It’s impossible to draw that image without round-tripping a similar path region at least four times if you intend to draw this without lifting your drawing device from the surface. I believe that this may be your issue in Illustrator – and hence AE.
Your best shot if to copy paste each mask’s path into a position property. Each time you paste into the position prop, ensure that you remove duplicate first/last position points. Then check for validity of the path before pasting each subsequent path.
Once you’ve copied all the paths, apply Roving Keyframes and adjust the duration accordingly.
Before starting out, you will still have to decide how the overlapped paths will be drawn.
Cheers
RoRK
Intensive mocha & AE Training in Singapore and Other Dangerous Locations
Imagineer Systems (mocha) Certified Instructor
& Adobe After Effects CS6 ACE/ACI