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 Rust (1.42.0) (judge verdict: Accepted)

const C: usize = 42;

fn main() {
    proconio::input! {
        n: usize,
        a: [u64; n],
    }

    let s = a[0] + a[1];
    let mut x = 0;
    for i in 2..n {
        x ^= a[i];
    }

    let mut dp = vec![vec![vec![a[0] + 1; 2]; 2]; C];
    dp[0][0][0] = 0;
    for i in 0..(C - 1) {
        let cx = (x >> i) & 1;
        let cs = (s >> i) & 1;
        let ca = (a[0] >> i) & 1;

        for j in 0..2 {
            for k in 0..2 {
                if dp[i][j][k] > a[0] {
                    continue;
                }
                for na in 0..2 {
                    for nb in 0..2 {
                        if (na ^ nb) != cx {
                            continue;
                        }
                        let ns = na + nb + j as u64;
                        if ns % 2 != cs {
                            continue;
                        }
                        let ni = i + 1;
                        let nj = if ns >= 2 { 1 } else { 0 };
                        let nk = if ca < na { 1 } else if ca == na { k } else { 0 };
                        let v = dp[i][j][k] | ((1 << i as u64) * na);
                        if dp[ni][nj][nk] > a[0] || v > dp[ni][nj][nk] {
                            dp[ni][nj][nk] =v;
                        }
                    }
                }
            }
        }
    }

    let r = dp[C - 1][0][0];
    if r <= a[0] && r > 0 {
        println!("{}", a[0] - r);
    } else {
        println!("-1");
    }
}