A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) The original answer is in the form \( k\sqrt{m} - n \). Please give the value of \( k + m + n \).

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