Pyramid Selling

Algebra Level 4

You are starting a direct-sale network with inviting \(2\) friends as your first members, who will each find \(2\) new members on their own. However, in order to be honored as new duo members, they must be invited and enrolled by the same person on top. In other words, the top member may not simply invite one person into the group; two newcomers are required for such membership.

For example, as shown in the above left, you (red), as the apex founder, have \(6\) (blue) members as the chain is complete. On the other hand, in the above right, you only have \(4\) members as one of your members "drops off" or fails to find \(2\) new members, as previously stated.

After some time, if you have \(42\) members under your chain network, what is the least possible number of people who "drop off", according to this rule?

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