
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
How many non-similar regular 500-pointed stars are there? Consider a regular \( n \)-pointed star to be the union of \( n \) line segments created by jumping over \( m \) points each time (with \( \gcd(m, n) = 1 \)), turning counterclockwise, where \( n = 500 = 2^2 \times 5^3 \).

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