Pi Day Challenge 2018 - Embryo

Geometry Level 5

Four spheres of diameters 314 , 159 , 265 , 358 are tangent to one another. These four spheres are inscribed in a larger sphere.

If the diameter $x$ of the larger sphere satisfies $ax^2+bx+c=0,$ where $|a|,|b|,$ and $|c|$ are coprime integers, find $|a|+|b|+|c|.$