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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#define IO ios::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define DB(x) cerr << __LINE__ << ": " << #x << " = " << (x) << "\n"
using namespace std;
using ll = long long;

signed main() {
    IO;
    ll n;
    cin >> n;
    vector<ll> a(n);
    ll xor_val = 0, and_val;
    for(int i = 0; i < n; ++i) {
        cin >> a[i];
        if(i > 1) xor_val ^= a[i];
    }
    if((a[0] + a[1] - xor_val < 0) || (a[0] + a[1] - xor_val) % 2 == 1) {cout << -1; return 0;}
    and_val = (a[0] + a[1] - xor_val) / 2;
    if((and_val & xor_val) != 0) {cout << -1; return 0;}
    
    ll a0 = 0, a1 = 0;
    bool pos = true;
    for(ll i = 39; i >= 0; --i) {
        ll two = 1ll << i;
        if(and_val & two) {
            a0 |= two; a1 |= two;
        }
    }
    for(ll i = 39; i >= 0; --i) {
        ll two = 1ll << i;
        if(xor_val & two) {
            ll new_a0 = a0 | two;
            if(new_a0 < a[0]) a0 = new_a0;
            else a1 |= two;
        }
    }

    if(a0 == 0) pos = false;
    if(a0 > a[0]) pos = false;

    if(!pos) cout << -1;
    else cout << a[0] - a0;
    return 0;
}