Determine the largest positive integer $n$ such that the following statement holds:

If $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$.

Let's think step by step and output the final answer within \boxed{}.
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