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 Common Lisp (SBCL 2.0.3) (judge verdict: Time Limit Exceeded)

(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx opt
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Queue with singly linked list
;;;

(defstruct (queue (:constructor make-queue
                      (&optional list &aux (tail (last list)))))
  (list nil :type list)
  (tail nil :type list))

(declaim (inline enqueue))
(defun enqueue (obj queue)
  "Pushes OBJ to the end of QUEUE."
  (symbol-macrolet ((list (queue-list queue))
                    (tail (queue-tail queue)))
    (if (null list)
        (setf tail (list obj)
              list tail)
        (setf (cdr tail) (list obj)
              tail (cdr tail))))
  queue)

(declaim (inline dequeue))
(defun dequeue (queue)
  "Removes and returns the element at the front of QUEUE. Returns NIL if QUEUE
is empty."
  (pop (queue-list queue)))

(declaim (inline queue-empty-p))
(defun queue-empty-p (queue)
  (null (queue-list queue)))

(declaim (inline queue-peek))
(defun queue-peek (queue)
  (car (queue-list queue)))

(declaim (inline enqueue-front))
(defun enqueue-front (obj queue)
  "Pushes OBJ to the front of QUEUE."
  (symbol-macrolet ((list (queue-list queue))
                    (tail (queue-tail queue)))
    (if (null list)
        (setf tail (list obj)
              list tail)
        (push obj list))
    queue))

(declaim (inline read-schar))
(defun read-schar (&optional (stream *standard-input*))
  (declare #-swank (sb-kernel:ansi-stream stream)
           (inline read-byte))
  #+swank (read-char stream nil #\Newline) ; on SLIME
  #-swank (code-char (read-byte stream nil #.(char-code #\Newline))))

(in-package :cl-user)

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(defconstant +inf+ #xffffffff)

(defun main ()
  (declare #.OPT)
  (let* ((h (read))
         (w (read))
         (k (read))
         (boundary (if (>= k 100000) 100000 0))
         (start-x (- (read) 1))
         (start-y (- (read) 1))
         (goal-x (- (read) 1))
         (goal-y (- (read) 1))
         (plan (make-array (list h w) :element-type 'bit :initial-element 0))
         ;; 既に〇〇進んだ
         (marked (make-array (list h w 4) :element-type 'int32 :initial-element -1))
         (dists (make-array (list h w) :element-type 'uint32 :initial-element +inf+))
         ;; x . y
         (que (make-queue)))
    (declare (uint31 h w k))
    (dotimes (i h)
      (dotimes (j w (read-schar))
        (ecase (read-schar)
          (#\@ (setf (aref plan i j) 1))
          (#\.))))
    (enqueue (cons start-x start-y) que)
    ;; up:0, down:1, left:2, right:3
    (setf (aref dists start-x start-y) 0)
    (setf (aref marked start-x start-y 0) k
          (aref marked start-x start-y 1) k
          (aref marked start-x start-y 2) k
          (aref marked start-x start-y 3) k)
    (loop until (queue-empty-p que)
          for (x . y) of-type (uint31 . uint31) = (dequeue que)
          for dist = (aref dists x y)
          do ;; up
             ;; #>marked
             (setf (aref marked x y 0) k
                   (aref marked x y 1) k
                   (aref marked x y 2) k
                   (aref marked x y 3) k)
             (loop for delta from 1 to k
                   for new-x = (- x delta)
                   while (and (>= new-x 0)
                              ;; あとk-deltaだけ進めるが、既にそれ以上に進んでいたら打ち切れる
                              (> (- (- k delta) boundary) (aref marked new-x y 0))
                              (zerop (aref plan new-x y)))
                   do (setf (aref marked new-x y 0) (- k delta))
                   when (< (+ dist 1) (aref dists new-x y))
                   do (setf (aref dists new-x y) (+ dist 1))
                      (enqueue (cons new-x y) que))
             ;; down
             (loop for delta from 1 to k
                   for new-x = (+ x delta)
                   while (and (< new-x h)
                              (>= (- (- k delta) boundary) (aref marked new-x y 1))
                              (zerop (aref plan new-x y)))
                   do (setf (aref marked new-x y 1) (- k delta))
                   when (< (+ dist 1) (aref dists new-x y))
                   do (setf (aref dists new-x y) (+ dist 1))
                      (enqueue (cons new-x y) que))
             ;; left
             (loop for delta from 1 to k
                   for new-y = (- y delta)
                   while (and (>= new-y 0)
                              (>= (- (- k delta) boundary) (aref marked x new-y 2))
                              (zerop (aref plan x new-y)))
                   do (setf (aref marked x new-y 2) (- k delta))
                   when (< (+ dist 1) (aref dists x new-y))
                   do (setf (aref dists x new-y) (+ dist 1))
                      (enqueue (cons x new-y) que))
             ;; right
             (loop for delta from 1 to k
                   for new-y = (+ y delta)
                   while (and (< new-y w)
                              (>= (- (- k delta) boundary) (aref marked x new-y 3))
                              (zerop (aref plan x new-y)))
                   do (setf (aref marked x new-y 3) (- k delta))
                   when (< (+ dist 1) (aref dists x new-y))
                   do (setf (aref dists x new-y) (+ dist 1))
                      (enqueue (cons x new-y) que)))
    (let ((res (aref dists goal-x goal-y)))
      (println
       (if (= res +inf+)
           -1
           res)))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3 5 2
3 2 3 4
.....
.@..@
..@..
"
    "5
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "1 6 4
1 1 1 6
......
"
    "2
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3 3 1
2 1 2 3
.@.
.@.
.@.
"
    "-1
")))