Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a} \times \mathbf{b}\| = 2.$  Compute
\[\mathbf{a} \cdot (\mathbf{b} \times (\mathbf{a} \times \mathbf{b})).\]

Let's think step by step and output the final answer within \boxed{}.
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