
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 $s$ can be expressed as a five-place decimal $0.abcde,$ where $a, b, c, d, e$ represent digits. It is desired to approximate $s$ by a fraction whose numerator is 1, 2, or 3 and whose denominator is an integer. The closest such fraction to $s$ is $\frac{3}{11}$. Determine how many possible values $s$ can take.

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