101705: [AtCoder]ABC170 F - Pond Skater

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