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)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let n = read::<usize>();
    let a = read_vec::<i64>();
    let x = a.iter().skip(2).fold(0, |s, &x| s ^ x);
    let mut s = a[0] + a[1];
    let mut dp = vec![vec![vec![-1i64; 2]; 2]; 43 + 1];
    dp[0][0][0] = 0;

    for ni in 0..43 {
        let cx = (x >> ni) & 1;
        let cs = (s >> ni) & 1;
        let ca = (a[0] >> ni) & 1;
        for nj in 0..2 {
            for nk in 0..2 {
                if dp[ni][nj][nk] == -1 {
                    continue;
                }
                for na in 0..2 {
                    let nb = cx ^ na;
                    if (nj as i64 + na + nb) % 2 != cs {
                        continue;
                    }
                    let cj = if nj as i64 + na + nb >= 2 { 1 } else { 0 };
                    let ck = if na == ca {
                        nk
                    } else if ca < na {
                        1
                    } else {
                        0
                    };
                    dp[ni + 1][cj][ck] = max(dp[ni + 1][cj][ck], dp[ni][nj][nk] | (na << ni));
                }
            }
        }
    }
    if dp[43][0][0] <= 0 {
        println!("-1");
        return;
    }
    println!("{}", a[0] - dp[43][0][0]);
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}