
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
Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, and $3b\cos A=c\cos A+a\cos C$.  
$(1)$ Find the value of $\tan A$;  
$(2)$ If $a=4\sqrt{2}$, find the maximum area of $\triangle ABC$.

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