
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 \( n \) be a positive odd number, and let the complex number \( z \) satisfy \( z^{2^n - 1} = 1 \). Find the value of \( Z_n = \prod_{k=0}^{n-1} \left( z^{2^k} + \frac{1}{z^{2^k}} - 1 \right) \).

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