
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  $f : \mathbb{N} \to \mathbb{N} $  such that  $f(a) + f(b)$  divides  $2(a + b - 1)$  for all  $a, b \in \mathbb{N}$ .

Remark:  $\mathbb{N} = \{ 1, 2, 3, \ldots \} $  denotes the set of the positive integers.

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