a, int b, int c, int d) {
			x = a;
			y = b;
			z = c;
			w = d;
		}

		public int compareTo(triple o) {
			return x - o.x;
		}
	}

	static int[] nxtarr() throws IOException {
		StringTokenizer st = new StringTokenizer(br.readLine());
		int[] a = new int[st.countTokens()];
		for (int i = 0; i < a.length; i++) {
			a[i] = Integer.parseInt(st.nextToken());
		}
		return a;
	}

	static long pow(long a, long e) // O(log e)
	{
		long res = 1;
		while (e > 0) {
			if ((e & 1) == 1)
				res *= a;
			a *= a;
			e >>= 1;
		}
		return res;
	}

	static int gcd(int a, int b) {
		if (a == 0)
			return b;
		return gcd(b % a, a);
	}
}