
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 \( \frac{1}{n} = 0.d_1d_2d_3d_4\ldots \), an infinite decimal that has the property that \( d_i = d_{i+10} \) for all positive integers \( i \). Assuming \( 9091 \) (a prime number) is in the factorization of \( 10^{10} - 1 \), how many positive integers are in \( T \)?

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