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 $S(B)>S(C)$.

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.