Lin Jumper
Lin is jumping on a segment $[0, 2^{10}-1]$ of the real line. It starts at $ 0 $ and visits only the whole numbers of the segment, and visits each of them no more than once. It turned out that the lengths of all his jumps are powers of two ($\{ 2^0, 2^1, 2^2, ...\}$). What is the maximum possible total distance traveled by Lin?
P.S. Lin can jump both to the right and left.
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