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My current solution (in Common Lisp) solves both of those (hopefully correctly!)
In my code I represented a quaternion as a four element list, and it can encode a 3D rotation around an axis. I wrote the basic functions for working with them almost straight from Wikipedia. There's only really four functions in my solution that use the basic functions: frame-forward, turn, rc-to-cube, and net-direction.
and managed to hack up a solution, even within the tight constraints i was working on this year (embedded device): https://github.com/itizir/advent-of-code-2022/blob/main/day22.c