
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 $0.defdefdef\ldots=0.\overline{def}$, where the digits $d$, $e$, and $f$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators exist if $d+e+f$ is not divisible by $3$?

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