TSTST 2018 Problem 7

Let $n$ be a positive integer. A frog starts on the number line at 0 . Suppose it makes a finite sequence of hops, subject to two conditions: - The frog visits only points in $\left\{1,2, \ldots, 2^{n}-1\right\}$, each at most once. - The length of each hop is in $\left\{2^{0}, 2^{1}, 2^{2}, \ldots\right\}$. (The hops may be either direction, left or right.) Let $S$ be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of $S$ ?