
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
For a positive integer \( n \), let \( \varphi(n) \) denote the number of positive integers less than and relatively prime to \( n \). Let \( S_{k} = \sum_{n} \frac{\varphi(n)}{n} \), where \( n \) runs through all positive divisors of \( 42^{k} \). Find the largest positive integer \( k < 1000 \) such that \( S_{k} \) is an integer.

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