Probability Level 4

For a positive integer $k$, let $N(k)$ denote the number of ordered quadruples of integers $(a,b,c,d)$ such that $0 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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