Suppose $a$, $b$, and $c$ are three complex numbers with product $1$. Assume that none of $a$, $b$, and $c$ are real or have absolute value $1$. Define
$$
p = (a + b + c) + \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \quad \text{and} \quad q = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
$$
Given that both $p$ and $q$ are real numbers, find all possible values of the ordered pair $(p, q)$.