As people have been discussing in the comments, there's a pretty good chance "blow the shit out of North Korea and doom everyone in Seoul to a horrific death" will be the way out of this. The media will play along, and very soberly talk about the War President who is Protecting Us. We have seen this movie before.
I mention Seoul not because I prioritize South Korean lives over North Korean ones. Blowing the shit out of any of them is bad, even if the North Korean state is bad (and it is very very bad, but that is not the fault of its people. Perhaps we are a bit to blame for blowing the shit out of it 60 years ago? maybe a little bit?). But South Korea is our "ally" (this is a dumb word but it is the word we use) and we are, in theory, footing the bill for a large military presence in order for them to not have the shit blown out of them (I am not naive, but it is occasionally worth pointing out why are are claiming to do the things we do even if those claims are bullshit).
I suppose I do care a bit about lives in South Korea more than other places because I have extended inlaw family there. While I don't know most of them, kinship, even obtained kinship, does have an effect on these things.