
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  $f:\mathbb{N} \rightarrow \mathbb{N}$  such that for every prime number  $p$  and natural number  $x$ ,  $$ \{ x,f(x),\cdots f^{p-1}(x) \}  $$  is a complete residue system modulo  $p$ . With  $f^{k+1}(x)=f(f^k(x))$  for every natural number  $k$  and  $f^1(x)=f(x)$ .

*Proposed by IndoMathXdZ*

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