A sequence of numbers is defined recursively by $a_1=2$, $a_2=\frac{4}{9}$, and $a_n=\frac{a_{n-2}\cdot a_{n-1}}{2a_{n-2}-a_{n-1}}$ for all $n\ge 3$. Then $a_{2023}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?