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 Ruby (2.7.1) (judge verdict: Wrong Answer)

# read https://img.atcoder.jp/abc172/editorial.pdf

def to_binary_array(num)
  bin = num.to_s(2).chars.map(&:to_i)
  bin = [0] * (ARRAY_SIZE - bin.size) + bin
  bin.reverse!
end

def binary_array_to_int(arr)
  ans = 0
  arr.size do |i|
    ans += 2**i if arr[i] == 1
  end
  ans
end

ARRAY_SIZE = 42
N = gets.chomp.to_i
AS = gets.chomp.split.map(&:to_i)

S = AS[0] + AS[1]
X = AS[2..-1].inject(0) { |result, n| result^n }

s_bin = to_binary_array(S)
a_x_bin = to_binary_array(X)

unless S > X || s_bin[0] == a_x_bin[0]
  puts "-1"
  exit
end

fix = [false] * ARRAY_SIZE
a = 0
0.upto(39) do |i|
  if s_bin[i].zero?
    fix[i] = true
    next if s_bin[i + 1] == a_x_bin[i + 1]
    a += 2**i
    a_x_bin = to_binary_array(a + (X^a))
  end
  if s_bin[i + 1] != a_x_bin[i + 1]
    puts "-1"
    exit
  end
end

if AS[0] < a
  puts "-1"
  exit
end

40.downto(0) do |i|
  next if fix[i] || a + 2**i > AS[0]
  a += 2**i
end

if a.zero?
  puts -1
  exit
end

puts AS[0] - a