Stopped distribution of a geometric brownian motion

Question

I have this question and do not know how to come close to a proof.

Let $X_t$ be a geometric Brownian motion, $$ \frac{dX_t}{X_t} = \mu dt + \sigma dW_t $$

And we have the first hitting time of $\tau = \inf\left\{t>0, X_t < s\right\}$, How could we derive $\mathbb{E}[X_{T\wedge\tau}]$ for any positive $T$ ?

I tried to used optional stopping time theorem but $X_t$ is not a martingale. Any references to get better at recovering stopped distribution or moments would be also helpful (self-study)

Thanks.