
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 cuboid with edge lengths $a$, $b$, and $c$, and the radius of its circumscribed sphere is $\frac{3}{2}$,  
(I) Find the maximum volume of the cuboid;  
(II) Let $\overrightarrow{m} = (1, 3, \sqrt{6})$, $\overrightarrow{n} = (a, b, c)$, find the maximum value of $\overrightarrow{m} \cdot \overrightarrow{n}$.

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