
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 all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.abcdabcdabcd\ldots=0.\overline{abcd}$, where the digits $a$, $b$, $c$, and $d$ are not necessarily distinct. To write the elements of $T$ as fractions in lowest terms, how many different numerators are required?

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