
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
By using equations, recurring decimals can be converted into fractions. For example, when converting $0.\overline{3}$ into a fraction, we can let $0.\overline{3} = x$. From $0.\overline{3} = 0.333\ldots$, we know that $10x = 3.333\ldots$. Therefore, $10x = 3 + 0.\overline{3}$. So, $10x = 3 + x$. Solving this equation, we get $x = \frac{1}{3}$, which means $0.\overline{3} = \frac{1}{3}$.

$(1)$ Convert $0.\overline{4}\overline{5}$ into a fraction, fill in the blanks below:

Let $0.\overline{4}\overline{5} = x$. From $0.\overline{4}\overline{5} = 0.4545\ldots$, we have $100x = 45.4545\ldots$.

So, $100x = 45 + 0.\overline{4}\overline{5}$. Therefore, ______. Solving this equation, we get $x = \_\_\_\_\_\_$.

$(2)$ Convert $0.2\overline{4}\overline{5}$ into a fraction.

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