
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 the maximum value of

$$
\int_{0}^{1}\left|f^{\prime}(x)\right|^{2}|f(x)| \frac{1}{\sqrt{x}} d x
$$

over all continuously differentiable functions \( f: [0,1] \rightarrow \mathbb{R} \) with \( f(0) = 0 \) and

$$
\int_{0}^{1}\left|f^{\prime}(x)\right|^{2} d x \leq 1
$$

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