3. (10%) A self-organizing ad-hoc satellite network consists of multiple satellites placed at arbitrary integer coordinates in 3-dimensional space. Each satellite is modeled as a point $A_i \in \mathbb{Z}^3$. A secure communication link may be established between satellites $A_i$ and $A_j$ only if the midpoint of the segment $A_iA_j$ also lies on an integer lattice point. Equivalently, all three coordinate-wise averages must be integers:

The Space Agency wants to guarantee that even without knowing the positions of the satellites beforehand, there must exist at least one secure link. (a) Determine the smallest integer $N$ such that: No matter how the $N$ satellites are placed in $\mathbb{Z}^3$, there is always at least one pair whose midpoint is a lattice point (i.e., at least one secure link always exists). (b) Now modify the secure-link condition: A link is allowed if the total coordinate sum of the two satellites is even, i.e.

Under this new rule, determine the new minimum number of satellites guaranteeing that at least one secure link must exist.