BG-MO 2023 Problem 1

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 $C$ .