
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
On the circle with center \( O \) and radius 1, the point \( A_{0} \) is fixed and points \( A_{1}, A_{2}, \ldots, A_{999}, A_{1000} \) are distributed such that \( \angle A_{0} OA_k = k \) (in radians). Cut the circle at points \( A_{0}, A_{1}, \ldots, A_{1000} \). How many arcs with different lengths are obtained?

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