
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 \(S\) be the set of integers \(n > 1\) for which \(\frac{1}{n} = 0.d_1d_2d_3d_4\ldots\), an infinite decimal that has the property that \(d_i = d_{i+16}\) for all positive integers \(i\). Given that \(65521\) (the largest Fermat prime) is prime, how many positive integers are in \(S\)?

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