
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 \( A_1, A_2, \dots, A_n \) chosen on the circumference of a circle \( O \), among the \(\binom{n}{2}\) angles \(\angle A_i O A_j \) (where \( 1 \leq i < j \leq n \)), there are at least 2007 angles that do not exceed \( 120^\circ \).

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