Set $ M= \{1,2,\cdots,2550\} $ and $\min A ,\ \max A $ represents the minimum and maximum values of the elements in the set $A$. For $ k \in \{1,2,\cdots 2006\} $define $$ x_k = \frac{1}{2008} \bigg (\sum_{A \subset M : n(A)= k} (\ min A + \max A) \, \bigg) $$. Find remainder from division $\sum_{i=1}^{2006} x_i^2$ with $2551$. (passer-by)