
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 four-place decimal $0.wxyz,$ where $w, x, y,$ and $z$ represent digits, any of which could be zero. It is desired to approximate $s$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $s$ is $\frac 27.$ Given that the nearby fractions with numerator 1 are $\frac 15$ and $\frac 13$, find the number of possible values for $s$.

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