
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 $f(x)= \frac { \sqrt {3}}{2}\sin 2x-\cos ^{2}x- \frac {1}{2}$.
$(1)$ Find the minimum value of $f(x)$, and write the set of values of the variable $x$ when the minimum value is attained.
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$, respectively, and $c= \sqrt {3}$, $f(C)=0$. If $\sin B=2\sin A$, find the values of $a$, $b$.

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