Find all positive integers $k$ with the following property:

For all positive integers $a,$ $b,$ and $c$ that make the roots of
\[ax^2 + bx + c = 0\]rational, the roots of $4ax^2 + 12bx + kc = 0$ will also be rational.

Enter all the possible values of $k,$ separated by commas.

Let's think step by step and output the final answer within \boxed{}.
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