
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 $\mathbf{Z}$ represent the set of all integers. Find all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ such that for any integers $a$ and $b$, the following holds:
$$
f(2a) + 2f(b) = f(f(a+b)).
$$

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