What form of verb is "thank" in "thank you"?

Is the word thank in Thank you! a verb? If not, what part of speech is it then? If it is a verb, is it in the imperative mood? I'm asking because I've seen someone write

Do thank you!

which sounds wrong to me¹. If it were an imperative verb, the above would be grammatically correct. That's why I started wondering what form it is morphologically.

^{1 Is it actually wrong? I understand one could say: I do thank you., but is Do thank you! valid? (I suppose it's meant to emphasize thank you)}

"Thank you." is an abbreviation of "I thank you." Thank is a performative verb — a verb where saying it performs the action itself.