
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 the function $f(x)=2\cos x(\cos x+\sqrt{3}\sin x)$, <br/>$(1)$ Find the extreme values of the function $f(x)$ on the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$; <br/>$(2)$ In triangle $ABC$, if $f(C)=2$, $2\sin B=\cos (A-C)-\cos (A+C)$, find the value of $\tan A$.

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