
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
Let $\alpha \in \left(0,\pi \right)$, given vectors $\overrightarrow{a}=({\sqrt{3}\sin\alpha,1})$ and $\overrightarrow{b}=(2,2\cos\alpha)$, and $\overrightarrow{a}⊥\overrightarrow{b}$.  
$(1)$ Find the value of $\sin \alpha$;  
$(2)$ Find the value of $\cos(2\alpha+\frac{7\pi}{12})$.

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