Inspired by Roberto Nicolaides

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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Inspiration .