Given the $\mathcal Z$-transform of input $x[n]$ and output $y[n]$, how can I find the ROC of the system function $H(z) = Y(z)/X(z)$? I have $$X(z) = \frac{2z\left(z-\frac{10}{3}\right)}{\left(z-\frac 13\right)(z-3)},\quad\text{ROC}: \frac 13 < |z| < 3$$ and $$Y(z) = \frac{-1}{4}\frac{z}{\left(z-\frac 13\right)\left(z-\frac 14\right)},\quad\text{ROC}: |z| > \frac 13$$
If I substitute $\hat{X}(z) = 1/X(z)$, then $ H(z) = Y(z)\hat{X}(z)$. Then, I know that the ROC of $H(z)$ contains the intersection of the ROCs of $Y(z)$ and $\hat{X}(z)$. But how can I find the ROC of $\hat{X}(z)$? I don't know if the corresponding sequence $\hat{x}[n]$ is causal or not.