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 PyPy3 (7.3.0) (judge verdict: Accepted)

import sys
sys.setrecursionlimit(10**6)

K = 43


def main(input, print):
    n = int(input())
    a = list(map(int, input().split()))
    a1 = a[0]
    a2 = a[1]
    s = a1 + a2
    x = 0
    for i in range(2, n):
        x ^= a[i]

    dp = [[[-1 for _ in range(2)] for _ in range(2)] for _ in range(K)]
    dp[0][0][0] = 0
    v = 1
    for i in range(K-1):
        cx = x & 1
        cs = s & 1
        ca = a1 & 1
        for j in range(2):
            for k in range(2):
                if dp[i][j][k] == -1:
                    continue
                for na in range(2):
                    for nb in range(2):
                        ni = i + 1
                        nj = 0
                        nk = k
                        if na ^ nb != cx:
                            continue
                        ns = na + nb + j
                        if ns % 2 != cs:
                            continue
                        if ns >= 2:
                            nj = 1
                        if ca < na:
                            nk = 1
                        elif ca == na:
                            nk = k
                        else:
                            nk = 0
                        dp[ni][nj][nk] = max(dp[ni][nj][nk], dp[i][j][k] | (v*na))
        x >>= 1
        s >>= 1
        a1 >>= 1
        v <<= 1

    ans = dp[K-1][0][0]
    if ans == -1 or ans == 0:
        print(-1)
    else:
        print(a[0]-ans)


if __name__ == '__main__':
    main(sys.stdin.readline, print)