
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
Observe that the fraction  $ \frac{1}{7}\equal{}0,142857$  is a pure periodical decimal with period  $ 6\equal{}7\minus{}1$ ,and in one period one has  $ 142\plus{}857\equal{}999$ .For  $ n\equal{}1,2,\dots$  find a sufficient and necessary
condition that the fraction  $ \frac{1}{2n\plus{}1}$  has the same properties as above and find two such fractions other than  $ \frac{1}{7}$ .

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