
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 $f(x)=\sin^2\left(\frac{x}{2}+\frac{\pi}{12}\right)+\sqrt{3}\sin\left(\frac{x}{2}+\frac{\pi}{12}\right)\cos\left(\frac{x}{2}+\frac{\pi}{12}\right)-\frac{1}{2}$.  
(I) Find the range of $f(x)$;  
(II) If the graph of $f(x)$ ($x > 0$) intersects with the line $y=\frac{1}{2}$, and the x-coordinates of the intersection points from smallest to largest are $x_1$, $x_2…$, $x_n$, find the sum of the first $2n$ terms of the sequence $\{x_n\}$.

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