
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
Let \( S = \{a_1, a_2, \cdots, a_r\} \) be a set of integers where \( r > 1 \). For a non-empty subset \( A \) of \( S \), define \( p(A) \) as the product of all integers in \( A \). Let \( m(S) \) represent the arithmetic mean of all \( p(A) \). Given that \( m(S) = 13 \), and there exists a positive integer \( a_{r+1} \) such that \( m\left(S \cup \{a_{r+1}\}\right) = 49 \), determine the values of \( a_1, a_2, \cdots, a_r \) and \( a_{r+1} \).

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