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: Accepted)

using System;
using CompLib.Util;

public class Program
{
    private int N;
    private long[] A;

    public void Solve()
    {
        var sc = new Scanner();
        N = sc.NextInt();
        A = sc.LongArray();

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

        // xorが0だと後手勝ち
        // 非0だと先手

        // A_0 xor A_1 がotherになるようにする

        // and はこれになる
        long and = (A[0] + A[1] - other);
        if (and % 2 != 0)
        {
            Console.WriteLine("-1");
            return;
        }

        and /= 2;

        if (and > A[0])
        {
            Console.WriteLine("-1");
            return;
        }

        // A_0を超えないかつ最大
        long f = A[0] + A[1] - and * 2;

        if ((f & and) > 0)
        {
            Console.WriteLine("-1");
            return;
        }

        long tmp = and;
        for (int i = 50; i >= 0; i--)
        {
            if ((f & (1L << i)) == 0) continue;
            if ((tmp | (1L << i)) > A[0]) continue;
            tmp |= (1L << i);
        }

        if (tmp == 0)
        {
            Console.WriteLine("-1");
            return;
        }

        long ans = A[0] - tmp;
        long ans2 = A[0] + A[1] - tmp;


        if ((tmp ^ ans2) != other)
        {
            Console.WriteLine("-1");
            return;
        }

        Console.WriteLine(ans);
    }

    public static void Main(string[] args) => new Program().Solve();
}

namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}