Specific volume of the coexisting phase at this first-order phase transition

Given the grand partition function
$$mathcal{Z}=(1+z)^V(1+z^{alpha V})$$
where $z=e^{mubeta}$. I’m asked to find the specific volume of the coexisting phase at the first-order phase transition.

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I don’t have any idea How to do that? Besides during the first-order phase transition, the volume suffer finite discontinuity so How come we can tell what’s the specific volume is?

So far I have found that
$$Pbeta=ln (1+z)+frac{z^{alpha V}}{1+z^{alpha V}}alphaln(z)$$
using
$$P=-left.frac{partial Phi_G}{partial V}right|_{T,mu}$$
I can use
$$frac{partial mu}{partial v}=frac{1}{v}frac{partial P}{partial v}$$
To find the specific volume, But How do I put the condition for coexisting phase?