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 11.0.6) (judge verdict: Accepted)

import java.io.*;
import java.util.*;

public class Main {
	static Scanner sc = new Scanner(System.in);
	static PrintWriter out = new PrintWriter(System.out);

	public static void main(String[] args) throws Exception {
		int n = sc.nextInt();
		long arr[] = new long[n], xor = 0;
		
		for(int i = 0; i < n; i++) {
			arr[i] = sc.nextLong();
			xor ^= i < 2 ? 0 : arr[i];
		}
		
		long sum = arr[0] + arr[1];
		long and = (sum - xor) >> 1;
		long a = 0, b = 0;
		
		for(int i = 60; i >= 0; i--) {
			boolean xi = (xor & (1L << i)) > 0;
			boolean ai = (and & (1L << i)) > 0;
			
			if(!xi && ai) {
				a |= 1L << i;
				b |= 1L << i;
			}
		}
		
		
		for(int i = 60; i >= 0; i--) {
			boolean xi = (xor & (1L << i)) > 0;
			boolean ai = (and & (1L << i)) > 0;
			
			if(xi && !ai) {
				if((a | 1L << i) <= arr[0]) {
					a |= 1L << i;
				}else {
					b |= 1L << i;
				}
			}
		}
		
		boolean flag = (a + b) == sum && (a ^ b) == xor;
		flag &= a >= 1 && a <= arr[0] && b >= arr[1];
		
		out.println(flag ? arr[0] - a : -1);
		out.close();
	}

}

class Scanner {
	StringTokenizer st;
	BufferedReader br;

	public Scanner(InputStream system) {
		br = new BufferedReader(new InputStreamReader(system));
	}

	public Scanner(String file) throws Exception {
		br = new BufferedReader(new FileReader(file));
	}

	public String next() throws IOException {
		while (st == null || !st.hasMoreTokens())
			st = new StringTokenizer(br.readLine());
		return st.nextToken();
	}

	public String nextLine() throws IOException {
		return br.readLine();
	}

	public int nextInt() throws IOException {
		return Integer.parseInt(next());
	}

	public double nextDouble() throws IOException {
		return Double.parseDouble(next());
	}

	public Long nextLong() throws IOException {
		return Long.parseLong(next());
	}
}