AMC 12 2017a Problem 18

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?

\textbf{(A)}\ 1 \textbf{(B)}\ 3 \textbf{(C)}\ 12 \textbf{(D)}\ 1239 \textbf{(E)}\ 1265