GAIC Math 2024 Problem 44

Let $S$ be a finite set of integers. Suppose that for any two distinct elements $p, q \in S$, there exist three elements $a$, $b$, $c \in S$ (not necessarily distinct, and $a \neq 0$) such that the polynomial $F(x) = \alpha x^2 + bx + c$ satisfies $F(p) = F(q) = 0$. The maximum number of elements in $S$ is \underline{\hspace{2cm}}