
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A set  $S$  of positive integers is  $\textit{sum-complete}$  if there are positive integers  $m$  and  $n$  such that an integer  $a$  is the sum of the elements of some nonempty subset of  $S$  if and only if  $m \le a \le n$ .

Let  $S$  be a sum-complete set such that  $\{1, 3\} \subset S$  and  $|S| = 8$ . Find the greatest possible value of the sum of the elements of  $S$ .

*Proposed by Michael Tang*

Present the answer in LaTex format: \boxed{Your answer}
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