Let \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) be unit vectors such that \(\|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{c}\| = 1\). The angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\alpha\), and the angle between \(\mathbf{a} \times \mathbf{b}\) and \(\mathbf{c}\) is \(\beta\). If \(\mathbf{a} \cdot \mathbf{c} = \frac{1}{2}\) and \(\mathbf{b} \cdot \mathbf{c} = \frac{\sqrt{3}}{2}\), find the value of \(\cos(\alpha + \beta)\). Let's think step by step and output the final answer within \boxed{}.
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