
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 circle is divided into  $n$  sectors ( $n \geq 3$ ). Each sector can be filled in with either  $1$  or  $0$ . Choose any sector  $\mathcal{C}$  occupied by  $0$ , change it into a  $1$  and simultaneously change the symbols  $x, y$  in the two sectors adjacent to  $\mathcal{C}$  to their complements  $1-x$ ,  $1-y$ . We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a  $0$  in one sector and  $1$ s elsewhere. For which values of  $n$  can we end this process?

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