
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
The  $ 2^N$  vertices of the  $ N$ -dimensional hypercube  $ \{0,1\}^N$  are labelled with integers from  $ 0$  to  $ 2^N \minus{} 1$ , by, for  $ x \equal{} (x_1,x_2,\ldots ,x_N)\in \{0,1\}^N$ ,
\[v(x) \equal{} \sum_{k \equal{} 1}^{N}x_k2^{k \minus{} 1}.\]
For which values  $ n$ ,  $ 2\leq n \leq 2^n$  can the vertices with labels in the set  $ \{v|0\leq v \leq n \minus{} 1\}$  be connected through a Hamiltonian circuit, using edges of the hypercube only?

*E. Bazavan & C. Talau*

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