
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 $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively, and $\frac{b}{\cos B}+\frac{c}{\cos C}=\frac{a}{\cos A}+\frac{3a}{\cos B\cos C}$.  
$(1)$ Find $\tan B \tan C$;  
$(2)$ Find the maximum value of $\tan A$.

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