
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 functions  $f:\mathbb{N}\to\mathbb{N}$  such that for all  $x,y\in\mathbb{N}$ :  $$ 0\le y+f(x)-f^{f(y)}(x)\le1 $$  that here  $$ f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots) $$  

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