
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
For the sequence $\{a_n\}$ with the first term $a_1$ and the general term $a_n$, and the sum of the first $n$ terms denoted as $S_n$, among the following statements:  
① If $S_n = n^2 + n$, then $\{a_n\}$ is an arithmetic sequence;  
② If $S_n = 2^n - 1$, then $\{a_n\}$ is a geometric sequence;  
③ If $2a_n = a_{n+1} + a_{n-1}$ (for $n \geq 2$), then $\{a_n\}$ is an arithmetic sequence;  
④ If $a_n^2 = a_{n+1} \cdot a_{n-1}$ (for $n \geq 2$), then $\{a_n\}$ is a geometric sequence;  
The correct statement(s) is/are.

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