If an undocumented immigrant refuses to say where they are from, where do they get deported to?

An undocumented immigrant is caught by the authorities.

They refuse to identity themselves, and they refuse to reveal which country they are originally from.

How do the authorities determine which country the illegal immigrant should be deported to?

Assume the undocumented immigrant entered illegally by foot without detection, did not bring their passport, burned off their fingerprints, and did plastic surgery on their face (so facial matching won't work)