In trapezoid \(ABCD\), sides \(AB\) and \(CD\) are parallel, and the lengths of sides \(AD\), \(AB\), and \(BC\) are equal.
If the perpendicular distance between \(AB\) and \(CD\) is \(8\) units, and the length of \(CD\) is equal to \(4\) more than half the sum of the other three side lengths, determine the area and perimeter of trapezoid \(ABCD.\)