Stolen car recovered, what are my options ?

Hey everyone, my car in was recently stolen and recovered, what are my options in term of dealing with insurance ?

I would love a write off but dont think that will happen.

I'm more concerned with the value of my vehicale going down to due to it being stolen and having it on the Carfax in the future,

Also more worried about the hidden damages a regular inspection wont find..

Has anyone been in this situation before ? Whats the best way to approach this ?

Thank you!