
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]


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A sequence \(\{ x_i \}\) is said to have a Cesaro limit if \(\lim_{n \to \infty} \frac{x_1 + x_2 + \cdots + x_n}{n}\) exists. Find all (real-valued) functions \(f\) on the closed interval \([0, 1]\) such that \(\{ f(x_i) \}\) has a Cesaro limit if and only if \(\{ x_i \}\) has a Cesaro limit.

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