How to show direct sum of free product of groups is not isomorphic to free product of direct sum of groups?

Question

I guess that $(\mathbb Z \times \mathbb Z) *\mathbb Z$ is not isomorphic to $\mathbb Z \times (\mathbb Z *\mathbb Z)$ since there's no such associative law. What I try: we can write these two groups as generators and relations such that $(\mathbb Z \times \mathbb Z) *\mathbb Z = \{a,b,c:ab = ba \}$ and $\mathbb Z \times (\mathbb Z *\mathbb Z) = \{a,b,c : abc = bca \}$. They all have 3 generators, and they should be quotient groups of $\mathbb Z * \mathbb Z * \mathbb Z$. But I don't know how to proceed. Any hint will be helpful.