2023 AMC 12A Problem 24

Let K be the number of sequences A_1, A_2, \ldots, A_n such that n is a positive integer less than or equal to 10, each A_i is a subset of \{1,2,3,\ldots,10\}, and A_{i-1} is a subset of A_i for each i between 2 and n, inclusive. For example, \{\}, \{5,7\}, \{2,5,7\}, \{2,5,7\}, \{2,5,6,7,9\} is one such sequence, with n=5. What is the remainder when K is divided by 10?

\textbf{(A)} 1 \qquad \textbf{(B)} 3 \qquad \textbf{(C)} 5 \qquad \textbf{(D)} 7 \qquad \textbf{(E)} 9

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