Ryan chooses five subsets $S_1, S_2, S_3, S_4, S_5$ of the set $\{1, 2, 3, 4, 5, 6, 7\}$ such that $|S_1| = 1$, $|S_2| = 2$, $|S_3| = 3$, $|S_4| = 4$, and $|S_5| = 5$. Moreover, for all $1 \le i < j \le 5$, either $S_i \cap S_j = S_i$ or $S_i \cap S_j = \emptyset$ (in other words, the intersection of $S_i$ and $S_j$ is either $S_i$ or the empty set). In how many ways can Ryan select the sets?

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