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 OCaml (4.10.0) (judge verdict: Time Limit Exceeded)

module Weighted01DirectedGraph
  (* 道の表現 *)
  (Path : sig
    type t
    (* 辺の名前 *)
    type edge
    (* 長さ0の道 *)
    val nil : t
    (* 道の後ろに辺を付け足した道 *)
    val snoc : t -> edge -> t
  end) :
sig
  val bfs01 :
    (* 頂点数 *)
    int ->
    (* 辺の名前が付いた隣接リスト *)
    (* 辺の重さが0ならfalse，1ならtrue *)
    ('v -> ('v * Path.edge * bool) list) ->
    (* 始点 *)
    'v ->
    (* 最短経路を返す関数 辿り着けなければNoneを返す） *)
    ('v -> Path.t option)
end =
struct
  let bfs01 n es s =
    (* 始点sからの経路 *)
    let d = Hashtbl.create n in
    (* 現在BFSで走査している頂点のリスト *)
    let vps = ref [(s, Path.nil)] in
    let rec bfs t =
      match Hashtbl.find_opt d t, !vps with
      (* もう既に全ての頂点までの経路が分かっている *)
      | None, [] -> None
      (* 既に終点までの経路が分かっているので返す *)
      | Some _ as ans, _ -> ans
      (* 終点までの経路が分かっていないので，01BFSを続行 *)
      | None, _ :: _ -> vps := List.fold_left dfs [] !vps; bfs t
    (* 重さ0の辺をDFSで縮約 *)
    and dfs vps (v, p) =
      if Hashtbl.mem d v then vps
      else begin
        Hashtbl.add d v p;
        List.fold_left (fun vps (u, e, b) ->
          if b
          then (u, Path.snoc p e) :: vps
          else dfs vps (u, Path.snoc p e)) vps (es v)
      end
    in bfs
end

module G = Weighted01DirectedGraph
(struct
  type t = int
  type edge = int
  let nil = 0
  let snoc = ( + )
end)

type dir = Neutral | Left | Right | Up | Down

let () = Scanf.scanf "%d %d %d\n%d %d %d %d\n" @@ fun h w k x1 y1 x2 y2 ->
  let x1 = x1 - 1 in
  let x2 = x2 - 1 in
  let y1 = y1 - 1 in
  let y2 = y2 - 1 in
  let css = Array.init h @@ fun _ -> Scanf.scanf "%s\n" Fun.id in
  let d = G.bfs01 (h * w * 40) (fun (i, j, s, dir) ->
    List.filter_map (fun (i, j, dir') ->
      if i < 0 || h <= i || j < 0 || w <= j || css.(i).[j] = '@'
      then None
      else if (dir = Neutral || dir = dir') && s < k
      then Some ((i, j, s + 1, dir'), 0, false)
      else Some ((i, j, 1, dir'), 1, true))
    [(i - 1, j, Up); (i + 1, j, Down); (i, j - 1, Left); (i, j + 1, Right)])
    (x1, y1, 0, Neutral) in
  Printf.printf "%d\n" @@
  max (-1) @@
  succ @@
  Array.fold_left (List.fold_left min) max_int @@
  Array.init k @@ fun s ->
    let s = s + 1 in
    List.map (fun o -> Option.value ~default:max_int o)
    [ d (x2, y2, s, Left); d (x2, y2, s, Right); d (x2, y2, s, Up); d (x2, y2, s, Down) ]