
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 decimal number $r$ can be represented as a four-place decimal $0.abcd$, where each of $a, b, c, d$ is a digit (possibly zero). Approximating $r$ with a fraction whose numerator is 1 or 2 and whose denominator is an integer, the closest such fraction to $r$ turns out to be $\frac{1}{5}$. Identify how many possible four-place decimal values of $r$ there are for which $\frac{1}{5}$ is the closest approximation.

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