
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 $f(x)= \sqrt {3}\sin 2x+2\cos ^{2}x-1$, where $x\in\mathbb{R}$.  
$(I)$ Find the smallest positive period and the intervals of monotonic decrease for the function $f(x)$.  
$(II)$ In $\triangle ABC$, the sides opposite to $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $c= \sqrt {3}$, $f(C)=1$, and $\sin B=2\sin A$, find the values of $a$ and $b$.

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