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 main
    use iso_fortran_env
    implicit none
    integer(int64) :: n, i
    integer(int64), allocatable :: a(:)
    integer(int64) :: s, x, d, ans
    logical :: bx, bd

    read *, n
    allocate(a(n))
    read *, a(1:n)

    x = 0
    d = 0
    s = a(1) + a(2)
    ans = 0

    if ( n>2 ) then
        do i = 3, n
            x = ieor(x, a(i))
        enddo
    endif

    if ( trailz(s-x) >= 1 ) then
        d = (s - x) / 2
    else
        print *, "-1"
        stop
    endif

    if ( iand(x, d) /= 0) then
        print *, "-1"
        stop
    endif

    do i = 0, 63
        bd = btest(d, i)
        if (bd) then
            ans = ibset(ans, i)
        endif
    enddo

    do i = 63, 0, -1
        bx = btest(x, i)
        if (bx) then
            if ( ibset(ans, i) <= a(1) ) then
                ans = ibset(ans, i)
            endif
        endif
    enddo

    if ((ans == 0) .or. (ans > a(1))) then
        print *, "-1"
        stop
    endif

    print *, a(1) - ans
 
end program main