There are $36$ participants at a BdMO event. Some of the participants shook hands with each other, but no two participants shook hands more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes had shaken hands with each other. Find the maximum possible number of handshakes at the party, providing proof.

(When two participants shake hands with each other, this is counted as one handshake.)

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