What do you call the first doctor that a patient meets?

Solution 1:

You might be thinking of 'first responder'

a person (as a police officer or an EMT) who is among those responsible for going immediately to the scene of an accident or emergency to provide assistance

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Or you might be thinking of a 'triage team'

triage in Medicine

noun A process for sorting injured people into groups based on their need for or likely benefit from immediate medical treatment. Triage is used on the battlefield, at disaster sites, and in hospital emergency rooms when limited medical resources must be allocated.

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