CF1834E.MEX of LCM

You are given an array $a$ of length $n$ . A positive integer $x$ is called good if it is impossible to find a subsegment $^{\dagger}$ of the array such that the least common multiple of all its elements is equal to $x$ .

You need to find the smallest good integer.

A subsegment $^{\dagger}$ of the array $a$ is a set of elements $a_l, a_{l + 1}, \ldots, a_r$ for some $1 \le l \le r \le n$ . We will denote such subsegment as $[l, r]$ .

Each test consists of multiple test cases. The first line of each test case contains a single integer $t$ ( $1 \le t \le 5 \cdot 10^4$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 3 \cdot 10^5$ ) — the length of the array $a$ .

The second line of each test case contains $n$ integers $a_1, a_2, \ldots , a_n$ ( $1 \leq a_i \leq 10^9$ ) — the elements of the array $a$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^5$ .

For each test case, output a single integer — the smallest good integer.

In the first test case, $4$ is a good integer, and it is the smallest one, since the integers $1,2,3$ appear in the array, which means that there are subsegments of the array of length $1$ with least common multiples of $1,2,3$ . However, it is impossible to find a subsegment of the array with a least common multiple equal to $4$ .

In the second test case, $7$ is a good integer. The integers $1,2,3,4,5$ appear explicitly in the array, and the integer $6$ is the least common multiple of the subsegments $[2, 3]$ and $[1, 3]$ .

In the third test case, $1$ is a good integer, since the least common multiples for the integer in the subsegments $[1, 1], [1, 2], [2, 2]$ are $2,6,3$ , respectively.