$Z(t) = A\cos(\omega t+\theta)$ where $A$~$\mathcal N(0,\sigma ^2) $ and $\theta $~$\mathcal U(0,2\pi)$ are independent.

I'm trying to figure out if $Z(t)$ is a Gaussian random process and whether it is strict sense stationary. It is easy to see that it is WSS, but I can't figure out about the SSS and whether it is a Gaussian process. It does look like to me that for every chosen time $t$ I get some kind of a normal random variable multiplied by a constant per $\theta$ and therefore it is a Gaussian process, but I can't seem to prove nor disprove it. Can someone clarify this for me? Thanks.