Johnny and His Hobbies
Contest: Codeforces Round 647 (Div. 2) - Thanks, Algo Muse!
Difficulty rating: 1200
Tags: bitmasks, brute force

Among Johnny's numerous hobbies, there are two seemingly harmless ones: applying bitwise operations and sneaking into his dad's office. As it is usually the case with small children, Johnny is unaware that combining these two activities can get him in a lot of trouble.

There is a set $$$S$$$ containing very important numbers on his dad's desk. The minute Johnny heard about it, he decided that it's a good idea to choose a positive integer $$$k$$$ and replace each element $$$s$$$ of the set $$$S$$$ with $$$s \oplus k$$$ ($$$\oplus$$$ denotes the exclusive or operation).

Help him choose such $$$k$$$ that Johnny's dad will not see any difference after his son is done playing (i.e. Johnny will get the same set as before playing). It is possible that no such number exists. It is also possible that there are many of them. In such a case, output the smallest one. Note that the order of elements in a set doesn't matter, i.e. set $$$\{1, 2, 3\}$$$ equals to set $$$\{2, 1, 3\}$$$.

Formally, find the smallest positive integer $$$k$$$ such that $$$\{s \oplus k | s \in S\} = S$$$ or report that there is no such number.

For example, if $$$S = \{1, 3, 4\}$$$ and $$$k = 2$$$, new set will be equal to $$$\{3, 1, 6\}$$$. If $$$S = \{0, 1, 2, 3\}$$$ and $$$k = 1$$$, after playing set will stay the same.

Input format:
In the first line of input, there is a single integer $$$t$$$ ($$$1 \leq t \leq 1024$$$), the number of test cases. In the next lines, $$$t$$$ test cases follow. Each of them consists of two lines.

In the first line there is a single integer $$$n$$$ ($$$1 \leq n \leq 1024$$$) denoting the number of elements in set $$$S$$$. Second line consists of $$$n$$$ distinct integers $$$s_i$$$ ($$$0 \leq s_i < 1024$$$), elements of $$$S$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases will not exceed $$$1024$$$.

Output format:
Print $$$t$$$ lines; $$$i$$$-th line should contain the answer to the $$$i$$$-th test case, the minimal positive integer $$$k$$$ satisfying the conditions or $$$-1$$$ if no such $$$k$$$ exists.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first test case, the answer is $$$1$$$ because it is a minimum positive integer and it satisfies all the conditions.

Example input:
6
4
1 0 2 3
6
10 7 14 8 3 12
2
0 2
3
1 2 3
6
1 4 6 10 11 12
2
0 1023

Example output:
1
4
2
-1
-1
1023

Editorial:
Consider $$$i$$$-th least significant bit ($$$0$$$ indexed). If it is set in $$$k$$$, but not in $$$s$$$, it will be set in $$$k \oplus s$$$. Hence $$$k \oplus s \geq 2^i$$$.
Consider such minimal positive integer $$$m$$$, that $$$2^m > s$$$ holds for all $$$s \in S$$$. $$$k$$$ cannot have the $$$i$$$-th bit set for any $$$i \geq m$$$. From this follows that $$$k < 2^m$$$. So there are only $$$2^m$$$ feasible choices of $$$k$$$. We can verify if a number satisfies the condition from the statement in $$$ \mathcal{O} \left(n \right) $$$ operations. This gives us a solution with complexity $$$\mathcal{O} \left(n \cdot 2^m \right)$$$. Note that in all tests $$$m$$$ is at most $$$10$$$.
There is also another solution possible. It uses the observation that if $$$k$$$ satisfies the required conditions, then for every $$$s \in S$$$ there exists such $$$t \in S$$$ ($$$t \neq s$$$) , that $$$t \oplus s = k$$$. This gives us $$$n-1$$$ feasible choices of $$$k$$$ and thus the complexity of this solution is $$$\mathcal{O} \left( n^2 \right)$$$.