SIMON-MARAIS 2021 A1

Let $a, b, c$ be real numbers such that $a \neq 0$ . Consider the parabola with equation $$y = ax^2 + bx + c,$$ and the lines defined by the six equations \begin{align*} &y = ax + b, & y = bx + c, & y = cx + a, &y = bx + a, & y = cx + b, & y = ax + c. \end{align*} Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$ .