OLYMPIC-REVENGE 2023 Problem 4

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$ . Is it possible for Gugu to cover all points of $S$ with his spheres?