TSTST 2023 Problem 5

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 $$p=(a+b+c)+\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \text { and } q=\frac{a}{b}+\frac{b}{c}+\frac{c}{a} .$$ Given that both $p$ and $q$ are real numbers, find all possible values of the ordered pair $(p, q)$.