
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  $\|x\|_*=(|x|+|x-1|-1)/2$ . Find all  $f:\mathbb{N}\to\mathbb{N}$  such that 
\[f^{(\|f(x)-x\|_*)}(x)=x, \quad\forall x\in\mathbb{N}.\]
Here  $f^{(0)}(x)=x$  and  $f^{(n)}(x)=f(f^{(n-1)}(x))$  for all  $n\in\mathbb{N}$ .

*Proposed by usjl*

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