2019 AMC 12A Problem 9

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. What is p+q ?

\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.