
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 $z_{1} \neq z_{2}$), the following inequality holds: 
\[
\left(z_{1}+\alpha\right)^{2}+\alpha \overline{z_{1}} \neq \left(z_{2}+\alpha\right)^{2}+\alpha \overline{z_{2}}.
\]

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