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

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