
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 differentiable functions  $f: (0,\infty)\mapsto (0,\infty)$  for which there is a positive real number  $a$  such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all  $x>0.$ 

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