
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]


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



- A shakes hands with B, and B shakes hands with C, or
- A doesn't shake hands with B, and B doesn't 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.

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