Usain is walking for exercise by zigzagging across an $80$-meter by $60$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. If the angle $\theta$ $\angle PAB=\angle QPC=\angle RQB=\cdots$ is $\arccos \left(\frac{2}{3}\right)$, what total length will Usain walk? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown. There could be more or fewer.)