A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[
a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}
\]
for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q$.

Let's think step by step and output the final answer within \boxed{}.
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