Consider a triangle $ABC$ with vertices $A(1, 2)$, $B(4, 6)$, and $C(7, 2)$. A permutation of the string "AAABBBCCC" is represented by the coordinates of points in the plane. We can only consider those permutations that transform the coordinates of points such that:
1. None of the first three characters is an $A$.
2. None of the next three characters is a $B$.
3. None of the last three characters is a $C$.

Let $N$ be the number of such valid permutations. Find the remainder when $N$ is divided by 1000. Let's think step by step and output the final answer within \boxed{}.
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