Given k \geq 1, let p_k denote the k-th smallest prime number. If N is the number of ordered 4-tuples (a, b, c, d) of positive integers satisfying abcd = \prod_{k=1}^{2023} p_k with a < b and c < d, find N(\bmod 1000).
Given k \geq 1, let p_k denote the k-th smallest prime number. If N is the number of ordered 4-tuples (a, b, c, d) of positive integers satisfying abcd = \prod_{k=1}^{2023} p_k with a < b and c < d, find N(\bmod 1000).