US-TST 2004 Problem 5

Let $A = (0, 0, 0)$ in 3D space. Define the weight of a point as the sum of the absolute values of the coordinates. Call a point a primitive lattice point if all of its coordinates are integers whose gcd is 1. Let square $ABCD$ be an unbalanced primitive integer square if it has integer side length and also, $B$ and $D$ are primitive lattice points with different weights. Prove that there are infinitely many unbalanced primitive integer squares such that the planes containing the squares are not parallel to each other.