101725: [AtCoder]ABC172 F - Unfair Nim

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.