
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
Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$ ($n\in\mathbb{N}^*$). There are four propositions about the sequence $\{a_n\}$:  
① If $a_{n+1}=a_n$ ($n\in\mathbb{N}^*$), then $\{a_n\}$ is both an arithmetic sequence and a geometric sequence;  
② If $S_n=a_n^2+b_n$ ($a, b\in\mathbb{R}$), then $\{a_n\}$ is an arithmetic sequence;  
③ If $S_n=1-(-1)^n$, then $\{a_n\}$ is a geometric sequence;  
④ If $\{a_n\}$ is an arithmetic sequence, then $S_n$, $S_{2n}-S_n$, $S_{3n}-S_{2n}$ ($n\in\mathbb{N}^*$) also form an arithmetic sequence;  
Among these propositions, the correct ones are \_\_\_\_\_\_ (fill in the correct numbers).

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