KöMaL 2023 C1765

The base of a regular four-sided pyramid $ABCDE$ is the square $ABCD$, and each edge of the pyramid is $32$ units long. A snail starts at vertex $E$ and crawls to vertex $A$ as follows: first moves along edge $EA$ to the point $P$ for which $EP=2$. Then continues to cross the face $ABE$ and arrives at point $Q$ of edge $EB$, where $EQ=4$. Then crosses the face $BCE$ to reach that point $R$ of edge $EC$ for which $ER=8$, and continues along face $CDE$ to point $S$ on edge $ED$, with $ES=16$. Finally, it crawls on the surface of face $DAE$, from $S$ to point $A$. What is the minimum distance covered by the snail altogether?