$ \text{Let } \alpha, \beta \text{ and } \gamma \text{ be three positive real numbers. Let } f(x) = \alpha x^{5} + \beta x^{3} + \gamma x,; x \in \mathbb{R} \text{ and } g : \mathbb{R} \to \mathbb{R} \text{ be such that } g(f(x)) = x \text{ for all } x \in \mathbb{R}. \text{ If } a_{1}, a_{2}, a_{3}, \ldots, a_{n} \text{ be in arithmetic progression with mean zero, then the value of } f!\left(g!\left(\frac{1}{n}\sum_{i=1}^{n} f(a_{i})\right)\right) \text{ is equal to:}$