
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 $\triangle ABC$, if $\sin^2A + \sin^2B < \sin^2C$, then the shape of $\triangle ABC$ is __________.

(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If the area of the triangle $S= \frac{\sqrt{3}}{4}(a^2+b^2-c^2)$, then angle $C=$ __________.

(3) In the sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. Given $a_1=1$ and $a_n=2S_{n-1}$ (for $n \geqslant 2$, $n \in \mathbb{N}^*$), the general formula for this sequence is __________.

(4) Given $S_n$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, and $S_6 > S_7 > S_5$, the correct statements among the following five are __________.

- ① $d < 0$;
- ② $S_{11} > 0$;
- ③ The maximum value of $n$ for which $S_n > 0$ is $12$;
- ④ The maximum term in the sequence $\{S_n\}$ is $S_{12}$;
- ⑤ $|a_6| > |a_7|$.

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