
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
For any positive integer \( k \), where \( 1 \leqslant k \leqslant 9 \), does there exist a corresponding quadratic function \( f_{k}(x) \) such that for any positive integer \( p \), we have \( f_{k}(\underbrace{\left(\frac{k k \cdots k}{}\right.}_{p \uparrow}))=\underbrace{\overline{k k \cdots k}}_{2 p \uparrow} \) (for example, \( k=3, p=2, \overline{k k}=33, f_{3}(33)= \) 3333)? If it exists, please provide the proof and the corresponding quadratic function \( f_{k}(x) \); if it does not exist, please provide the reasoning.

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