
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
(1) In the arithmetic sequence $\{a_n\}$, if $a_2=9$ and $a_5=33$, then the common difference of $\{a_n\}$ is _______.

(2) In $\triangle ABC$, if $a=3$, $b= \sqrt{3}$, and $\angle A= \frac{\pi}{3}$, then the measure of $\angle C$ is _______.

(3) Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=n^2+2n-1$, then the general term $a_n=$ _______.

(4) Given the sequence $\{a_n\} (n\in \mathbb{N}^*)$ with the sum of its first $n$ terms denoted as $S_n$, consider the following four propositions:

① If $\{a_n\}$ is an arithmetic sequence, then the points $(10,\frac{S_{10}}{10})$, $(100,\frac{S_{100}}{100})$, and $(110,\frac{S_{110}}{110})$ are collinear;

② If $\{a_n\}$ is an arithmetic sequence, and $a_1=-11$, $a_3+a_7=-6$, then among the numbers $S_1, S_2, \ldots, S_n$, there must exist a maximum;

③ If $\{a_n\}$ is a geometric sequence, then $S_m$, $S_{2m}-S_m$, $S_{3m}-S_{2m} (m\in \mathbb{N}^*)$ also form a geometric sequence;

④ If $S_{n+1}=a_1+qS_n$ (where the constants $a_1,q\neq 0$), then $\{a_n\}$ is a geometric sequence.

The correct propositions are _______.

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