
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]


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Given the function $f(x)=\cos{x}\sin{x}-\sqrt{3}\cos^2{x}+\frac{\sqrt{3}}{2}, x \in \mathbb{R}$.  
$(1)$ Find the smallest positive period of $f(x)$;  
$(2)$ Find the minimum value of $f(x)$ on the closed interval $[-\frac{\pi}{4}, \frac{\pi}{4}]$ and the corresponding value of $x$.

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