
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
Given a complex number \( z \) that satisfies the equation \( |z - \mathrm{i}| = 1 \) and \( z \neq 0, z \neq 2\mathrm{i} \), and another complex number \( \omega \) such that \( \frac{\omega}{\omega - 2\mathrm{i}} \cdot \frac{z - 2\mathrm{i}}{z} \) is a real number, what is the shape of the set of points corresponding to \( z \) in the complex plane? Provide a reasoning for your answer.

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