
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 $\mathrm{z}_{1}$, $\mathrm{z}_{2}$ (where $\left|\mathrm{z}_{1}\right| < 1$, $\left|\mathrm{z}_{2}\right| < 1$, $\mathrm{z}_{1} \neq \mathrm{z}_{2}$), the following inequality holds:
$$(\mathrm{z}_{1} + \alpha)^{2} + \alpha \overline{\mathrm{z}_{1}} \neq (\mathrm{z}_{2} + \alpha)^{2} + \alpha \overline{\mathrm{z}_{2}}.$$

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