This problem combines concepts from conic sections (parabolas and their tangents) and circles (family of circles, conditions for tangency to axes, and radius calculation). We need to find circles that satisfy two distinct tangency conditions and then sum their diameters.

1. Key Concepts and Strategy

The core idea revolves around two main concepts:

1. Tangent to a Parabola: Finding the slope of the tangent at a given point on the parabola using differentiation.
2. Family of Circles:
   • The general equation of a family of circles touching a line $L=0$ at a specific point $(x_1, y_1)$ is given by $$(x-x_1)^2 + (y-y_1)^2 +