
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  $a\in\mathbb{N}$  such that exists a bijective function  $g :\mathbb{N} \to \mathbb{N}$  and a function  $f:\mathbb{N}\to\mathbb{N}$ , such that for all  $x\in\mathbb{N}$ ,   $$ f(f(f(...f(x)))...)=g(x)+a $$   where  $f$  appears  $2009$  times.

*(tatari/nightmare)*

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