
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 $z$ be an imaginary number, and let $\omega = z + \frac{1}{z}$ be a real number, with $-1 < \omega < 2$.
(1) Find the value of $|z|$ and the range of the real part of $z$;
(2) Find the range of values for $|z-2|$.

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