Imagine there are $N$ people throwing a party. For any two of them, the time before they meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\lambda = 1$. After they stick together, the newly formed group acts as an individual, and can merge with other individuals/groups following the same rule.
In other words, the time before any two groups (consisting of 1 or more people) meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\lambda = 1$. Obviously, these $N$ people will (almost surely) form a group of size $N$ in the end.
The question is, what's the average time an individual finds their first group? Here "find their first group" includes (1) form a group with another individual, and (2) join an existing group.
I have tried considering all people except this individuals as an whole, but the later is having internal merges, so it becomes a "compound non-homogeneous Poisson process" (not sure if that's the correct term) which I cannot solve.