
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 $\overrightarrow{a}=(\sqrt{3},1)$, $\overrightarrow{b}=(\cos \alpha,\sin \alpha)$ ($\alpha \in \mathbb{R}$)  
$(I)$ If $\alpha=-\frac{\pi}{6}$, express the vector $\overrightarrow{c}=(2\sqrt{3},0)$ using the basis $\overrightarrow{a}$, $\overrightarrow{b}$;  
$(II)$ If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\alpha$.

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