
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
\[
\text{There are } 30 \text{ different complex numbers } z \text{ such that } z^{30}=1. \text{ For how many of these is } z^5 \text{ a real number?}
\]

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