AMC 12 2020a Problem 22

Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that $$$$ (2 + i)^n = a_n + b_ni $$$$for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is$$\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?$$ $\textbf{(A) }\frac 38 \textbf{(B) }\frac7{16} \textbf{(C) }\frac12 \textbf{(D) }\frac9{16} \textbf{(E) }\frac47$