
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 to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $b^{2}+c^{2}=3bc\cos A$.  
$(1)$ If $B=C$ and $a=2$, find the area of $\triangle ABC$.  
$(2)$ Find the value of $\frac{\tan A}{\tan B}+\frac{\tan A}{\tan C}$.

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