A circle of radius \( 2 \) unit passes through the vertex and the focus of the parabola \( y^{2} = 2x \) and touches the parabola \( y = \left( x - \dfrac{1}{4} \right)^{2} + \alpha \), where \( \alpha > 0 \). Then \( (4\alpha - 8)^{2} \) is equal to _______.
