
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
Given a real number $\alpha$, determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ depending on $\alpha$ such that

$$
f(f(x+y) f(x-y))=x^{2}+\alpha y f(y)
$$

for all $x, y \in \mathbb{R}$.

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