
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
Determine all complex numbers \(\alpha\) such that for any complex numbers \(z_1, z_2\) (with \(|z_1|, |z_2| < 1\) and \(z_1 \ne z_2\)), the following holds:
$$
(z_1 + \alpha)^2 + \alpha \overline{z_1} \ne (z_2 + \alpha)^2 + \alpha \overline{z_2}.
$$

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