USAMO 2016 — Days 1 and 2 Question 1

Let $X_1$, $X_2$, …, $X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i \cap X_{i+1} = \emptyset$ and $X_i \cup X_{i+1} \neq S$, for all $i \in \{1, …, 99\}$. Find the smallest possible number of elements in $S$.

Proposed by Iurie Boreico (credited by Evan Chen).