
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  $\tfrac{S}{n}.$ 

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