
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
Find the smallest positive integer $n$ that satisfies the following condition: For any $n$ points $A_{1}, A_{2}, \cdots, A_{n}$ selected on the circumference of a circle $\odot O$, there are at least 2007 angles $\angle A_{i} O A_{j}$ (where $1 \leq i < j \leq n$) that are no greater than $120^{\circ}$.

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