
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
The number $r$ can be expressed as a five-place decimal $0.abcde,$ where $a, b, c, d,$ and $e$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 2 or 3 and whose denominator is an integer. The closest such fraction to $r$ is $\frac{3}{11}.$ What is the number of possible values for $r$?

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