Let $(B_t)_{t\ge 0}$ be a standard Brownian motion. Consider $$\sup_{\tau} \mathbb{E}\frac{|B_{\tau}|}{1+\tau},$$ where supremum is taken over stopping times $\tau$ adapted to the natural filtration of $B$. Is there an easy way of finding this value?
I know that there is a whole theory devoted to optimal stopping of stochastic processes (Snell envelope, etc.) but I am pretty sure that there should be a straightforward and short solution - it was given as a one point (out of 5) of a warmup question in lecture notes - the rest of them were more or less obvious....