Question: Can Maple expand in Laurent series for specific regions?

Is there a way to tell Maple to expand a complex function in Laurent series around a point, and have it show the series expansion that are valid for different regions? Either by the user telling it which region to use, or it automatically shows all regions?

For example, given

f(z):= z-> (3*z+1)/((z-1)*(z+1))

This has a pole at z=1 and at z=-1. I want to expand this around z=1. Hence it will have Laurent series in the annulus between z=1 and z=-1, but there is also region outside z=-1 that goes to infinity. So there are two regions.

When I do

with(numapprox);
laurent((3*z+1)/(z^2-1),z=1);

Maple gives the correct Laurent series for the region in the annulus |Z-1|<2, i.e. the first region (the one inside the two singularities).

But I want to see the expansion for different region, for |z-1|>2 (to check if I did it ok). This is what I get btw

3/(z-1) - 2/(z-1)^2 + 4/(z-1)^3 - 8/(z-1)^4 +.... 

Do I need a special package for this?

Fyi, I found this question here but it does not really answer my question. I want to speficy both the point of expansion, and also the region itself.