A Condition on Double Zeros

Let $f(x)=ax^2+bx+c$ ($a,b,c\in\mathbb R$, $a>0$). Let $p$ be the statement "both $f(x)$ and $f(f(x))$ have exactly two zeros," and $q$ the statement "$f\!\left(f\!\left(-\dfrac b{2a}\right)\right)<0$." Determine the logical relationship: is $p$ a sufficient condition, a necessary condition, both, or neither, for $q$? Justify.