
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
If  $S$  is a sequence of positive integers let  $p(S)$  be the product of the members of  $S$ . Let  $m(S)$  be the arithmetic mean of  $p(T)$  for all non-empty subsets  $T$  of  $S$ . Suppose that  $S'$  is formed from  $S$  by appending an additional positive integer. If  $m(S) = 13$  and  $m(S') = 49$ , find  $S'$ .

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