
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  $\mathbb{N}$  denote the set of positive integers. For how many positive integers  $k\le 2018$  do there exist a function  $f: \mathbb{N}\to \mathbb{N}$  such that  $f(f(n))=2n$  for all  $n\in \mathbb{N}$  and  $f(k)=2018$ ?

[i]Proposed by James Lin

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