Let \(\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}\), and \(\mathbf{w} = \begin{pmatrix} 0 \\ 3 \\ -1 \end{pmatrix}\). Given that the vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are linearly independent, find the value of the scalar \(k\) such that:
\[ (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = k \cdot (\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})). \] Let's think step by step and output the final answer within \boxed{}.
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