
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 rational numbers \( r \) where \( 0 < r < 1 \), and \( r \) can be expressed as a repeating decimal \( \overline{0.abcabcabc\cdots} = \overline{0.a\dot{b}\dot{c}} \), where \( a, b, \) and \( c \) are not necessarily distinct. How many distinct numerators can elements of \( S \) have when expressed in simplest fractional form?

(10th Annual American Mathematics Invitational, 1992)

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