Minimal additive energy given upper bound on doubling constant

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For a set $A$ of positive integers, let $D(A)$ be the number of distinct positive integers that can be written as the sum of two (not necessarily distinct) elements of $A$. Also let $E(A)$ be the number of ordered tuples $(a,b,c,d)$ of elements of $A$ such that $a+b=c+d$. For example, if $A=\{1,2,4,5,7,8\}$, then $D(A)=15$ and $E(A)=114$.

Let $S$ be a set of $6$ positive integers such that $D(S)\leq18$. Determine, with proof, the minimum possible value of $E(S)$.

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