
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:
\[ 
(1) \quad 0<r<1 ;
\]
(2) \( r=0.abcabcabc\cdots=0.ab\dot{c} \), where \( a, b, c \) are not necessarily distinct.

When the numbers \( r \) in \( s \) are written as irreducible fractions, how many different numerators are there?

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