Find the dimension of the subspace $S \subset \mathbb{R}^4$ consisting of all vectors $x$ satisfying $Ax = 0$ where $A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}$.