Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a\diamondsuit b=a^{\log_7(b)}\qquad\text{and}\qquad a\heartsuit b=a^{\frac{1}{\log_7(b)}}$$

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 $n\ge 4$. To the nearest integer, what is $\log_7(a_{2019})$?