
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 \( n \) be the product of the first 10 primes, and let
\[ S = \sum_{xy \mid n} \varphi(x) \cdot y, \]
where \( \varphi(x) \) denotes the number of positive integers less than or equal to \(x\) that are relatively prime to \( x \), and the sum is taken over ordered pairs \((x, y)\) of positive integers for which \( xy \) divides \( n \). Compute \(\frac{S}{n}\).

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