How do I get through the security gates?

The gate system here is basically a complicated maze.

You need to collect three black cards to get to the Pre-Approval room.

Notice what happens whenever you walk into a different room here.

You have to give up some of your security cards, but you get back cards of a
different color.

All of the gates leading into the same room function the same way -- the
price for entering a room is the same, regardless of how you get there.