The projects headquarters are After fleeing Seattle, The Project sets up its headquarters at a military base somewhere in the Rocky mountains.Mountains (Rockies).
This map shows the Rockies, with the military base highlighted:
It turns out that the base is most likely the Cheyenne Mountain Complex, part of NORAD.
It's from this base that teams Alpha, Beta and Gamma set of on their missions to find pre-collapse technology artefacts:
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