Variance of a normal with random mean and scale

Let $M$ and $S$ be independent $\text{Exp}(1)$ random variables. Conditional on $M=\mu$ and $S=\sigma$, the pair $(X,Y)$ is bivariate normal with both means equal to $\mu$, both variances equal to $\sigma^2$, and correlation $\rho$, where $\rho\in[-1,1]$ is a fixed constant. Find $\text{Var}(Y)$.

$3$. By the law of total variance, $\text{Var}(Y)=\mathbb{E}[\text{Var}(Y\mid M,S)]+\text{Var}(\mathbb{E}[Y\mid M,S])=\mathbb{E}[S^2]+\text{Var}(M)$. For $\text{Exp}(1)$ these are $2$ and $1$; $\rho$ plays no role.