There are $24$ participants who attended a meeting. Each pair of them either shook hands once or did not. A total of $216$ handshakes occurred during the meeting. For any two participants who have shaken hands, at most $10$ among the remaining $22$ participants have shaken hands with exactly one of these two persons. Define a *friend circle* as a group of $3$ participants where each person has shaken hands with the other two. Find the minimum possible number of friend circles.

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