
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
If the function $f(x)=\sqrt{3}\sin(\frac{π}{2}-2x)+2\sin{x}\cos{x}$.  
$(1)$ Find the smallest positive period of the function $f(x)$.  
$(2)$ If the graph of the function $f(x)$ is shifted to the right by $\frac{π}{6}$ units to obtain the graph of the function $g(x)$, when $x∈[-\frac{π}{6},\frac{π}{2}]$, find the range of $g(x)$.

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