The meaning of "connect" in this question is:
If you have four cities, "connecting" them means finding two pairs of cities, whose geodesics intersect, and finding the intersection point. To put it another way, we use the cities to make an X shape, and then find the center of the X.
If we use this 'connection' process for each one of the columns, we get:
So now, we get four cities. The obvious next step is to use the connection process again. If we do this , we have
"You'll have to do the same 5 times"
We use this connection process 5 times.
"If you find the city, you will be able to check if it's indeed the one we are looking for"
If we take the initials of every one of the cities obtained from our first four connection processes, we get
Great thanks to @Stiv for helping me with the idea of connections, and the OP for suggesting round lines.