
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
Let the function $f(x)=4\cos^2 \pi x-4\sqrt{3}\sin \pi x\cdot\cos \pi x$ have a minimum positive period of $\pi (\pi > 0)$.  
$(1)$ Find the value of $\pi$;  
$(2)$ If the domain of $f(x)$ is $\left[-\frac{\pi}{3}, \frac{\pi}{6}\right]$, find the maximum and minimum values of $f(x)$ and the corresponding values of $x$.

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