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 $C$.
1. **Consider the maximal path \( P \) in the graph \( G \)**:
   - Let \( P \) be a path with vertices \( T_1, T_2, \dots, T_k \).
   - By the maximality of \( P \), \( T_1 \) does not have connections with any vertices outside of \( P \).

2. **Analyze the connections of \( T_1 \)**:
   - Since \( T_1 \) has a degree of at least 3, it must be connected to at least two other vertices in \( P \). Let these vertices be \( T_i \) and \( T_j \) where \( 1 < i < j \leq k \).

3. **Form cycles using \( T_1 \)**:
   - The connections of \( T_1 \) to \( T_i \) and \( T_j \) form cycles in the graph. Specifically, the cycles have lengths \( i \), \( j \), and \( j - i + 2 \).

4. **Determine the gcd of the cycle lengths**:
   - We need to find the greatest common divisor (gcd) of the cycle lengths \( i \), \( j \), and \( j - i + 2 \).
   - If there exists a prime \( p \) such that \( p \mid i \), \( p \mid j \), and \( p \mid (j - i + 2) \), then \( p \) must divide their linear combinations. Specifically, \( p \mid 2 \) because:
     \[
     p \mid (j - i + 2) \implies p \mid (j - i + 2) - (j - i) = 2
     \]
   - Therefore, the only possible values for \( p \) are 1 and 2.

5. **Construct examples to verify the possible gcd values**:
   - For \(\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|) = 1\):
     - Consider a regular octagon where each vertex is connected to its two adjacent vertices and its antipode. This graph has cycles of lengths 8, 4, and 2, and their gcd is 1.
   - For \(\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|) = 2\):
     - Consider a regular hexagon where each vertex is connected to its two adjacent vertices and its antipode. This graph has cycles of lengths 6, 3, and 2, and their gcd is 2.

The final answer is \( \boxed{ \gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|) } \) can be either 1 or 2.