Dr$.$ Sasso needs the Calculus students to design an open$-$top box with a square base that can hold this year's Attendance Madness prize. The prize is $256$ cubic inches of rice pudding. To minimize the cost of the box, the surface area has to be minimized. What dimensions will minimize the cost of the box? $(S=4xy+x^2$ and $V=x^{2}y)$

Complete all $5$ steps of the optimization process in order to justify your response.