Extremes of a Partial-Sum Sequence

A geometric sequence $\{a_n\}$ with first term $\dfrac{3}{2}$ is not non-increasing, and its partial sums $S_n$ satisfy: $S_3 + a_3$, $S_5 + a_5$, $S_4 + a_4$ form an arithmetic sequence.

1. Find a closed formula for $a_n$. 2. Let $T_n = S_n - \dfrac{1}{S_n}$ for $n \in \mathbb{N}^*$. Find the largest and smallest terms of the sequence $\{T_n\}$.