Eccoci.

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:

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

If $S(A)$ is minimised for a given $n$, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.

• $n = 1$: $\{1\}$
• $n = 2$: $\{1, 2\}$
• $n = 3$: $\{2, 3, 4\}$
• $n = 4$: $\{3, 5, 6, 7\}$
• $n = 5$: $\{6, 9, 11, 12, 13\}$

It seems that for a given optimum set, $A = \{a_1, a_2, \dots, a_n\}$, the next optimum set is of the form $B = \{b, a_1 + b, a_2 + b, \dots, a_n + b\}$, where $b$ is the "middle" element on the previous row.

By applying this "rule" we would expect the optimum set for $n = 6$ to be $A = \{11, 17, 20, 22, 23, 24\}$, with $S(A) = 117$. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for $n = 6$ is $A = \{11, 18, 19, 20, 22, 25\}$, with $S(A) = 115$ and corresponding set string: 111819202225.

Given that $A$ is an optimum special sum set for $n = 7$, find its set string.

NOTE: This problem is related to Problem 105 and Problem 106.