
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
<span style="color:darkred">Find all differentiable functions  $f\colon [0,\infty)\to [0,\infty)$  for which  $f(0)=0$  and  $f^{\prime}(x^2)=f(x)$  for any  $x\in [0,\infty)$  .</span>

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