Box Mystery

Geometry Level 4

There are 6 identical square sheets of paper, each with an area of \(a^2\) for some positive integer \(a\).

Instead of putting them together as a simple cube, these sheets are cut and taped to construct a cuboid of dimensions \(a\times b\times c\) for some distinct positive integers \(a, b, c\).

If the volume of the box equals \(9a^2\), compute \(a+b+c\).

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