
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
A function  $f: \N\rightarrow\N$  is circular if for every  $p\in\N$  there exists  $n\in\N,\ n\leq{p}$  such that  $f^n(p)=p$  ( $f$  composed with itself  $n$  times) The function  $f$  has repulsion degree  $k>0$  if for every  $p\in\N$   $f^i(p)\neq{p}$  for every  $i=1,2,\dots,\lfloor{kp}\rfloor$ . Determine the maximum repulsion degree can have a circular function.**Note:** Here  $\lfloor{x}\rfloor$  is the integer part of  $x$ .

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