Expected value of a function of the stopping time for sum martingale of exponential random variables.

Question

Let $(X_n)$ be i.i.d random variables with exponential distribution with parameter 1. Let $S_n=X_1+\cdots+X_n$ and $\tau=\inf\{n\geq 1: S_n\geq 1\}$ be a stopping time. For $a,b>0$ the problem asks to compute $\mathbb{E}(e^{-a\tau+(1-b)S_{\tau}})$. Using the product martingale $M_n=e^{(1-b)S_n}b^{-n}=e^{-an+(1-b)S_n}b^{-n}e^{an}$ along with the Doob's Stopping Theorem / Dominated Convergence Theorem I got $\mathbb{E}(e^{-a\tau+(1-b)S_{\tau}})=\mathbb{E}\left(b^\tau e^{-a\tau}\right)$, but I could not evaluate the RHS. Any help is appreciated. Thanks in advance!