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: Accepted)

(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
;;;
;;; Binary heap
;;;

(define-condition heap-empty-error (error)
  ((heap :initarg :heap :reader heap-empty-error-heap))
  (:report
   (lambda (condition stream)
     (format stream "Attempted to pop empty heap ~W" (heap-empty-error-heap condition)))))

(defmacro define-binary-heap (name &key (order '#'>) (element-type 'fixnum))
  "Defines a binary heap specialized for the given order and the element
type. This macro defines a structure of the name NAME and relevant functions:
MAKE-<NAME>, <NAME>-PUSH, <NAME>-POP, <NAME>-REINITIALIZE, <NAME>-EMPTY-P,
<NAME>-COUNT, and <NAME>-PEEK."
  (check-type name symbol)
  (let* ((string-name (string name))
         (fname-push (intern (format nil "~A-PUSH" string-name)))
         (fname-pop (intern (format nil "~A-POP" string-name)))
         (fname-reinitialize (intern (format nil "~A-REINITIALIZE" string-name)))
         (fname-empty-p (intern (format nil "~A-EMPTY-P" string-name)))
         (fname-count (intern (format nil "~A-COUNT" string-name)))
         (fname-peek (intern (format nil "~A-PEEK" string-name)))
         (fname-make (intern (format nil "MAKE-~A" string-name)))
         (acc-position (intern (format nil "~A-POSITION" string-name)))
         (acc-data (intern (format nil "~A-DATA" string-name))))
    `(progn
       (locally
           ;; prevent style warnings
           (declare #+sbcl (muffle-conditions style-warning))
         (defstruct (,name
                     (:constructor ,fname-make
                         (size
                          &aux (data ,(if (eql element-type '*)
                                          `(make-array (1+ size))
                                          `(make-array (1+ size) :element-type ',element-type))))))
           (data #() :type (simple-array ,element-type (*)))
           (position 1 :type (integer 1 #.most-positive-fixnum))))

       (declaim #+sbcl (sb-ext:maybe-inline ,fname-push))
       (defun ,fname-push (obj heap)
         "Adds OBJ to HEAP."
         (declare (optimize (speed 3))
                  (type ,name heap))
         (symbol-macrolet ((position (,acc-position heap)))
           (when (>= position (length (,acc-data heap)))
             (setf (,acc-data heap)
                   (adjust-array (,acc-data heap)
                                 (min (- array-total-size-limit 1)
                                      (* position 2)))))
           (let ((data (,acc-data heap)))
             (declare ((simple-array ,element-type (*)) data))
             (labels ((update (pos)
                        (declare (optimize (speed 3) (safety 0)))
                        (unless (= pos 1)
                          (let ((parent-pos (ash pos -1)))
                            (when (funcall ,order (aref data pos) (aref data parent-pos))
                              (rotatef (aref data pos) (aref data parent-pos))
                              (update parent-pos))))))
               (setf (aref data position) obj)
               (update position)
               (incf position)
               heap))))

       (declaim #+sbcl (sb-ext:maybe-inline ,fname-pop))
       (defun ,fname-pop (heap)
         "Removes and returns the element at the top of HEAP."
         (declare (optimize (speed 3))
                  (type ,name heap))
         (symbol-macrolet ((position (,acc-position heap)))
           (let ((data (,acc-data heap)))
             (declare ((simple-array ,element-type (*)) data))
             (labels ((update (pos)
                        (declare (optimize (speed 3) (safety 0))
                                 ((integer 1 #.most-positive-fixnum) pos))
                        (let* ((child-pos1 (+ pos pos))
                               (child-pos2 (1+ child-pos1)))
                          (when (<= child-pos1 position)
                            (if (<= child-pos2 position)
                                (if (funcall ,order (aref data child-pos1) (aref data child-pos2))
                                    (unless (funcall ,order (aref data pos) (aref data child-pos1))
                                      (rotatef (aref data pos) (aref data child-pos1))
                                      (update child-pos1))
                                    (unless (funcall ,order (aref data pos) (aref data child-pos2))
                                      (rotatef (aref data pos) (aref data child-pos2))
                                      (update child-pos2)))
                                (unless (funcall ,order (aref data pos) (aref data child-pos1))
                                  (rotatef (aref data pos) (aref data child-pos1))))))))
               (when (= position 1)
                 (error 'heap-empty-error :heap heap))
               (prog1 (aref data 1)
                 (decf position)
                 (setf (aref data 1) (aref data position))
                 (update 1))))))

       (declaim (inline ,fname-reinitialize))
       (defun ,fname-reinitialize (heap)
         "Makes HEAP empty."
         (setf (,acc-position heap) 1)
         heap)

       (declaim (inline ,fname-empty-p))
       (defun ,fname-empty-p (heap)
         "Returns true iff HEAP is empty."
         (= 1 (,acc-position heap)))

       (declaim (inline ,fname-count))
       (defun ,fname-count (heap)
         "Returns the current number of the elements in HEAP."
         (- (,acc-position heap) 1))

       (declaim (inline ,fname-peek))
       (defun ,fname-peek (heap)
         "Returns the topmost element of HEAP."
         (if (= 1 (,acc-position heap))
             (error 'heap-empty-error :heap heap)
             (aref (,acc-data heap) 1))))))


(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)

;; dist x y . dir
(define-binary-heap heap
  :order (lambda (node1 node2)
           (< (the uint32 (car node1))
              (the uint32 (car node2))))
  :element-type list)

(defun main ()
  (declare #.OPT)
  (let* ((h (read))
         (w (read))
         (k (read))
         (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))
         (dists (make-array (list h w 4) :element-type 'uint32 :initial-element +inf+))
         ;; x . y
         (que (make-heap (* h w 4))))
    (declare (uint31 h w k))
    (dotimes (i h)
      (dotimes (j w (read-schar))
        (ecase (read-schar)
          (#\@ (setf (aref plan i j) 1))
          (#\.))))
    (heap-push (list* 0 start-x start-y 0) que)
    ;; up:0, down:1, left:2, right:3
    (setf (aref dists start-x start-y 0) 0)
    (labels ((visit (new-x new-y new-dir new-dist)
               (when (and (<= 0 new-x (- h 1))
                          (<= 0 new-y (- w 1))
                          (zerop (aref plan new-x new-y))
                          (< new-dist (aref dists new-x new-y new-dir)))
                 (heap-push (list* new-dist new-x new-y new-dir) que)
                 (setf (aref dists new-x new-y new-dir) new-dist))))
      (loop until (heap-empty-p que)
            for (dist x y . dir) of-type (uint32 uint32 uint32 . uint32) = (heap-pop que)
            for ceil = (* k (ceiling dist k))
            when (= dist (aref dists x y dir))
            do (ecase dir
                 (0 (visit (- x 1) y 0 (+ dist 1)))
                 (1 (visit (+ x 1) y 1 (+ dist 1)))
                 (2 (visit x (- y 1) 2 (+ dist 1)))
                 (3 (visit x (+ y 1) 3 (+ dist 1))))
               (dotimes (new-dir 4)
                 (dbg x y new-dir ceil)
                 (visit x y new-dir ceil))))
    (let ((res (loop for dir below 4
                     minimize (ceiling (aref dists goal-x goal-y dir) k))))
      #>dists
      (println
       (if (>= res 10000000)
           -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
")))