GPS runs off broadcasted signal from precisely positioned satellite. I'm curious, how were these satellite positioned to begin with and what keeps their positions correctly synched?
They weren't precisely positioned. They are not in synchronous orbits, so from the ground they appear to be moving all the time.
The orbits, however, are known exactly.
There are ground stations that track the satellites. The orbits are corrected if needed, but most often they just update the orbit data so that the receivers can calculate the current positions of the satellites correctly.
I recommend you take a look at the Wikipedia GPS page. It is simplified, but still more detailed than what I can write here as an answer.