AMC 12 2014a Problem 18

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

$\textbf{(A) }19 \textbf{(B) }31 \textbf{(C) }271 \textbf{(D) }319 \textbf{(E) }511$