Straight From P To Q. Well, Not Really

Consider a quadrilateral $ABCD$ with $AB=16$ , $BC=8$ , $CD=8$ and $\angle ABC = \angle BCD = 90^{\circ}$ . Let $P$ be a point on $AB$ such that $AP=2$ and let $Q$ be a point on $CD$ such that $DQ=3$ .

Find the length of the shortest path which:

• Begins at $P$ , then
• Meets the side $DA$ at a point $W$ , then
• Meets the side $BC$ at a point $X$ , then
• Meets the side $DA$ again at a point $Y$ , then
• Meets the side $AB$ at a point $Z$ , and then
• Ends (finally) at $Q$ .