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Finding non-constant, distinct $f,g,h : \mathbb{R} \to \mathbb{R}$ such that $f \circ g = h$, $g \circ h = f$, and $h \circ f = g$

I've been stumped by this problem:

Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that $$f\circ g=h$$ $$g\circ h=f$$ $$h\circ f=g$$ or prove that no three such functions exist.

I highly suspect, by now, that no non-trivial triplet of functions satisfying the stated property exists, but I don't know how to prove it.

How do I prove this, or how do I find these functions if they do exist?

BONUS POINTS: The functions should also be continuous.