
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 arithmetic-geometric sequence $\{a\_n\}$ has a sum of its first $n$ terms denoted as $S\_n$. It is known that $S\_2$, $S\_4$, and $S\_3$ form an arithmetic sequence.

(1) Find the common ratio $q$ of the sequence $\{a\_n\}$.
(2) If $a\_1 - a\_3 = 3$, determine the number of terms for which the sum is $\frac{21}{8}$.

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