I don't accept liars in my real life (had too many of them) so I ain't gonna have sympathy for an attention seeking kid who was lying and caught out sorry.
Its simple,
Step 1. Accept his apology
Step 2. Leave
That's it you don't need to deal with them. Lairs are everywhere we just have to deal with it and accept the fact. PS this guy is saying sorry so he's doesn't want you to fell bad about it.
NO! You don't have to accept it at all. If someone lies and is caught out they will always try and apologise because they have no other option. You can accept it, doesn't mean I have to.