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

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