Chat Six

Chat Six is an application that allows you to invite 6 friends to chat in your private chat room online. Your friend then can create his or her new chat room and invite any 6 friends (including you) into it. Therefore, it is possible for you and your online friend to have at most 5 mutual friends in 2 different chat rooms.

As the community grows larger, every member eventually has 1 chat room of their own (each consisting of 7 persons), and if we pick any 2 persons out of this network (friends or not), they will have 2 mutual friends in common.

If \(n\) is a number of people in this entire social network, what is the least possible value of \(n\)?

Assume there is no deletion of any room or member along the way.

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