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 Fortran (GNU Fortran 9.2.1) (judge verdict: Accepted)

program unfair_nim
  implicit none
  integer :: n, i
  integer(8) :: a(300) = 0, s, x = 0, d, y = 0
  read(*,*) n
  read(*,*) a(1:n)
  s = a(1) + a(2)
  do i = 3, n
    x = xor(x, a(i))
  end do
  if (s - x < 0 .or. mod(s - x, 2) /= 0) then
    write(*,'(i0)') -1
    stop
  end if
  d = (s - x) / 2
  if (and(d, x) /= 0 .or. d > a(1)) then
    write(*,'(i0)') -1
    stop
  end if
  do i = 62, 0, -1
    if (.not.btest(x, i)) cycle
    y = ibset(y, i)
    if (xor(d, y) > a(1)) y = ibclr(y, i)
  end do
  write(*,'(i0)') merge(-1_8, a(1) - xor(d, y), xor(d, y) == 0)
end program unfair_nim