20
23
21
22
24

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Based on the inequality, the terms are already in order from least to greatest.

Since the median is $n$ , $\frac{(m+10)+(n+1)}{2} = \frac{m+n+11}{2} = n \implies m+11 = n$ .

Since the mean is $n$ , $\frac{m+(m+4)+(m+10)+(n+1)+(n+2)+2n}{6} = \frac{3m+4n+17}{6} = n \implies 3m+17 = 2n$ .

Substituting, we get $3m+17 = 2(m+11) = 2m+22 \implies m=5$ .

$n = 5+11 = 16$ , so our final answer is $5+16=\boxed{21}$ .