Variance under random parameters

Let $M$ and $S$ be independent $\text{Exponential}(1)$ random variables. Given $M = \mu$ and $S = \sigma$, the pair $(X, Y)$ is bivariate normal with both means $\mu$, both variances $\sigma^2$, and correlation $\rho$ (a fixed constant with $|\rho| \le 1$). Compute $\operatorname{Var}(Y)$.