The fundamental group of the union of three convex open subsets of $ \mathbb{R}^n$.

Question

I have to prove that the fundamental group of the union of three open convex subsets of $\mathbb{R}^n$ is trivial or $\mathbb{Z}$. I can show that it has only one generator, but I can't prove that if the generator is not null-homotopic it is not nilpotent.