Has my iMac a faulty GPU?

Hey out there,

I recently bought a 27" iMac for about 330€. Works quite well but unfortunately I had some troubles with installing Sierra on it. I had to restart the computer several times to get to the installer - the screen stayed black or white and sometimes it even get blue with black lines.

As I finally managed to get Sierra on the hard drive the computer worked well. But after several minutes/hours the computer restarts randomly.

After some research I found out that this computer model (i5 3.1GHz, serial number) had kind of a recall by apple. As we all know this recall expired.

Are my described symptoms a signal for a faulty GPU? Beside this can anyone tell me what exactly was the problem with this GPU?

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Furthermore, can I replace the Radeon 6970 only with original apple parts or can I achieve, somehow, to get a good replacement with an Alienware 6970 card?

Thanks!

Well, in the mean time, the computer decided to fail every single boot attempt with showing a blank screen.

At least the apple shows up and the bar goes about the half.