AMC 12 2025b Problem 14

Consider a decreasing sequence of $n$ positive integers $x_1>x_2>\cdots>x_n$ that satisfies the following conditions:

• The average of the first $3$ terms in the sequence is $2025.$
• For all $4 \leq k \leq n,$ the average of the first $k$ terms is $1$ less than the average of the first $k - 1$ terms.

What is the greatest possible value of ${n?}$

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