A 18. Setting up breakout rooms

A Zoom session has N = 2^k people. There are n breakout sessions, each into two equal-sized breakout rooms (i.e., N/2 people in each) The facilitator wishes that every person in the session meets with every other person in a breakout room at least once. Show this can be done if n = k + 1.

Note: This is the simplest version. If you want more of a challenge, you could try the case where N is not a power of 2. Obviously if N is odd, the breakout rooms cannot be exactly equal in size, but there is always an efficient solution where the numbers of people in each breakout room are always within 2 of one another.