
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
Let  $\alpha$  be a fixed real number. Find all functions  $f:\mathbb R \to \mathbb R$  such that   $$ f(f(x + y)f(x - y)) = x^2 + \alpha yf(y) $$  for all  $x,y \in \mathbb R$ .

*Proposed by Walther Janous*

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