
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 the following propositions:  
① For the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n = 3n^2 - n + 1$, the sequence is an arithmetic sequence;  
② In a geometric sequence $\{a_n\}$ where all terms are positive and the common ratio $q > 1$, then the geometric sequence $\{a_n\}$ is increasing;  
③ The sum of the first $n$ terms of the geometric sequence $1, a, a^2, a^3, \ldots$ (where $a \neq 0$) is $S_n = \frac{1-a^n}{1-a}$;  
④ For an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms as $S_n$, if $S_9 < 0$ and $S_{10} > 0$, then the sum of the first 5 terms of this sequence is the smallest.  
Among these propositions, the correct ones are __________ (fill in the numbers of all correct propositions).

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