Let \( \alpha, \beta \) and \( \gamma \) be three positive real numbers. Let \( f(x)=\alpha x^5+\beta x^3+\gamma x, \, x\in \mathbb{R} \) and \( g:\mathbb{R}\to\mathbb{R} \) be such that \( g\big(f(x)\big)=x \) for all \( x\in \mathbb{R} \). If \( a_1,a_2,a_3,\ldots,a_n \) be in arithmetic progression with mean zero, then the value of \( f\!\left(g\!\left(\dfrac{1}{n}\displaystyle\sum_{r=1}^{n} f(a_i)\right)\right) \) is equal to:
