
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
Find all real-valued functions \( f(x) \) defined on \([0, \infty)\), such that:
1. \( f \) is continuous on \([0, \infty)\),
2. \( f(x) > 0 \) for \( x > 0 \),
3. For all \( x_0 > 0 \), the centroid of the region under the curve \( y = f(x) \) between 0 and \( x_0 \) has y-coordinate equal to the average value of \( f(x) \) on \([0, x_0] \).

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