Genie and Hats

A bunch of men are on an island.
A genie comes down and gathers everyone together and places a magical hat on some people’s heads (i e , at least one person has a hat).

The hat is magical: it can be seen by other people, but not by the wearer of the hat himself. To remove the hat, those (and only those who have a hat) must dunk themselves underwater at exactly midnight. If there are n people and c hats, how long does it take the men to remove the hats?
The men cannot tell each other (in any way) that they have a hat.

Note: Genie does not tell how many hats she has put.
All the men are perfect logicians and know that each other are perfect logicians

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