
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The following conclusions are given:
1. If $A > B > C$, then $\sin A > \sin B > \sin C$;
2. There exist $A$, $B$, and $C$ such that $\tan A\tan B\tan C < \tan A+\tan B+\tan C$;
3. If $\sin ^{2}A+\sin ^{2}B > \sin ^{2}C$, then $\triangle ABC$ is an obtuse triangle;
4. If $\frac{a}{\cos \frac{A}{2}}=\frac{b}{\cos \frac{B}{2}}=\frac{c}{\cos \frac{C}{2}}$, then $\triangle ABC$ is an equilateral triangle.
Find the correct proposition number(s).

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