A15912. Lunatic Never Content

You have an array $a$ of $n$ non-negative integers. Let's define $f(a, x) = [a_1 \bmod x, a_2 \bmod x, \dots, a_n \bmod x]$ for some positive integer $x$ . Find the biggest $x$ , such that $f(a, x)$ is a palindrome.

Here, $a \bmod x$ is the remainder of the integer division of $a$ by $x$ .

An array is a palindrome if it reads the same backward as forward. More formally, an array $a$ of length $n$ is a palindrome if for every $i$ ( $1 \leq i \leq n$ ) $a_i = a_{n - i + 1}$ .

The first line contains a single integer $t$ ( $1 \leq t \leq 10^5$ ) — the number of test cases.

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 10^5$ ).

The second line of each test case contains $n$ integers $a_i$ ( $0 \leq a_i \leq 10^9$ ).

It's guaranteed that the sum of all $n$ does not exceed $10^5$ .

For each test case output the biggest $x$ , such that $f(a, x)$ is a palindrome. If $x$ can be infinitely large, output $0$ instead.

In the first example, $f(a, x = 1) = [0, 0]$ which is a palindrome.

In the second example, $f(a, x = 2) = [1, 0, 1, 0, 0, 1, 0, 1]$ which is a palindrome.

It can be proven that in the first two examples, no larger $x$ satisfies the condition.

In the third example, $f(a, x) = [0]$ for any $x$ , so we can choose it infinitely large, so the answer is $0$ .