
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 $f(x)=4\tan x\sin \left( \frac{\pi}{2}-x\right)\cos \left(x- \frac{\pi}{3}\right)- \sqrt{3}$,

$(1)$ Find the domain and the smallest positive period of $f(x)$;

$(2)$ Discuss the monotonicity of $f(x)$ in the interval $\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]$.

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