Problem

The function $f$ is defined by $f(x) = x^2 + 4ax + a$ for $x \in \mathbb{R}$, where $a$ is a constant.

The function $g$ is such that $g^{-1}(x) = \sqrt[3]{2x - 4}$ for $x \in \mathbb{R}$.

(a) Given that the range of $f$ is $f(x) \geq -33$, find the possible values of $a$.

(b) Given instead that $fgg(0) = 96$, find the value of $a$.