Arranging numbers in a 3X9 grid

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In how many different ways can you arrange the numbers 1-9 in a 3X9 grid such that each number appears exactly once in each row: and each number appears exactly once in each of the left, middle, and right 3X3 grids? Here is the grid along with one possible arrangement:

2 7 1 3 5 9 6 4 8

4 3 8 6 7 2 1 5 9

5 6 9 1 4 8 2 3 7

Here is what I am thinking: There are 9! ways to arrange the first row; then for the second row, first three columns we have 6 choose 3 ways to choose the numbers times 3! ways to arrange them. Now for the second row column four we have two cases:

If the number appearing in column four row one was used in row 2 columns one through three, we have 6 choices for row 2 column 4, otherwise we have 5 choices. If the number appearing in row one column 5 appeared in the first three columns of row 2, then we have 5 choices for row 2 column 5, otherwise we have 4 choices. The same for column six, we have either 4 choices or 3. For the final three columns we have 3! ways to arrange the final three numbers. The final row should have 3*3! ways to arrange the numbers.

This seems a little overly complicated. Can anyone critique my logic and maybe come up with a better approach?