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.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.Comparator;
import java.util.InputMismatchException;
import java.util.Iterator;
import java.util.NoSuchElementException;
import java.util.PriorityQueue;

public class Main {
    int h;
    int w;
    int k;
    char[][] c;
    void solve(MyScanner in, MyWriter out) {
        h = in.nextInt();
        w = in.nextInt();
        k = in.nextInt();
        int x1 = in.nextInt() - 1;
        int y1 = in.nextInt() - 1;
        int x2 = in.nextInt() - 1;
        int y2 = in.nextInt() - 1;
        c = in.next2dCharArray(h, w);
        Vertex[][][] dist = dijkstra(x1, y1);
        long ans = Long.MAX_VALUE;
        for (int i = 0; i < 4; i++) {
            Vertex v = dist[x2][y2][i];
            if (v != null)
                ans = Math.min(ans, v.dist);
        }
        out.println(ans == Long.MAX_VALUE ? -1 : ans);
    }
    private static final int[] di = {-1, 0, 1, 0};
    private static final int[] dj = {0, 1, 0, -1};
    private Vertex[][][] dijkstra(int startI, int startJ) {
        var done = new boolean[h][w][4];
        var dist = new Vertex[h][w][4];
        var q = new PriorityQueue<Vertex>();
        for (int dir = 0; dir < 4; dir++) {
            var v = new Vertex(startI, startJ, dir, k, 0);
            dist[startI][startJ][dir] = v;
            q.offer(v);
        }
        while (!q.isEmpty()) {
            Vertex v = q.poll();
            if (done[v.i][v.j][v.dir])
                continue;
            done[v.i][v.j][v.dir] = true;
            for (int dir = 0; dir < 4; dir++) {
                if (dir == v.dir) {
                    int ni = v.i + di[dir];
                    int nj = v.j + dj[dir];
                    if (nj < 0 || nj >= w || ni < 0 || ni >= h)
                        continue;
                    if (c[ni][nj] == '@')
                        continue;
                    if (done[ni][nj][v.dir])
                        continue;
                    var newv = new Vertex(ni,
                                          nj,
                                          v.dir,
                                          v.currentK == k ? 1 : v.currentK + 1,
                                          v.dist + (v.currentK == k ? 1 : 0));
                    Vertex oldv = dist[ni][nj][v.dir];
                    if (oldv == null || oldv.compareTo(newv) > 0) {
                        dist[ni][nj][v.dir] = newv;
                        q.offer(newv);
                    }
                } else {
                    for (int nextDir = 0; nextDir < 4; nextDir++) {
                        if (dir == nextDir)
                            continue;
                        if (done[v.i][v.j][nextDir])
                            continue;
                        var newv = new Vertex(v.i, v.j, nextDir, 0, v.dist + 1);
                        Vertex oldv = dist[v.i][v.j][nextDir];
                        if (oldv == null || oldv.compareTo(newv) > 0) {
                            dist[v.i][v.j][nextDir] = newv;
                            q.offer(newv);
                        }
                    }
                }
            }
        }
        return dist;
    }
    private static final class Vertex implements Comparable<Vertex> {
        final int i;
        final int j;
        final int dir;
        final int currentK;
        final long dist;
        Vertex(int i, int j, int dir, int currentK, long dist) {
            this.i = i;
            this.j = j;
            this.dir = dir;
            this.currentK = currentK;
            this.dist = dist;
        }
        int currentK() {
            return currentK;
        }
        long dist() {
            return dist;
        }
        static final Comparator<Vertex> COMP =
                Comparator.comparingLong(Vertex::dist)
                          .thenComparingInt(Vertex::currentK);
        @Override
        public int compareTo(Vertex o) {
            return COMP.compare(this, o);
        }
    }
    public static void main(String[] args) {
        MyWriter w = new MyWriter(System.out);
        new Main().solve(new MyScanner(System.in), w);
        w.flush();
    }
    static final class MyScanner {
        static final int BUFFER_SIZE = 8192;
        private final InputStream in;
        private final byte[] buffer = new byte[BUFFER_SIZE];
        private int point;
        private int readLength;

        MyScanner(InputStream in) {
            this.in = in;
        }
        private int readByte() {
            if (point < readLength) {
                int result = buffer[point];
                point += 1;
                return result;
            }
            try {
                readLength = in.read(buffer);
            } catch (IOException e) {
                throw new AssertionError(null, e);
            }
            if (readLength == -1) {
                return -1;
            }
            point = 1;
            return buffer[0];
        }
        private static boolean isVisibleChar(int c) {
            return 33 <= c && c <= 126;
        }
        char nextChar() {
            int c = readByte();
            while (!(c == -1 || isVisibleChar(c))) {
                c = readByte();
            }
            if (c == -1) {
                throw new NoSuchElementException();
            }
            return (char)c;
        }
