Let random variables $(X_n)$ be i.i.d, $\mathbb{E}X_n = a > 0$, $\operatorname{Var}X_n = \sigma^2 > 0$. Let $S_n = X_1+X_2+\cdots+X_n$. Show that the sequence of random variables
$$ \eta_n = \frac{2\sqrt{a}}{\sigma}\left(\sqrt{|S_n|} - \sqrt{na}\right)$$
converges by law to standard normal distribution.
I was trying to use CLT, but the absolute value in $S_n$ is quite problematic. I was also trying to look at cumulative distribution function convergence, but also not result. Some hints are greatly appreciated.