how do I find all $k \in \mathbb{R} $ such that $\lim_{n \to \infty} a_n = +\infty$ for a given sequence without using approximations

I have a sequence $(a_n)_{n \in \mathbb{N}}$ of positive terms defined by the recurrence relation

$\frac{a_{n+1}}{a_n} = \left(\frac{2n}{2n + k + 4}\right)^{2n},$

where $k \in \mathbb{R}$ and $k \neq -4$. I need to determine all values of $k$ for which $\lim_{n \to \infty} a_n = +\infty$.

I would like to avoid using Taylor series approximations, binomial approximations.

Ratio-test, cauchy's root-test aswell as euler limit and subsequences are the tools that have been taught, how should I proceed with this problem?