MATL, 28 bytes
Inspired by Sanchises' answer to a loosely related challenge. I think the approach is also similar to that in Jonah's answer to this challenge.
1JO2Bli:"6Mh]htn:Eq*^YphYsXG The output is rotated 90 degrees with respect to the examples. That is, the first segment is right, not up.
The code plots more steps than necessary. With input 11 it gives F14 as shown in the challenge (rotated):
