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 Java (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        try (Scanner sc = new Scanner(System.in)) {
            int N = sc.nextInt();
            long[] A = new long[N];
            for (int i = 0; i < N; i++) {
                A[i] = sc.nextLong();
            }

            long l = 0;
            for (int i = 2; i < N; i++) {
                l ^= A[i];
            }

            long solution = 0;
            int score = Long.bitCount((A[0] - solution) ^ (A[1] + solution) ^ l);

            long bestSolution = A[0];
            int bestScore = 100;

            for (int it = 0; it < 1e7; ++it) {
                double temperature = 10.0 / (it + 1);

                int shift = it % 64;
                long bit = 1L << shift;

                solution ^= bit;

                if (solution >= A[0]) {
                    solution ^= bit;
                    continue;
                }

                int deltaScore = Long.bitCount((A[0] - solution) ^ (A[1] + solution) ^ l) - score;
                if (Math.random() < Math.exp(-deltaScore * temperature)) {
                    score += deltaScore;
                    if (score < bestScore || (score == bestScore && solution < bestSolution)) {
                        bestScore = score;
                        bestSolution = solution;
                    }
                } else {
                    solution ^= bit;
                }
            }

            if (bestScore > 0) {
                System.out.println(-1);
            } else {
                System.out.println(bestSolution);
            }
        }
    }
}