
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 $S_n$ ($n \in \mathbb{N}^*$), and consider the following three propositions about the sequence $\{a_n\}$:  
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;  
Among these propositions, the numbers of the true propositions are.

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