
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
Write down a set \( S \) of positive integers, all greater than 1, such that for each \( x \in S \), \( x \) is a proper divisor of \( \left(\frac{P}{x}\right) + 1 \), where \( P \) is the product of all the elements of \( S \). Your score is \( 2n \), where \( n=|S| \).

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