Inverse $\mathcal Z$-transform when region of convergence goes outwards from the inner pole? I am looking for the inverse $\mathcal Z$-transform of the following:

$$
\frac{1}{1-\frac 12 z^{-1}}+\frac{1}{1+\frac 13 z^{-1}}
$$

When the region of convergence is $z &gt; 1/3$. I have found the $\mathcal Z$-transform for when $z &gt; 1/2$ as a right sided signal:

$$
\left(\frac{1}{2}\right)^n u[n] + \left(-\frac{1}{3}\right)^n u[n]
$$

I can't seem to find a inverse $\mathcal Z$-transform when the ROC goes outwards from the inner pole. Does the inverse $\mathcal Z$-transform not exist for $z&gt;1/3$?
There exist $3$ sequences with the given expression as their $\mathcal{Z}$-transform. These $3$ sequences correspond to $3$ different regions of convergence (ROCs):


$|z|&gt;\frac12$: right-sided
$\frac13&lt;|z|&lt;\frac12$: two-sided
$|z|&lt;\frac13$: left-sided


It's important that you figure out both right-sided and left-sided inverse $\mathcal{Z}$-transforms of the basic transform

$$X(z)=\frac{1}{1+az^{-1}}$$

and their respective ROCs. Then it's easy to find all $3$ inverse transforms for the given example.