AMC 10 2022b Problem 14

Suppose that $S$ is a subset of $\left\{ 1, 2, 3, \cdots , 25 \right\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S.$ What is the maximum number of elements $S$ may contain? $\textbf{(A)}\ 12 \textbf{(B)}\ 13 \textbf{(C)}\ 14 \textbf{(D)}\ 15 \textbf{(E)}\ 16$