KöMaL 2022 A826

An antelope is a chess piece which moves similarly to the knight: two cells $(x_1, y_1)$ and $(x_2, y_2)$ are joined by an antelope move if and only if $\big\{|x_1-x_2|, |y_1-y_2|\big\} = \{3, 4\}.$ The numbers from $1$ to $10^{12}$ are placed in the cells of a $10^6 \times 10^6$ grid. Let $D$ be the set of all absolute differences of the form $|a-b|$, where $a$ and $b$ are joined by an antelope move in the arrangement. How many arrangements are there such that $D$ contains exactly four elements?