Olympiad Geometry · Superchallenge problem

Angle $A$ measures $12^\circ$. Starting at $A$, a zigzag of segments of equal length is drawn between the two sides of the angle: $P_1$ is on one side with $AP_1 = 1$, then $P_2$ on the other side with $P_1P_2 = 1$, then $P_3$ back on the first side with $P_2P_3 = 1$, and so on, each new point farther from $A$ than the previous point on the same side. What is the largest possible number of segments of length $1$ in the zigzag, counting $\overline{AP_1}$?