For a positive integer $$k$$, let $$N(k)$$ denote the number of ordered quadruples of integers $$(a,b,c,d)$$ such that $$0<d<c<b<a<11$$ and the sum of the six pairwise positive differences between $$a,b,c,d$$ is $$k$$.

Find the sum of all values of $$k,$$ such that $$N(k)$$ is the largest possible.

This problem is posed by Daniel C.

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