
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 $f(x)=\sqrt{3}\sin{x}\cos{x}-\frac{1}{2}\cos{2x}-1$, $x\in \mathbb{R}$.  
$(1)$ Find the minimum value and the smallest positive period of the function $f(x)$.  
$(2)$ Given triangle $ABC$ with sides $a$, $b$, $c$, where $c=3$ and $f(C)=0$. If the vectors $\overrightarrow{m}=({1,\sin{A}})$ and $\overrightarrow{n}=({2,\sin{B}})$ are collinear, find the values of $a$ and $b$.

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