2014 AMC 12A 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\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.