A permutation \pi:\{1,2, \ldots, N\} \rightarrow\{1,2, \ldots, N\} is very odd if the smallest positive integer k such that \pi^{k}(a)=a for all 1 \leq a \leq N is odd, where \pi^{k} denotes \pi composed with itself k times. Let X_{0}=1, and for i \geq 1, let X_{i} be the fraction of all permutations of \{1,2, \ldots, i\} that are very odd. Let \mathcal{S} denote the set of all ordered 4-tuples (A, B, C, D) of nonnegative integers such that A+B+C+D=2023. Find the last three digits of the integer
2023 \sum_{(A, B, C, D) \in \mathcal{S}} X_{A} X_{B} X_{C} X_{D}