
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 $f(x)= \sqrt {2}\sin \frac {x}{2}\cos \frac {x}{2}- \sqrt {2}\sin ^{2} \frac {x}{2}$.
(1) Find the smallest positive period of $f(x)$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c= \sqrt {3}$, $f(C)=0$. If vectors $\overrightarrow{m}=(1,\sin A)$ and $\overrightarrow{n}=( \sqrt {2},\sin B)$ are collinear, find $a$ and $b$.

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