
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  $$ \int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx $$  over all continuously differentiable functions  $f:[0,1]\to\mathbb R$  with  $f(0)=0$  and  $$ \int^1_0|f'(x)|^2dx\le1. $$  

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