A circle of radius \(1\) is internally tangent to two circles of radius \(2\) at points \(A\) and \(B\), where \(AB\) is a diameter of the smaller circle. Find the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles. The original answer is in the form \(\frac{k}{m} \pi - n\sqrt{p}\), where \(\frac{k}{m}\) and \(n\sqrt{p}\) are in their simplest forms. Please provide the value of k + m + n + p.

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