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Function “Love Triangle”

I've been stumped by this problem:

Find three non-constant, 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?

All help is appreciated!