Inspired by Roberto Nicolaides

Geometry Level 5

How many cubes with edge length 7 are there in $$\mathbb{R}^3$$ such that all the vertices are in $$\mathbb{Z}^3$$ and the origin is one of the vertices?

If you come to the conclusion that there are infinitely many such cubes, enter 666.

Clarifications:

$$\mathbb{R}^3$$ is the set of all (ordered) triples $$(x,y,z)$$ of real numbers and $$\mathbb{Z}^3$$ is the set of all (ordered) triples$$(x,y,z)$$ of integers.

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