Is the product of ergodic process and gaussian RV still ergodic?

Question

Let $\boldsymbol{S}=\{S_n\}_{n=1}^{\infty}$ be a stationary and ergodic process and $X_n,Z_n,n=1,2,...$ be i.i.d. Gaussian RVs. My question is if $\{Y_n\}_{n=1}^{\infty}$, where $Y_n:=S_n\cdot X_n + Z_n$ is still a stationary and ergodic process? I don't have much experience in ergodic theory, and I found some books/slides on the internet but the properties/results in them are abstract and I have no idea which one is related to my question.