2015 Cayley Problem 21

The average of a list of three consecutive odd integers is 7. When a fourth positive integer, m, different from the first three, is included in the list, the average of the list is an integer. What is the sum of the three smallest possible values of m?

\textbf{(A)}\ 6\quad \textbf{(B)}\ 9\quad \textbf{(C)}\ 21\quad \textbf{(D)}\ 29\quad \textbf{(E)}\ 33

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.