Allowing the user to choose a new font from a list on iOS

I have a text field where I would like the user to be able to set the font from a drop down box. I can't seem to find any documentation related to doing anything like this, but it seems like something that should be fairly trivial so I assume I'm just using poor search terms or something.

I can obviously build the drop down box myself, but I have no idea how to get a list of all available fonts prefrably including my own custom fonts.