
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
For any given \( n \geq 3 \) (\( n \in \mathbb{Z}_{+} \)), randomly select \( n \) points \( A_{1}, A_{2}, \cdots, A_{n} \) on the circumference of a circle \( \odot O \). Among the \( \binom{n}{2} \) angles \( \angle A_{i} O A_{j}(1 \leq i < j \leq n) \), there are \( S(n) \) angles that are no greater than \( 120^\circ \). Find the minimum value of \( S(n) \).

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