
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
Determine all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that
\[ f^{a^{2}+b^{2}}(a+b) = a f(a) + b f(b) \quad \text{for every } a, b \in \mathbb{Z}. \]

Here, \( f^{n} \) denotes the \( n^{\text{th}} \) iteration of \( f \), i.e., \( f^{0}(x) = x \) and \( f^{n+1}(x) = f\left(f^{n}(x)\right) \) for all \( n \geqslant 0 \).

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