
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 the distance from $\alpha^2$ to 1 is three times the distance from $\alpha$ to 1, while the distance from $\alpha^4$ to 1 is five times the distance from $\alpha$ to 1. Additionally, $\cos(\text{arg}(\alpha)) = \frac{1}{2}$. Determine all possible values of $\alpha$.

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