
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 a sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n\), where
\[ a_1 = 3, \quad S_n = 2a_n + \frac{3}{2}((-1)^n - 1). \]

If there exist three terms \(a_1, a_p, a_q\) (\(p, q \in \mathbb{Z}_+\), \(1 < p < q\)) that form an arithmetic sequence, find \( q - p \).

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