
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 $f(x)=3\sin\left(\frac{x}{2} + \frac{\pi}{6}\right) + 3$, where $x \in \mathbb{R}$.
- (I) Find the intervals of increase for the function $f(x)$.
- (II) If $x \in \left[\frac{\pi}{3}, \frac{4\pi}{3}\right]$, find the maximum and minimum values of $f(x)$, and specify the values of $x$ for which $f(x)$ attains these extreme values.

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