Again With The Tangent Circles?

Consider the following circles: $\Gamma_1$ is centered at $(1,0)$ with radius 1, $\Gamma_2$ is centered at $(-1,0)$ with radius 1, and $\Gamma_3$ is centered at $(0,4)$ with radius 2.

Now consider all circles that are tangent to $\Gamma_1$ , $\Gamma_2$ , and $\Gamma_3$ . Find the sum of the distinct curvatures of these tangent circles.

Details and assumptions

• The curvature of a circle is the reciprocal of its radius.
• We are only summing distinct curvatures, so if there are two circles with the same curvature, only add their curvature once.