Tangent sphere to a cube

Geometry Level 5

A sphere of radius $$3\sqrt{2}$$ is tangent to the edges $$AB,$$ $$AD,$$ $$AA_1,$$ and the face diagonal $$CD_1$$ of the cube $$ABCDA_1B_1C_1D_1$$.

The volume of the cube can be written as $$a+b\sqrt{c}$$, where $$a,$$ $$b$$ are integers and $$c$$ is a square-free positive integer. What is the value of $$a+b+c$$?

Details and assumptions

The order of the vertices in the cube is shown on the picture below.

Face diagonal simply means the diagonal of a face. It is technically not an edge of the cube, which is why we denoted $$CD_1$$ separately.

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