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 residual = a.iter().skip(2).fold(0, |s, &x| s ^ x);
    let mut a2 = a[0] + a[1];

    let max_bit = 43;
    let mut cands = vec![-1; max_bit];
    for i in (0..max_bit).rev() {
        if (residual >> i) & 1 == 0 {
            if a2 >= 1 << (i + 1) {
                cands[i] = 1;
                a2 -= 1 << (i + 1);
            } else if a2 >= 1 << i {
                cands[i] = 0;
            } else {
                cands[i] = 0;
            }
        } else {
            if a2 >= 1 << (i + 1) {
                a2 -= 1 << i;
            } else if a2 >= 1 << i {
                a2 -= 1 << i;
            } else {
                println!("-1");
                return;
            }
        }
    }
    if a2 > 0 {
        println!("-1");
        return;
    }
    debug!(cands);
    let mut a0 = 0;
    for i in (0..max_bit).rev() {
        if cands[i] == 1 {
            a0 += 1 << i;
        }
    }
    if a0 > a[0] {
        println!("-1");
        return;
    }
    for i in (0..max_bit).rev() {
        if cands[i] == -1 {
            if a0 + (1 << i) <= a[0] {
                a0 += 1 << i;
            }
        }
    }
    if a0 == 0 {
        println!("-1");
        return;
    }
    println!("{}", a[0] - a0);
}

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()
}