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(
    non_snake_case,
    unused_variables,
    unused_assignments,
    unused_mut,
    dead_code
)]
//use permutohedron::LexicalPermutation as _;
use proconio::input;
//use proconio::marker::Chars;
//use proconio::marker::Usize1;
//use std::collections::HashSet;
//use std::collections::HashMap;
//use std::collections::BTreeMap;
use std::cmp::*;
// use std::num;
fn main() {
    input! {N:usize, A:[u64;N]}
    let mut sum = 0u64;
    for i in 2..N {
        sum ^= A[i];
    }
    //println!("{}", sum);
    // 残りの個数のXOR合計はsum。
    //    let mut ans = 0u64;
    //  let mut i = 0;
    let mut a0 = A[0];
    let mut a1 = A[1];
    let ans = sub(A[0], A[1], sum);
    if ans < A[0] {
        println!("{}", ans);
    } else {
        println!("{}", -1);
    }
}
fn sub(a: u64, b: u64, c: u64) -> u64 {
    if (a & 1) ^ (b & 1) != c & 1 {
        return std::u64::MAX;
    }
    if c == 0 {
        if a < b {
            return std::u64::MAX;
        } else {
            return (a - b) / 2;
        }
    }
    // case 0
    let mut x = sub(a / 2, b / 2, c / 2);
    // case 1
    let mut y = if a > 0 {
        sub((a - 1) / 2, (b + 1) / 2, c / 2)
    } else {
        std::u64::MAX
    };
    //println!("{},{},{},{},{}", a, b, c, x, y);
    if x < std::u64::MAX {
        x = x * 2;
    }
    if y < std::u64::MAX {
        y = y * 2 + 1;
    }
    return min(x, y);
}