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A maybe simpler way to present the beautiful answer: $2a+3b=-35\cdot 1539\cdot(36N+7)$, $2b+3c=35\cdot 608\cdot(18N+13)$, $2c+3a=35\cdot 72\cdot(18N+13)\cdot(36N+7)$, where I shifted $N$ by $1$ to make the numbers smaller.
@YCor Could one perhaps argue that the intent is "the property (of being) weakly archimedean", rather than "the weak (version of the) archimedean property"?
Replacing $\lfloor n/2 \rfloor$ with $\lfloor n/e \rfloor$ or $\lceil n/e \rceil$ would make the problem much harder to deal with euristically, if not impossible. And replacing instead with $\lfloor n/3 \rfloor$ may make it fun to explore, since all or all orbits should then be finite, euristically.
@TomSolberg : I don't understand this last question. Nowhere is there any mention of right or left hand turns... There are paths with the minimal number $n-2$ of turns containing only left turns, or only right turns, or anything in between.
@LSpice Since you spent time on this thread already, I'd like to ask your opinion: wouldn't it be best if I decluttered it by deleting this and another one of my several non-answers?
What if we only know that all the sections through a fixed point are centrally symmetric - is there a non-degenerate counterexample then? (If we take only the sections through a fixed internal segment, a triangular prism aligned with that segment is an easy counterexample: all the sections are rectangles.) By non-degenerate I mean: other than a flat 2-dimensional convex "solid" with the fixed point outside the plane containing it.
