
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  $G$  be a graph on  $n\geq 6$  vertices and every vertex is of degree at least 3. If  $C_{1}, C_{2}, \dots, C_{k}$  are all the cycles in  $G$ , determine all possible values of  $\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|)$  where  $|C|$  denotes the number of vertices in the cycle  $C$ .

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