Flights are arranged between 13 countries. For $k \ge 2$, the sequence $A_1, A_2, \ldots, A_k$ is said to be a cycle if there exists a flight from $A_1$ to $A_2$, from $A_2$ to $A_3$, $\ldots$, from $A_{k-1}$ to $A_k$, and from $A_k$ to $A_1$. Find the smallest possible number of flights such that, regardless of how the flights are arranged, there exists a cycle.

Let's think step by step and output the final answer within \boxed{}.
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