2025 AMC 10B Problem 17

Consider a decreasing sequence of n positive integers x_1 \gt x_2 \gt x_3 \gt \cdots \gt x_n that satisfies the following conditions. The average of the first 3 terms in the sequence is 2025. For all 4 \le k \le n, the average of the first k terms in the sequence is 1 less than the average of the first k - 1 terms in the sequence.

What is the greatest possible value of n?

\textbf{(A)} 1013 \qquad \textbf{(B)} 1014 \qquad \textbf{(C)} 1016 \qquad \textbf{(D)} 2016 \qquad \textbf{(E)} 2025

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