
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 $f(x) = \sqrt{3}\cos\frac{\omega x}{2}\sin\frac{\omega x}{2} + \cos^2\frac{\omega x}{2} + \cos\omega x$ ($\omega > 0$) passes through the point $\left(\frac{\pi}{3}, \frac{1}{2}\right)$, and the minimum distance from the center of symmetry of the function $f(x)$ to the axis of symmetry is greater than $\frac{\pi}{10}$.  
(I) Find the analytical expression of $y=f(x)$ and its interval of monotonic increase;  
(II) What transformations of the graph of $y= \sqrt{3}\sin x + \frac{1}{2}$ can result in the graph of $y=f(x)$?

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