Products That Stay Inside

A finite increasing sequence $a_1 < a_2 < \cdots < a_n$ ($n \ge 3$) has the property that for any indices $i < j < k$, at least one of the products $a_ia_j$, $a_ja_k$, $a_ia_k$ is itself a term of the sequence.

1. If $n = 4$ and $a_1 = 1$, $a_2 = 2$, $a_3 = a$, $a_4 = 6$, find $a$. 2. Prove that $2, 3, 5$ cannot all be terms of such a sequence. 3. Find the maximum possible $n$.