
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 \( a, b, c \) be digits from 0 to 9 (repetition allowed). If all recurring decimals of the form \( 0 . \dot{a} \dot{c} \) are converted to their simplest fractional forms, how many different values can the numerator take?

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