For certain real constants $ p, q, r$, we are given a system of equations $$\begin{cases} a^2 + b + c = p \\ a + b^2 + c = q \\ a + b + c^2 = r \end{cases}$$What is the maximum number of solutions of real triplets $(a, b, c)$ across all possible $p, q, r$? Give an example of the $p$, $q$, $r$ that achieves this maximum.