Let $u=\{a_1, \ldots, a_5\}$, where all $a_i \in \{0,1\}$. Let $D(u)$ be the set of all subsequences of $u$ with length 4. Let $A$ be a set of sequences $u_1, \ldots, u_{|A|}$ with the property that $|D(u_i) \cap D(u_j)| = 0$ for all $i \neq j$. What is the maximum value of $|A|$?

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