C. Kanade's Perfect Multiples
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

We've carved those memories into ourselves... No matter how hard they were, they're the lives we carried out!
— Angel Beats!

In the afterlife school, Kanade studies a peculiar number game.

She gives you two integers $$$n$$$ and $$$k$$$, as well as an array $$$a$$$ consisting of $$$n$$$ integers, where $$$1 \le a_i \le k$$$ holds.

For an integer set $$$B = \{b_1, b_2, \ldots, b_m\}$$$ where $$$1\le b_i\le k$$$, we call it complete if and only if both of the following hold:

  • For each $$$1\le i\le n$$$, at least one divisor of $$$a_i$$$ is contained in $$$B$$$;
  • For each $$$1\le j\le m$$$, all positive multiples of $$$b_j$$$ which are less than or equal to $$$k$$$ appear in the array $$$a$$$ at least once.

You have to find a complete set $$$B$$$ with minimum possible size, or determine that no such set exists.

Input

Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1\le n\le 2\cdot 10^5$$$, $$$1\le k\le 10^9$$$) — the length of $$$a$$$ and the upper bound of elements of $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1\le a_i\le k$$$) — the elements of $$$a$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\cdot 10^5$$$.

Output

For each test case:

  • If no complete set $$$B$$$ exists, print a single integer $$$-1$$$ in the only line of output.
  • Otherwise:
    • First print a single integer $$$m$$$ ($$$1\le m\le n$$$) in the first line of output — the size of $$$B$$$. Note that you have to minimize the size of $$$B$$$.
    • Then output $$$m$$$ integers $$$b_1, b_2, \ldots, b_m$$$ ($$$1 \le b_i \le k$$$) in the second line — the set you constructed.

If there are multiple answers, you may print any of them.

Example
Input
4
4 6
3 2 4 6
5 5
1 2 3 4 5
3 6
2 3 6
1 2
2
Output
2
2 3 
1
1 
-1
1
2
Note

In the first test case, $$$B=\{2,3\}$$$ works. For $$$b=2$$$, all multiples $$$\{2,4,6\}$$$ (up to $$$k=6$$$) appear in $$$a$$$; for $$$b=3$$$, multiples $$$\{3,6\}$$$ appear. Every $$$a_i$$$ is divisible by $$$2$$$ or $$$3$$$. No single $$$b$$$ can satisfy both conditions, so $$$m=2$$$ is minimal.

In the second test case, $$$B=\{1\}$$$ satisfies both rules since every number is divisible by $$$1$$$ and all multiples of $$$1$$$ up to $$$k$$$ appear.