For any cardinal $\kappa$ we set $2^{<\kappa} = \big|\{A\subseteq \kappa: |A|<\kappa\}\big|$. Let $\kappa$ be an infinite cardinal. We say $\kappa$ is exact if $\kappa = 2^{<\kappa}$, and we say that $\kappa$ is reachable if there is $\lambda < \kappa$ such that $2^\lambda \geq \kappa$.

Note that $\aleph_0$ is exact, but not reachable. Moreover $2^{\aleph_0}$ is reachable, but exactness is not clear to me - it possibly depends on ${\sf CH}$.

For what combinations of "exact" / "reachable" are there examples of cardinals, and/or consistency results?