
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 numbers  $a$  for which there exists a non-constant function  $f :\Bbb R \to \Bbb R$  satisfying the following two equations for all  $x\in \Bbb R:$ i)  $f(ax) = a^2f(x)$  and
ii)  $f(f(x)) = a f(x).$ 

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