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As an American, we've got it similar but expensive. We've got emergency rooms which triage and cost like a thousand dollars or more, then there are urgent cares which usually cost $50 (copay) and tell you whether or not to go to the er (usually you can get in somewhat quickly and occasionally they can even help), then there's your regular doctor and that'll typically be about a month if you're a returning patient though some like mine keep an appointment slot either every day or every week open for same day appointments so you can call first thing after they open in the hopes of being seen.

Oh, and insurance tells you who you get to have as a doctor. And every January the list may suddenly change