Problem:
Score : 600 points

Problem Statement

There are N piles of stones. The i-th pile has A_i stones.

Aoki and Takahashi are about to use them to play the following game:

Starting with Aoki, the two players alternately do the following operation:

Operation: Choose one pile of stones, and remove one or more stones from it.

When a player is unable to do the operation, he loses, and the other player wins.

When the two players play optimally, there are two possibilities in this game: the player who moves first always wins, or the player who moves second always wins, only depending on the initial number of stones in each pile.

In such a situation, Takahashi, the second player to act, is trying to guarantee his win by moving at least zero and at most (A_1 - 1) stones from the 1-st pile to the 2-nd pile before the game begins.

If this is possible, print the minimum number of stones to move to guarantee his victory; otherwise, print -1 instead.

Constraints

2 \leq N \leq 300

1 \leq A_i \leq 10^{12}

Input

Input is given from Standard Input in the following format:

N
A_1 \ldots A_N

Output

Print the minimum number of stones to move to guarantee Takahashi's win; otherwise, print -1 instead.

Sample Input 1

2
5 3

Sample Output 1

1

Without moving stones, if Aoki first removes 2 stones from the 1-st pile, Takahashi cannot win in any way.

If Takahashi moves 1 stone from the 1-st pile to the 2-nd before the game begins so that both piles have 4 stones, Takahashi can always win by properly choosing his actions.

Sample Input 2

2
3 5

Sample Output 2

-1

It is not allowed to move stones from the 2-nd pile to the 1-st.

Sample Input 3

3
1 1 2

Sample Output 3

-1

It is not allowed to move all stones from the 1-st pile.

Sample Input 4

8
10 9 8 7 6 5 4 3

Sample Output 4

3

Sample Input 5

3
4294967297 8589934593 12884901890

Sample Output 5

1

Watch out for overflows.

Solution in Common Lisp (SBCL 2.0.3) (judge verdict: Wrong Answer)

(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
(defmacro xor (form1 form2)
  (let ((f1 (gensym)) (f2 (gensym)))
    `(let ((,f1 ,form1)
           (,f2 ,form2))
       (if ,f1 (not ,f2) ,f2))))

(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
;;;

(defun test (a1 a2)
  (loop for x from 0 to a1
        for value = (logxor (- a1 x) (+ a2 x))
        do ;; (format t "~10,'0B~%" value)
           (println value)
           ;; (format t "~D ~D: ~8,'0B~%" (- a1 x) (+ a2 x) value)
        ))

(defun solve (a1 a2 rhs &optional (length 40))
  (declare #.OPT
           (uint62 a1 a2 rhs))
  (labels ((dfs (pos value1)
             (declare ((integer -1 40) pos)
                      (uint62 value1))
             (dbg pos value1)
             (let* ((k (- a1 value1))
                    (value2 (+ a2 k)))
               (declare (uint62 value2))
               (when (= pos -1)
                 (if (and (> value1 0)
                          (= (logxor value1 value2) rhs))
                     (return-from solve value1)
                     (return-from dfs)))
               (let* ((cut (dpb 0 (byte pos 0) value2))
                      (mask (ldb (byte 64 0) (lognot (- (ash (logand cut (- cut)) 1) 1)))))
                 ;; (format t "~64,'0B~%" value1)
                 ;; (format t "~64,'0B~%" value2)
                 ;; (format t "~64,'0B~%" mask)
                 (unless (= (logand mask (logxor value1 value2))
                            (logand mask rhs))
                   (return-from dfs)))
               ;; 次を1
               (let* ((next-value1 (+ value1 (ash 1 pos))))
                 ;; (dbg next-value1 next-value2)
                 (when (< next-value1 a1)
                   (dfs (- pos 1) next-value1)))
               (dbg value1 value2 k)
               ;; 次を0
               (dfs (- pos 1) value1))))
    (dfs (- length 1) 0)))

(defun main ()
  (let* ((n (read))
         (as (make-array n :element-type 'uint62 :initial-element 0)))
    (dotimes (i n)
      (setf (aref as i) (read)))
    (let ((a1 (aref as 0))
          (a2 (aref as 1))
          (d (reduce #'logxor as :start 2 :initial-value 0)))
      (dbg a1 a2 d)
      (let ((res (solve a1 a2 d 40)))
        (println
         (if res
             (- a1 res)
             -1))))))

#-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 "2
5 3
"
    "1
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "2
3 5
"
    "-1
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3
1 1 2
"
    "-1
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "8
10 9 8 7 6 5 4 3
"
    "3
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3
4294967297 8589934593 12884901890
"
    "1
")))