Step 1: The given vector equation \( \mathbf{a} \times \mathbf{c} = \mathbf{a} \times \mathbf{b} \) implies that \( \mathbf{c} \) lies in the plane formed by \( \mathbf{a} \) and \( \mathbf{b} \). We will use this condition to express \( \mathbf{c} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).
Step 2: The dot product equation \( (\mathbf{a} + \mathbf{c}) \cdot (\mathbf{b} + \mathbf{c}) = 168 \) provides a second condition to find \( \mathbf{c} \). Expand the dot product and solve for \( | \mathbf{c} |^2 \).
Step 3: By solving these equations, we find that the maximum value of \( | \mathbf{c} |^2 \) is 308. Thus, the correct answer is (3).