
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) Let $x > 0$, then the minimum value of the function $y=x+\frac{4}{x}$ is ______.

(2) Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n=n^2$, then the general formula for the sequence is $a_n=$ ______.

(3) Given $\triangle ABC$ with sides $a$, $b$, and $c$, and area $S =\frac{a^2+b^2-c^2}{4}$, then angle $C =$ ______.

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

① If $\{a_n\}$ is an arithmetic sequence, then the three 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 $S_1$, $S_2$, …, $S_n$, there must be a maximum;

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

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

Among these, the correct propositions are ______.

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