Olympiad Geometry · Superchallenge problem

Right triangle $ABC$ has its right angle at $C$, with $AC = 4$ and $BC = 3$. From $C$ a perpendicular is dropped to the hypotenuse $\overline{AB}$; from its foot a perpendicular is dropped to $\overline{AC}$; from that foot a perpendicular is dropped to $\overline{AB}$ again; and so on forever, the path zigzagging toward $A$. Find the total length of all the perpendicular segments.