If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$? 

$ 
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the preceding}
$