A matrix $A$ is nilpotent of index $k$ if $A^k = 0$ and $A^{k-1} \neq 0$. For $A = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$, what is the nilpotency index?