Find the UMVUE of $b^{\mu}$

Question

Let $X_1,X_2,..X_n$ be a random sample from Cauchy$(\mu,1)$ population. Find the UMVUE of $b^{\mu}$ where $b$ is any positive real number. Now actually calculating sample mean won't work here because its distribution is also Cauchy$(\mu,1)$,so expectation won't exist.And all I know is that the sample median asymptotically goes to $\mu$,so it won't work here either.Is there any trick other than the normal procedures like Lehmann Scheffe to do this?