
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 $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\cdots = 0.\dot{a}b\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many different numerators are there when the elements of $s$ are written as reduced fractions?

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