Three mutually tangent spheres of radius \(1\) rest on a horizontal plane. A sphere of radius \(2\) rests on them. Find the distance from the plane to the top of the larger sphere. The answer is in the form \(k + \frac{\sqrt{m}}{n}\), where \(n\) is not divisible by any square number other than \(1\). Please calculate the value of \(k + m + n\).

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