Given vectors \(\mathbf{u} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}\), and \(\mathbf{w} = \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix}\), verify the triple product identity \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) and solve for the scalar result. Let's think step by step and output the final answer within \boxed{}.
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