Let $S=k[x,y,z,w]$ be the polynomial ring over a field $k$ and $P=(x,y,z,w)S$. We set $R=S_{P}/[(x,y)\cap (z,w)]S_{P}$. Prove that $dim(R)=2$ but $depth(R)=1$.

I try to solve this exercise by myself but I truly get stuck. I think it starts from this point: $$0\to R/(I\cap J)\to R/I \times R/J \to R/(I+J)$$

Replace $R=S_{P}$, $I=(x,y)S_{P}$, $J=(z,w)S_{P}$. However, that is all I have.