Five people are gathered in a meeting. Some pairs of people shake hands. An ordered triple of people $(A,B,C)$ is called a *trio* if one of the following conditions is true:

1. $A$ shakes hands with $B$, and $B$ shakes hands with $C$, or
2. $A$ does not shake hands with $B$, and $B$ does not shake hands with $C$.

If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.

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