
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 that $\{a_n\}$ is a geometric sequence, where $a_1$ and $a_8$ are the roots of the equation $x^2-2x\sin \alpha-\sqrt{3}\sin \alpha=0$, and $(a_1+a_8)^2=2a_3a_6+6$, calculate the value of acute angle $\alpha$.

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