At a party, there are $2011$ people, each with a glass of fruit juice, sitting around a circular table. Every second, they clink glasses according to the following two rules:
1. They do not clink glasses crosswise.
2. At each point in time, everyone can clink glasses with at most one other person.

How many seconds are needed, at a minimum, for everyone to have clinked glasses with everyone else?

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