
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
A student mistakenly calculated the square of the number $0.3$ instead of properly squaring the repeating decimal $0.\overline{3}$. As a result, his calculation was $0.2$ less than the actual value when squaring $0.\overline{3}$. Find the accurate decimal representation of the repeating decimal $0.\overline{3}$ squared.
- **A)** $0.3$
- **B)** $0.09$
- **C)** $0.111$
- **D)** $0.111\overline{1}$

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