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)

// -*- coding:utf-8-unix -*-

use proconio::input;

// based on the pdf explanation

fn main() {
    input! {
        n: usize,
        a_vec: [usize; n],
    }

    let a1 = a_vec[0];
    let a2 = a_vec[1];
    let mut x = 0;
    for i in 2..n {
        x ^= a_vec[i];
    }
    let x = x;
    let s = a1 + a2;

    if s < x || (s - x) % 2 != 0 {
        println!("-1");
        return;
    }
    let d = (s - x) / 2;
    if d & x != 0 {
        println!("-1");
        return;
    }

    if d > a1 {
        println!("-1");
        return;
    }

    let mut y = 0;
    let mut bit = 2usize.pow(40);
    while bit != 0 {
        if bit & x == 0 {
            bit /= 2;
            continue;
        }
        if d ^ y ^ bit <= a1 {
            y ^= bit;
        }
        bit /= 2;
    }

    if d ^ y == 0 {
        println!("-1");
        return;
    }

    println!("{}", a1 - (d ^ y));
}