
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
Define a regular n-pointed quadratic star as the union of n line segments \( P_1P_2, P_2P_3, \ldots, P_nP_1 \) such that:
- The points \( P_1, P_2, \ldots, P_n \) are coplanar and no three of them are collinear,
- Each of the n line segments intersects at least one of the other segments at a point other than an endpoint,
- All angles at \( P_1, P_2, \ldots, P_n \) are congruent,
- All line segments \( P_2P_3, \ldots, P_nP_1 \) are congruent,
- At each vertex, the path turns counterclockwise at an angle less than 180 degrees,
- The points are joined in a quadratic pattern i.e., starting from a vertex \( P_i \), the next vertex is \( P_{(i^2 \mod n)} \).

How many non-similar regular 1000-pointed quadratic stars are there, assuming no regular 3-pointed, 4-pointed, or 6-pointed quadratic stars exist and that all regular 5-pointed quadratic stars are similar?

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