
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 a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$. It is known that $B$ is an acute angle. The vectors $\overrightarrow{m} = (2\sin B, -\sqrt{3})$ and $\overrightarrow{n} = (\cos 2B, 2\cos^2 \frac{B}{2} - 1)$ are given, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(I) Find the measure of angle $B$ and the range of the radius $R$ of the circumcircle of $\triangle ABC$ when $b \in [\sqrt{3}, 2\sqrt{3}]$.
(II) If $b = 2$, find the maximum value of the area of $\triangle ABC$.

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