Diagonalization & Matrix Powers · Hard problem

A $4\times 4$ matrix $A$ has characteristic polynomial $(\lambda - 2)^2(\lambda - 3)^2$, with $\operatorname{rank}(A - 2I) = 2$ and $\operatorname{rank}(A - 3I) = 3$. What is the largest number of linearly independent eigenvectors of $A$?