
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that the first term of a geometric sequence $\{a\_n\}$ is $\frac{3}{2}$, and the sum of the first $n$ terms is $S\_n$, where $n \in \mathbb{N}^*$. Also, $-2S\_2$, $S\_3$, and $4S\_4$ form an arithmetic sequence.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. For a sequence $\{A\_n\}$, if there exists an interval $M$ such that $A\_i \in M$ for all $i = 1, 2, 3, ...$, then $M$ is called the "range interval" of sequence $\{A\_n\}$. Let $b\_n = S\_n + \frac{1}{S\_n}$, find the minimum length of the "range interval" of sequence $\{b\_n\}$.

Present the answer in LaTex format: \boxed{Your answer}
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