
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 sequence $\{a_n\}$ have a sum of the first $n$ terms denoted as $S_n$ ($n\in\mathbb{N}^*$). Regarding the sequence $\{a_n\}$, the following four propositions are given:  
① If $\{a_n\}$ is both an arithmetic sequence and a geometric sequence, then $a_n=a_{n+1}$ ($n\in\mathbb{N}^*$);  
② If $S_n=an^2+bn$ ($a, b\in\mathbb{R}$), then $\{a_n\}$ is an arithmetic sequence;  
③ If $S_n=1-(-1)^n$, then $\{a_n\}$ is a geometric sequence;  
④ If $S_1=1$, $S_2=2$, and $S_{n+1}-3S_n+2S_{n-1}=0$ ($n\geq2$), then the sequence $\{a_n\}$ is a geometric sequence.  
Among these propositions, the true ones are numbered \_\_\_\_\_\_.

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