
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 subset \( S \) of the set \(\{1,2, \ldots, 10\}\) is chosen randomly, with all possible subsets being equally likely. Compute the expected number of positive integers which divide the product of the elements of \( S \). (By convention, the product of the elements of the empty set is 1.)

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