
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 $\alpha \neq 1$ be a complex number such that $|\alpha^2 - 1| = 3 |\alpha - 1|$, $|\alpha^3 - 1| = 8 |\alpha - 1|$, and the argument of $\alpha$ is $\frac{\pi}{6}$. Find all possible values of $\alpha$.

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