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 Kotlin (1.3.71) (judge verdict: Time Limit Exceeded)

import java.io.BufferedReader
import java.io.IOException
import java.io.InputStreamReader
import java.io.Reader
import java.util.*

private val scanner = BufferedScanner()
private fun readLn() = scanner.nextLine() // string line
private fun readInt() = readLn().toInt() // single int
private fun readLong() = readLn().toLong()
private fun readStrings() = readLn().split(" ") // list of strings
private fun readInts() = readStrings().map { it.toInt() } // list of ints
private fun readLongs() = readStrings().map { it.toLong() }

const val INF = 1e9.toInt()

@ExperimentalStdlibApi
fun main() {
    val timer = System.nanoTime()
    val (h, w, k) = readInts()
    val (x1, y1, x2, y2) = readInts().map { it - 1 }
    val grid = Array(h) { "" }
    repeat(h) {
        grid[it] = readLn()
    }
    val shortest = Array(h) { IntArray(w) { INF } }
    val q = mutableListOf<Pair<Int, Int>>()
    var p = 0
    q.add(x1 to y1)
    shortest[x1][y1] = 0
    val rows = Array(h) { r ->
        (0 until w).filterNot { c -> grid[r][c] == '@' }.toTreeSet()
    }
    val cols = Array(w) { c ->
        (0 until h).filterNot { r -> grid[r][c] == '@' }.toTreeSet()
    }
    val rowPref = Array(h) { r ->
        (0 until w).scan(0, { acc, c ->
            acc + if (grid[r][c] == '@') 1 else 0
        })
    }
    val colPref = Array(w) { c ->
        (0 until h).scan(0, { acc, r ->
            acc + if (grid[r][c] == '@') 1 else 0
        })
    }
    System.err.println("init finished: " + ((System.nanoTime() - timer) / 1e9))
    val timer2 = System.nanoTime()
    rows[x1].remove(y1)
    cols[y1].remove(x1)
    while (p < q.size) {
        val (x, y) = q[p++]
        if (x == x2 && y == y2) {
            break
        }
        val it1 = rows[x].tailSet(y, false).iterator()
        while (it1.hasNext()) {
            val yy = it1.next()
            if (y + k < yy || rowPref[x][y] != rowPref[x][yy + 1]) {
                break
            }
            q.add(x to yy)
            shortest[x][yy] = shortest[x][y] + 1
            it1.remove()
            cols[yy].remove(x)
        }
        val it2 = rows[x].headSet(y, false).descendingIterator()
        while (it2.hasNext()) {
            val yy = it2.next()
            if (yy + k < y || rowPref[x][y + 1] != rowPref[x][yy]) {
                break
            }
            q.add(x to yy)
            shortest[x][yy] = shortest[x][y] + 1
            it2.remove()
            cols[yy].remove(x)
        }
        val it3 = cols[y].tailSet(x, false).iterator()
        while (it3.hasNext()) {
            val xx = it3.next()
            if (x + k < xx || colPref[y][x] != colPref[y][xx + 1]) {
                break
            }
            q.add(xx to y)
            shortest[xx][y] = shortest[x][y] + 1
            it3.remove()
            rows[xx].remove(y)
        }
        val it4 = cols[y].headSet(x, false).descendingIterator()
        while (it4.hasNext()) {
            val xx = it4.next()
            if (xx + k < x || colPref[y][x + 1] != colPref[y][xx]) {
                break
            }
            q.add(xx to y)
            shortest[xx][y] = shortest[x][y] + 1
            it4.remove()
            rows[xx].remove(y)
        }
    }
    println(if (shortest[x2][y2] == INF) -1 else shortest[x2][y2])
    System.err.println("job finished: " + ((System.nanoTime() - timer2) / 1e9))
}

private fun <T> Iterable<T>.toTreeSet(): TreeSet<T> {
    val s = TreeSet<T>()
    forEach { s.add(it) }
    return s
}

class BufferedScanner @JvmOverloads constructor(reader: Reader? = InputStreamReader(System.`in`)) {
    var br: BufferedReader = BufferedReader(reader, 1 shl 20)
    var st: StringTokenizer? = null
    operator fun next(): String {
        while (st == null || !st!!.hasMoreElements()) {
            try {
                st = StringTokenizer(br.readLine())
            } catch (e: IOException) {
                e.printStackTrace()
            }
        }
        return st!!.nextToken()
    }

    fun nextInt(): Int {
        return next().toInt()
    }

    fun nextLong(): Long {
        return next().toLong()
    }

    fun nextDouble(): Double {
        return next().toDouble()
    }

    fun nextLine(): String {
        var str = ""
        try {
            str = br.readLine()
        } catch (e: IOException) {
            e.printStackTrace()
        }
        return str
    }

    fun close() {
        try {
            br.close()
        } catch (e: IOException) {
            e.printStackTrace()
        }
    }
}