CMO 1993 Problem 5

Let $y_{1}, y_{2}, y_{3},\ldots$ be a sequence such that $y_{1}=1$ and, for $k>0,$ is defined by the relationship: $$y_{2k}=\begin{cases}2y_{k}& \text{if}~k~ \text{is even} 2y_{k}+1 & \text{if}~k~ \text{is odd}\end{cases}$$$$y_{2k+1}=\begin{cases}2y_{k}& \text{if}~k~ \text{is odd} 2y_{k}+1 & \text{if}~k~ \text{is even}\end{cases}$$Show that the sequence takes on every positive integer value exactly once.