
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 $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 0$ and for all real numbers $x$ and $y$, establish the following:
\[f(x^2 + y^2) = (x + y)(f(x) + f(y)).\]
Find the possible values for $f(2)$.

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