A solid disk of radius \(1\) and uniform density has a hole drilled through it, in the shape of a small circle of radius \(\frac{1}{2}\). This small circle is tangent on the edge of the large circle.

The distance from the geometric center of the large circle to the object's center of mass can be expressed as \(\frac{1}{a}\), where \(a\) is a positive integer. Find \(a\).

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