The circular base of a hemisphere of radius \(2\) rests on the base of a square pyramid of height \(6\). The hemisphere is tangent to the other four faces of the pyramid. The original answer is in the form of \(\sqrt{k}\). What is the value of \(k\) if it represents the edge-length squared of the base of the pyramid?

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