Daniel's quadruple differences

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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