
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
Find all function  $f:\mathbb{Z}\to\mathbb{Z}$  satisfying $\text{(i)}$   $f(f(m)+n)+2m=f(n)+f(3m)$  for every  $m,n\in\mathbb{Z}$ , $\text{(ii)}$   there exists a  $d\in\mathbb{Z}$  such that  $f(d)-f(0)=2$ , and

  $\text{(iii)}$   $f(1)-f(0)$  is even.

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