
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 nonzero real number. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$
f(f(x+y))=f(x+y)+f(x) f(y)+\alpha x y
$$

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

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