A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three rectangular sides of the box meet at a corner of the box. The center points of those three rectangular sides are the vertices of a triangle with area $30$ square inches. Find $m + n.$

Let's think step by step and output the final answer within \boxed{}.
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