
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
During the midterm review, Xiaobin had doubts about the definition of rational numbers in the textbook, which states that "integers and fractions are collectively referred to as rational numbers." He asked Teacher Wang, "Finite decimals can be converted into fractions, but can infinite repeating decimals be converted into fractions? Why are they considered rational numbers?" Using the example of the infinite repeating decimal $0.\overline{7}$, Teacher Wang conducted the following verification with Xiaobin: Let $x=0.\overline{7}$, since $0.\overline{7}=0.777\ldots$ with a repeating cycle of $1$ digit, therefore $10\times 0.\overline{7}=10\times 0.777\ldots$, which leads to $10x=7.\overline{7}$, $10x=7+0.\overline{7}$, $10x=7+x$, $10x-x=7$, hence $x=\frac{7}{9}$. Through Teacher Wang's explanation, Xiaobin realized that infinite repeating decimals with a repeating cycle of $1$ digit can be expressed in the form of a fraction. He then raised a new question, "Can infinite repeating decimals with a repeating cycle of $2$ or $3$ digits also be expressed as fractions?"

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