AIME 2023b Problem 10

Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ $$\begin{array}{|c|c|c|c|c|c|} \hline 1 & 3 & 5 & 7 & 9 & 11 \\ \hline 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \end{array}$$