Two neighboring towns, MWMTown and NIMOTown, have a strange relationship with regard to weather. On a certain day, the probability that it is sunny in either town is $\frac{1}{23}$ greater than the probability of MWMTown being sunny, and the probability that it is sunny in MWMTown, given that it is sunny in NIMOTown, is $\frac{12}{23}$. [Find the probability that it is sunny in NIMOTown.]
Let the probability that it is sunny in MWMTown be $m$ and in NIMOTown be $n$. Then, the probability that MWMTown or NIMOTown is sunny is $m+n-m\cap n=\frac{1}{23}+m\implies n=\frac{1}{23}+p$, for $p=m\cap n$.
We're also given that $m | n = \frac{p}{n}=\frac{12}{23}\implies n=\frac{12}{23}p=\frac{1}{23}+p$, but this is obviously impossible.
I'm considering that I'm wrong about $m$ and $n$ being dependent, but it seems like they are because of the first condition; otherwise, I think my reasoning with the equations is fine. What did I do wrong?