Z-transform and region of convergence of $x(n)=\left(\frac{1}{3}\right)^{|n|}-\left[\left(\frac{1}{2}\right)^{n} u(n)\right]$

Question

How can we find the Z-transform of $$ x(n)=\left(\frac{1}{3}\right)^{|n|}-\left[\left(\frac{1}{2}\right)^{n} u(n)\right] $$ and it's region of convergence (ROC) ?? where the Z-transform of x(n) is $X(z)=\sum_{n=-\infty}^{\infty} x[n] z^{-n}$ .and u(n)=1, n>=0 and 0 otherwise.

For the second part involving $\left(\frac{1}{2}\right)^{n} u(n)$, we know how to solve it and find its region of convergence by simply transforming it into geometric series. |z|>1/2. but my problem is regarding the first term with the absolute value of n.