HMMT 2018 U24

Find the largest positive integer $n$ for which there exist $n$ finite sets $X_{1}, X_{2}, \ldots, X_{n}$ with the property that for every $1 \leq a<b<c \leq n$, the equation

$$\left|X_{a} \cup X_{b} \cup X_{c}\right|=\lceil\sqrt{a b c}\rceil$$

holds.