
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 its first $n$ terms denoted as $S\_n$, there are four propositions about the sequence $\{a\_n\}$:
1. If $\{a\_n\}$ is both an arithmetic sequence and a geometric sequence, then $S\_n = n a\_1$;
2. If $S\_n = 2 + (-1)^n$, then $\{a\_n\}$ is a geometric sequence;
3. If $S\_n = a n^2 + b n$ ($a, b \in \mathbb{R}$), then $\{a\_n\}$ is an arithmetic sequence;
4. If $S\_n = p^n$, then regardless of the value of $p$, $\{a\_n\}$ is never a geometric sequence.
Find the correct proposition(s) number(s).

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