
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 \( \mathbb{Z} \) be the set of integers. Determine all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that, for all integers \( a \) and \( b \),
$$
f(2a) + 2f(b) = f(f(a+b)).
$$

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