
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
Consider a nonzero integer number  $n$  and the function  $f:\mathbb{N}\to \mathbb{N}$  by 
\[ f(x) = \begin{cases}
		\frac{x}{2} & \text{if } x \text{ is even} 
		\frac{x-1}{2} + 2^{n-1} & \text{if } x \text{ is odd}
\end{cases}. 
\] Determine the set: \[ 
	A = \{ x\in \mathbb{N} \mid \underbrace{\left( f\circ f\circ ....\circ f \right)}_{n\ f\text{'s}}\left( x \right)=x \}. 
	\]

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