
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 largest real number \( c \) such that the following conclusion holds for all integers \( n \geq 3 \): Let \( A_{1}, A_{2}, \ldots, A_{n} \) be arcs on a circle (each arc includes its endpoints). If there are at least \( \frac{1}{2} \mathrm{C}_{n}^{3} \) triplets \( (i, j, k) \) such that
\[ 1 \leq i < j < k \leq n \text{ and } A_{i} \cap A_{j} \cap A_{k} \neq \varnothing, \]
then there exists a subset \( I \subset \{1,2, \ldots, n\} \) 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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