Let \Gamma = \{\varepsilon, 0, 00, \ldots\} be the set of all finite strings consisting of only zeroes. We consider six-state unary \text{DFAs} D = \left(F, q_{0}, \delta\right) where F is a subset of Q = \{1, 2, 3, 4, 5, 6\}, not necessarily strict and possibly empty; q_{0} \in Q is some start state; and \delta: Q \rightarrow Q is the transition function. For each such \text{DFA} D, we associate a set F_{D} \subseteq \Gamma as the set of all strings w \in \Gamma such that
\underbrace{\delta\left(\cdots\left(\delta\left(q_{0}\right)\right) \cdots\right)}_{|w| \text{ applications }} \in F,
We say a set \mathcal{D} of \text{DFAs} is diverse if for all D_{1}, D_{2} \in \mathcal{D} we have F_{D_{1}} \neq F_{D_{2}}. What is the maximum size of a diverse set?