CMO 2020 Problem 4

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.