Special Subset Sums: Testing

Let $S (A)$ represent the sum of elements in set $A$ of size $n$ . We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$ , the following properties are true:

$S (B) \neq S (C)$ ; that is, sums of subsets cannot be equal.
If $B$ contains more elements than $C$ then $S (B) > S (C)$ .

For example, ${81, 88, 75, 42, 87, 84, 86, 65}$ is not a special sum set because $65 + 87 + 88 = 75 + 81 + 84$ , whereas ${157, 150, 164, 119, 79, 159, 161, 139, 158}$ satisfies both rules for all possible subset pair combinations and $S (A) = 1286$ .

Using sets.txt (right click and "Save Link/Target As..."), a 4K text file with one-hundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, $A_{1}, A_{2}, \dots, A_{k}$ , and find the value of $S (A_{1}) + S (A_{2}) + \dots + S (A_{k})$ .

NOTE: This problem is related to Problem 103 and Problem 106.

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.