
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
Let \( R \) be the reals. Find all \( f : K \to R \), where \( K \) is \([0, \infty)\) or a finite interval \([0, a)\), such that

\[ \left( \frac{1}{k} \int_{0}^{k} f(x) \, dx \right)^2 = f(0) \cdot f(k) \]

for all \( k \) in \( K \).

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