
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 $a_1, a_8$ are the two roots of the equation $x^2 - 2x\sin \alpha -\sqrt{3}\sin \alpha =0$, and $(a_1 + a_8)^2 = 2a_3a_6 + 6$, find the value of the acute angle $\alpha$.

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