
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 the complex number \( z \) satisfy the condition \( |z - i| = 1 \), with \( z \neq 0 \) and \( z \neq 2i \). Given that the complex number \( \omega \) is such that \( \frac{\omega - 2i}{\omega} \cdot \frac{z}{z - 2i} \) is a real number, determine the principal value range of the argument of the complex number \( \omega - 2 \).

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