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.util.*;
import java.io.*;
public class Main{
   public static void main(String[] main) throws Exception{
      BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
      StringTokenizer st = new StringTokenizer(br.readLine());
      PrintWriter out = new PrintWriter(System.out);
      int n = Integer.parseInt(st.nextToken());
      st = new StringTokenizer(br.readLine());
      long a1 = 0, a2 = 0;
      long xor = 0;
      for(int i = 0; i < n; i++) {
    	  long a = Long.parseLong(st.nextToken());
    	  if(i == 0)
    		  a1 = a;
    	  else if(i == 1)
    		  a2 = a;
    	  else
    		  xor = a ^ xor;
      }
      long sum = a1 + a2;
      long res = 0;
      PriorityQueue<Long> one = new PriorityQueue<Long>();
      boolean works = true;
      long pow = 1;
      while(sum > 0) {
    	  if(sum%2 == xor%2) {
    		  if(sum%2 == 1)
    			  one.add(-pow);
    		  else {
    			  if(sum%4 != xor%4) {
	    			  res += pow;
	    			  sum -= 2;
    			  }
    		  }
    	  }
    	  else {
    		  works = false;
    		  break;
    	  }
    	  sum /= 2;
    	  xor /= 2;
    	  pow *= 2;
      }
      if(!works || res > a1 || xor > 0)
    	  out.println(-1);
      else {
    	  while(one.size() > 0) {
    		  pow = -one.remove();
    		  if(res + pow <= a1)
    			  res += pow;
    	  }
    	  if(res == 0)
    		  out.println(-1);
    	  else
    		  out.println(a1 - res);
      }
      out.close();
   }
}