What is the maximum volume of a rectangular box with a square base and no top if its total surface area is $507$ square inches?

$($For a diagram related to this problem refer to the pdf document in course content labeled: Calculus I Final Exam diagrams$).$

None of the given answers.

$256$ cubic inches

$1098\mathrm{.}5$ cubic inches

$665\mathrm{.}5$ cubic inches

$32$ cubic inches

$62\mathrm{.}5$ cubic inches

$600$ cubic inches

$($H$)216$ cubic inches

$171\mathrm{.}5$ cubic inches

$364\mathrm{.}5$ cubic inches

$($K$)864$ cubic inches