Let $ \mathbf{A} $ be a square matrix such that $ \det(\mathbf{xI} - \mathbf{A}) = \mathbf{x}^4 (\mathbf{x} - 1)^2 (\mathbf{x} - 2)^3 $, where $ \det(\mathbf{M}) $ denotes the determinant of a square matrix $ \mathbf{M} $.
If $ \text{rank}(\mathbf{A}^2) < \text{rank}(\mathbf{A}^3) = \text{rank}(\mathbf{A}^4) $, then the geometric multiplicity of the eigenvalue $ 0 $ of $ \mathbf{A} $ is __________
