Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x) = \log _ { \mathrm { e } } ( x ^ { 2 } + 1 ) - e ^ { - x } + 1$ and $g(x) = \frac { 1 - 2 e ^ { 2 x } } { e ^ { x } }$. Then, for which of the following range of $\alpha$, the inequality $f\left( g\left( \frac { ( \alpha - 1 ) ^ { 2 } } { 3 } \right) \right) > f\left( g\left( \alpha - \frac { 5 } { 3 } \right) \right)$ holds?\\
(1) $( - 2 , - 1 )$\\
(2) $(2, 3)$\\
(3) $(1, 2)$\\
(4) $( - 1, 1 )$