
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
Define $\mathcal{T}$ as the set of integers $n > 1$ such that $\frac{1}{n} = 0.e_1e_2e_3\ldots$ is an infinite decimal with the property that $e_i = e_{i+18}$ for all positive integers $i$. Determine how many positive integers are in $\mathcal{T}$, assuming all prime factors for $10^{18}-1$ are known.

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