Let $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ be unit vectors such that $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 1.$ Additionally, suppose $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are linearly independent. Compute $(\mathbf{u} \times \mathbf{v}) \cdot (\mathbf{v} \times \mathbf{w})$. Let's think step by step and output the final answer within \boxed{}.
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