
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 positive constant  $C$  such that the following is satisfied: Given  $n$  arcs (containing their endpoints)  $A_1,A_2,\ldots ,A_n$  on the circumference of a circle, where among all sets of three arcs  $(A_i,A_j,A_k)$   $(1\le i< j< k\le n)$ , at least half of them has  $A_i\cap A_j\cap A_k$  nonempty, then there exists  $l>Cn$ , such that we can choose  $l$  arcs among  $A_1,A_2,\ldots ,A_n$ , whose intersection is nonempty. 

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