complex analysis - Multiple annuli of laurent series expansion

Let $A_z(R_1,R_2)$ denote the open annulus about $z$ with inner radius $R_1$ and outer radius $R_2$.

According to the basic theorem of Laurent series, if $f\in Hol(A_z(R_1,R_2))$ then $f$ has a laurent series expansion about $z$ which converges locally uniformly in the annulus.

But what about cases where $f$ is holomorphic in multiple, mutually disjoint annuli?

For instance, consider $$f=\frac{1}{(z-1)z}$$

It has singularities both at $z=0$ and $z=1$ and is holomorphic in both the anuulus $A_0(0,1)$ and $A_0(1,\infty)$.

Does that mean that $f$ can be represented as two different Laurent series about $0$? each converging in a different annulus? If so, what can we tell about the relation between their coefficients? And what can the series tell us about $f$?

If there is only a single Laurent expansion of $f$ about $0$, then where does it converge?

Perhaps most importantly though, do the answers to these questions depend on the specific example of $f$ at all?