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 C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;

namespace AtCoder.Abc172
{
    class F
    {
        public static void Main()
        {
            var n = int.Parse(Console.ReadLine());
            var a = Console.ReadLine().Split(' ').Select(x => long.Parse(x)).ToArray();

            var xor = a.Skip(2).Aggregate(0L, (acc, x) => acc ^= x);
            var sum = a[0] + a[1];
            var and = (sum - xor) >> 1; // a + b = a ^ b + ((a & b) << 1)より
            var or = and | xor;
            int bitN = 0;
            while (or > 0)
            {
                bitN++;
                or >>= 1;
            }

            long a0 = and & ~xor;
            long a1 = a0;
            long bit = 1L << (bitN - 1);
            while (bit > 0)
            {
                var iand = and & bit;
                var ixor = xor & bit;

                if (iand > 0)
                {
                    if (ixor == 0)
                    {
                        a0 |= bit;
                        a1 |= bit;
                    }
                    else
                    {
                        Console.WriteLine(-1);
                        return;
                    }
                }
                else if (ixor > 0)
                {
                    if ((a0 | bit) <= a[0])
                        a0 |= bit;
                    else
                        a1 |= bit;
                }
                bit >>= 1;
            }

            if (a0 > a[0] || a0 == 0)
                Console.WriteLine(-1);
            else
                Console.WriteLine(a[0] - a0);
        }
    }
}