
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 vectors $\overrightarrow{a} = (\cos \frac{3x}{2}, \sin \frac{3x}{2})$, $\overrightarrow{b} = (\cos \frac{x}{2}, -\sin \frac{x}{2})$, $\overrightarrow{c} = (\sqrt{3}, -1)$, where $x \in R$.
(I) When $\overrightarrow{a} \perp \overrightarrow{b}$, find the set of values for $x$;
(II) Find the maximum value of $|\overrightarrow{a} - \overrightarrow{c}|$.

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