
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 \( S \). \( S' \) 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}
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