Let \( f:\mathbb{R}\to \mathbb{R} \) and \( g:\mathbb{R}\to \mathbb{R} \) be two functions defined by \( f(x)=\log_{e}(x^{2}+1)-e^{-x}+1 \) and \( g(x)=\dfrac{1-2e^{2x}}{e^{x}} \). Then, for which of the following range of \( \alpha \), the inequality \( f\!\left(g\!\left(\dfrac{(\alpha-1)^{2}}{3}\right)\right)>f\!\left(g\!\left(\alpha-\dfrac{5}{3}\right)\right) \) holds?
