
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. Consider the following propositions:  
$(1)$ If $A > B > C$, then $\sin A > \sin B > \sin C$;  
$(2)$ If $\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}$, then $\triangle ABC$ is an equilateral triangle;  
$(3)$ If $\sin 2A = \sin 2B$, then $\triangle ABC$ is an isosceles triangle;  
$(4)$ If $(1+\tan A)(1+\tan B) = 2$, then $\triangle ABC$ is an obtuse triangle;  
$(5)$ There exist $A$, $B$, and $C$ such that $\tan A\tan B\tan C < \tan A + \tan B + \tan C$ holds.  
Among these propositions, the correct ones are __________ (Write down the numbers of all correct propositions).

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