
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, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?

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