AMC 12 2018a Problem 6

For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?

$\textbf{(A) }20 \textbf{(B) }21 \textbf{(C) }22 \textbf{(D) }23 \textbf{(E) }24$