
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
(1) Suppose $z$ is a complex number, $\omega = z + \frac{1}{z}$ is a real number, and $-1 < \omega < 2$, find the value of $|z|$ and the range of the real part of $z$;

(2) Calculate the value of $C_{5}^{4} + C_{6}^{4} + C_{7}^{4} + C_{8}^{4} + C_{9}^{4} + C_{10}^{4}$.

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