AMC 12 2020a Problem 10

There is a unique positive integer $n$ such that$$\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.$$What is the sum of the digits of $n?$

$\textbf{(A) } 4 \textbf{(B) } 7 \textbf{(C) } 8 \textbf{(D) } 11 \textbf{(E) } 13$