2023 AIME II 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.