
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 four-place decimal $0.abcd$, where $a, b, c,$ and $d$ represent digits. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer less than 10. The closest such fraction to $r$ is found to be $\frac 27$. Determine the number of possible values for $r$ given that the competing fractions are now $\frac{1}{5}$ and $\frac{1}{6}$.

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