
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 $f(x)=\sin \frac {x}{2}+ \sqrt {3}\cos \frac {x}{2}$, where $x\in\mathbb{R}$.  
$(1)$ Find the smallest positive period of the function $f(x)$, and find the intervals of monotonic increase for $f(x)$ on $x\in[-2\pi,2\pi]$.  
$(2)$ Describe the translations and scalings needed to transform the graph of the function $f(x)=\sin x$ ($x\in\mathbb{R}$) into the graph of the given function $f(x)$.

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