
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 $f(x) = \sqrt{3}\sin x\cos x - \cos^2 x - \frac{1}{2}$, where $x \in \mathbb{R}$.

(1) Find the minimum value of the function $f(x)$ and its smallest positive period.

(2) Given a triangle $\triangle ABC$ with the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, satisfying $\sin B - 2\sin A = 0$ and $c = 3$, and $f(C) = 0$, find the values of $a$ and $b$.

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