        String next() {
            return next(16);
        }
        String next(int n) {
            int c = readByte();
            while (!(c == -1 || isVisibleChar(c))) {
                c = readByte();
            }
            if (c == -1) {
                throw new NoSuchElementException();
            }
            StringBuilder b = new StringBuilder(n);
            do {
                b.append((char)c);
                c = readByte();
            } while (c != -1 && isVisibleChar(c));
            return b.toString();
        }
        long nextLong() {
            int c = readByte();
            while (!(c == -1 || isVisibleChar(c))) {
                c = readByte();
            }
            if (c == -1) {
                throw new NoSuchElementException();
            }
            boolean minus = false;
            long limit = -Long.MAX_VALUE;
            if (c == '-') {
                minus = true;
                limit = Long.MIN_VALUE;
                c = readByte();
            }
            long n = 0L;
            do {
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                if (n < limit / 10L) {
                    throw new InputMismatchException();
                }
                n *= 10L;
                int digit = c - '0';
                if (n < limit + digit) {
                    throw new InputMismatchException();
                }
                n -= digit;
                c = readByte();
            } while (c != -1 && isVisibleChar(c));
            return minus ? n : -n;
        }
        int nextInt() {
            long n = nextLong();
            if (n < Integer.MIN_VALUE || n > Integer.MAX_VALUE) {
                throw new InputMismatchException();
            }
            return (int)n;
        }
        double nextDouble() {
            return Double.parseDouble(next());
        }
        int[] nextIntArray(int n) {
            int[] result = new int[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextInt();
            }
            return result;
        }
        long[] nextLongArray(int n) {
            long[] result = new long[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextLong();
            }
            return result;
        }
        char[] nextCharArray(int n) {
            char[] result = new char[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextChar();
            }
            return result;
        }
        char[][] next2dCharArray(int n, int m) {
            char[][] result = new char[n][m];
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < m; j++) {
                    result[i][j] = nextChar();
                }
            }
            return result;
        }
        int[][] nextVerticalIntArrays(int arrayCount, int arrayLength) {
            int[][] result = new int[arrayCount][arrayLength];
            for (int i = 0; i < arrayLength; i++) {
                for (int j = 0; j < arrayCount; j++) {
                    result[j][i] = nextInt();
                }
            }
            return result;
        }
        long[][] nextVerticalLongArrays(int arrayCount, int arrayLength) {
            long[][] result = new long[arrayCount][arrayLength];
            for (int i = 0; i < arrayLength; i++) {
                for (int j = 0; j < arrayCount; j++) {
                    result[j][i] = nextLong();
                }
            }
            return result;
        }
        char[][] nextVerticalCharArrays(int arrayCount, int arrayLength) {
            char[][] result = new char[arrayCount][arrayLength];
            for (int i = 0; i < arrayLength; i++) {
                for (int j = 0; j < arrayCount; j++) {
                    result[j][i] = nextChar();
                }
            }
            return result;
        }
    }
    static final class MyWriter extends PrintWriter {
        MyWriter(OutputStream out) {
            super(out);
        }
        void println(int[] x) {
            println(x, " ");
        }
        void println(int[] x, String delimiter) {
            if (x.length > 0) {
                print(x[0]);
                for (int i = 1; i < x.length; i++) {
                    print(delimiter);
                    print(x[i]);
                }
            }
            println();
        }
        void println(long[] x) {
            println(x, " ");
        }
        void println(long[] x, String delimiter) {
            if (x.length > 0) {
                print(x[0]);
                for (int i = 1; i < x.length; i++) {
                    print(delimiter);
                    print(x[i]);
                }
            }
            println();
        }
        void println(Iterable<?> iterable) {
            println(iterable, " ");
        }
        void println(Iterable<?> iterable, String delimiter) {
            Iterator<?> i = iterable.iterator();
            if (i.hasNext()) {
                print(i.next());
                while (i.hasNext()) {
                    print(delimiter);
                    print(i.next());
                }
            }
            println();
        }
        void printLines(int[] x) {
            println(x, System.lineSeparator());
        }
        void printLines(long[] x) {
            println(x, System.lineSeparator());
        }
        void printLines(Iterable<?> iterable) {
            println(iterable, System.lineSeparator());
        }
    }
}