
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 \(T\) be the set of integers \(n > 1\) for which \(\frac{1}{n} = 0.e_1e_2e_3e_4\ldots\), an infinite decimal that has the property that \(e_i = e_{i+10}\) for all positive integers \(i\). Given that \(101\) is prime, how many positive integers are in \(T\)?

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