MEX of LCM

Description:

Input:

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 $$$3 \cdot 10^5$$$.

Output:

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

Sample Input:

6
3
1 2 3
5
1 2 3 4 5
2
2 3
1
1000000000
12
1 8 4 2 3 5 7 2 9 10 11 13
12
7 2 5 4 2 1 1 2 3 11 8 9

Sample Output:

4
7
1
1
16
13

Note:

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.