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OP:

OK, you got that far.

Here's the next thing to try:

Boot to internet recovery AGAIN.

THIS TIME, when you get to the utilities, open the OS installer.
You want to re-install the OS.
DOING THIS WILL NOT HARM DATA THAT IS ON THE DRIVE.
It will just re-install the OS.

Give this a try. It may work, or it may not.
But you have to at least try it.

Be aware that the Mac will restart one or more times, and the screen may go dark for a minute or two with no other indication of activity. Just be patient.

Give this a try and get back to us.
Again, it may or may not work.
There might be something wrong with the drive that prevents the install.

If it doesn't work, there remain more things to try.
But get going on this.
 
Well, the machine certainly is capable of running.

If you try to start it normally, without the command-option-R, do you get a startup chime but a black screen? If so, it might possibly be a dead CMOS (PRAM) battery. Said battery is a small CR2032 hearing-aid style battery. Unfortunately I imagine that you have to take the screen off to get at it, which IIRC isn't entirely trivial on that model.
 
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