AMC 12 2019a Problem 23

Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a \diamondsuit b = a^{\log_{7}(b)} \text{and} a \heartsuit b = a^{\frac{1}{\log_{7}(b)}}$$for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3 \heartsuit 2$ and $$a_n = (n \heartsuit (n-1)) \,\diamondsuit a_{n-1}$$for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$?

$\textbf{(A) } 8 \textbf{(B) } 9 \textbf{(C) } 10 \textbf{(D) } 11 \textbf{(E) } 12$