
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]


system
A conference is attended by 24 delegates. Each pair of delegates either shakes hands once or does not shake hands at all. By the end of the conference, it is found that there were a total of 216 handshakes. Moreover, for any two delegates \( P \) and \( Q \) who have shaken hands with each other, no more than 10 of the remaining 22 delegates have shaken hands with either \( P \) or \( Q \). A "circle of friends" is defined as a group of three delegates where each pair within the group has shaken hands. Determine the minimum possible number of such circles of friends among these 24 delegates.

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