
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
An geometric sequence $\{a_n\}$ has a first term of $\frac{3}{2}$ and a common ratio of $-\frac{1}{2}$. The sum of the first $n$ terms is denoted as $S_n$. When $n \in \mathbb{N}^*$, find the ratio of the maximum and minimum values of $S_n - \frac{1}{S_n}$.

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