
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 $d_i = d_{i+10}$ for all positive integers $i$. Determine how many positive integers are in $S$.

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