Cookies

Elsa’s elder brother Eric has got n cookies in his icebox and each cookie has a special number written on it. Let’s denote the number written on the $i^{th}$ cookie by $f_i$. $f_i$ is defined as follows:

Here, divmed(x) states the median value of divisors of x. Let x has k divisors, then divmed(x) is $\lceil{k/2}\rceil^{th}$ smallest divisor. For example, divmed(100)=10, divmed(10)=2, divmed(1)=1.

One day, Eric opened the icebox and recognized that some of his cookies are missing. His sister Elsa had eaten some of them.

On the first day, she ate cookies that have multiples of $p_1$ as indexes and then, re-indexed them starting from 1 according to their original order. In the same way, she ate cookies with multiples of $p_i$ as indexes on the $i^{th}$ day and re-indexed them. If there were less than $p_i$ cookies left, she ate none of them. Elsa continued to do so for m days.

As Elsa is keen on math and Eric didn’t want to blame his sister, Eric asked her to find out the number written on the $k^{th}$ of remaining cookies.

The first line of the input contains an integer T(1≤T≤60), denoting the number of test cases. The first line of each test case contains 3 positive integers n(1≤n≤$10^{10}$), m(1≤m≤$10^5$) and k(1≤k≤$10^{10}$). The next line contains m integers $p_1$,$p_2$,$ \cdots $,$p_m$ (1≤$p_i$≤$10^{10}$).

For each test case, output one line containing the answer. If the $k^{th}$ cookie doesn’t exist, then output -1 in a single line.

2
25 5 9
3 5 8 12 10
25 5 2
3 5 8 12 10

31
2

2020 Multi-University Training Contest 1