Causality and ROC of a stable LTI system So I am looking at a stable LTI system whose input is $x[n]$ and output is $y[n]$. The equation relating the two is here:
$$
y[n-1]-\frac{10}{3}y[n]+y[n+1]=x[n]
$$

I was able to compute its system function $H(z)$ to be:
$$
H(z)=\frac{z^{-1}}{z^{-2}-\frac{10}{3}z^{-1}+1}
$$

I then performed partial fraction expansion to end up with:
$$
H(z)=\frac{\frac 38}{\left(1-3z^{-1}\right)} - \frac{\frac 38}{\left(1-\frac 13z^{-1}\right)}
$$

What I am struggling with is I can't decide what its ROC should be since the system is specified to be stable which means it has to include the $j\omega$-axis. I also need to determine if the system is causal or not. Since a system can only be causal and stable when all its poles are on the left hand plane, I am not sure. 

Any help determining both the ROC and causality of the system would be very helpful.
This looks like homework, so I'll give you a hint to help you figure this out by yourself.

There are 2 poles: one at $z=3$ (i.e., outside the unit circle) and one at $z=\frac13$ (i.e., inside the unit circle). For such a system there are three possible ROCs; one corresponds to an anti-causal system, another one to a causal system, and a third one to a non-causal system with a two-sided impulse response. But only one of those ROCs includes the unit circle, and that's the ROC corresponding to a stable system.