Let \( A \) and \( B \) be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^{2} = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between \( A \) and \( B \), and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^{2} = 4x \), then the value of \( (a + d) \) is ________.
