Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear transformation such that \[ T \left( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}, T^2 \left( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, T^2 \left( \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}. \] Then the rank of \( T \) is \(\underline{\hspace{2cm}}\) .