
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 maximum real number \( c \) such that the following conclusion holds for all integers \( n \geq 3 \): Let \( A_{1}, A_{2}, \cdots, A_{n} \) be arcs on a circle (each arc includes its endpoints). If there are at least \( \frac{1}{2} \binom{n}{3} \) triples \( (i, j, k) \) satisfying 
\[ 
1 \leq i < j < k \leq n \text{, and } A_{i} \cap A_{j} \cap A_{k} \neq \varnothing,
\]
then there exists a set \( I \subset \{1, 2, \cdots, n\} \) such that
\[ 
|I| > c n \text{, and } \bigcap_{i \in I} A_{i} \neq \varnothing.
\]

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