A white screen is indicative of a bad hard drive. In your case it sounds like you have a fusion drive set up so it's harder to diagnose. Since you have a bootable USB device, I would start with it. Disconnect the hard drive and remove the SSD. Now see if you can boot just from the stick. If so, then install the most likely failure candidate, the hard drive and attempt a repair using Disk Utilities from the USB. Even better would be to put the suspect drive in an external enclosure.
'''UPDATE 8/7/17'''
I would really feel better about this if I knew for a certainty that that USB flash drive would completely boot on another Mac. (Then try it on the iMac with both internal drives out. That said, I've done reflows on the GPUs on this machine and the 21.5" with about a 70% success rate. You don't need a high end hot air work station to get the job done. iFixit does not have a guide for the 27" but does for the 21.5" and the are going to be almost identical when it comes to the card. Here is that guide, if you decide to do the flow let me know and I'll recommend an economical hot air gun. [guide|5980]
'''The video card for your 27" machine is Apple Part # iMac & 661-5969-R'''
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The part is not cheap. I'd like to see you try a reflow of the card before buying the card. An inexpensive station is about $100 and after use you could resell it for 3/4 of what you paid for it. http://www.ebay.com/itm/AT8586-Hot-Air-Gun-2IN1-REWORK-STATION-Soldering-Desoldering-Iron-WorkStation-/172134892184?epid=1081670654&hash=item28140a0298:g:7CEAAOSwv9hW6A~~
A white screen is indicative of a bad hard drive. In your case it sounds like you have a fusion drive set up so it's harder to diagnose. Since you have a bootable USB device, I would start with it. Disconnect the hard drive and remove the SSD. Now see if you can boot just from the stick. If so, then install the most likely failure candidate, the hard drive and attempt a repair using Disk Utilities from the USB. Even better would be to put the suspect drive in an external enclosure.
'''UPDATE 8/7/17'''
I would really feel better about this if I knew for a certainty that that USB flash drive would completely boot on another Mac. (Then try it on the iMac with both internal drives out. That said, I've done reflows on the GPUs on this machine and the 21.5" with about a 70% success rate. You don't need a high end hot air work station to get the job done. iFixit does not have a guide for the 27" but does for the 21.5" and the are going to be almost identical when it comes to the card. Here is that guide, if you decide to do the flow let me know and I'll recommend an economical hot air gun. [guide|5980]
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'''The video card for your machine is Apple Part # iMac & 661-5969-R'''
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'''The video card for your 27" machine is Apple Part # iMac & 661-5969-R'''
A white screen is indicative of a bad hard drive. In your case it sounds like you have a fusion drive set up so it's harder to diagnose. Since you have a bootable USB device, I would start with it. Disconnect the hard drive and remove the SSD. Now see if you can boot just from the stick. If so, then install the most likely failure candidate, the hard drive and attempt a repair using Disk Utilities from the USB. Even better would be to put the suspect drive in an external enclosure.
'''UPDATE 8/7/17'''
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I would really feel better about this if I knew for a certainty that that USB flash drive would completely boot on another Mac. (Then try it on the iMac with both internal drives out. That said, I've done reflows on the GPUs on this machine and the 21.5" with about a 70% success rate. You don't need a high end hot air work station to get the job done. iFixit does not have a guide for the 27" but does for the 21.5" and the are going to be almost identical when it comes to the card. Here is that guide, if you decide to do the flow let me know and I'll recommend an economical hot air gun. https://www.ifixit.com/Guide/iMac+Intel+21.5-Inch+EMC+2428+GPU+Card+Replacement/5980
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I would really feel better about this if I knew for a certainty that that USB flash drive would completely boot on another Mac. (Then try it on the iMac with both internal drives out. That said, I've done reflows on the GPUs on this machine and the 21.5" with about a 70% success rate. You don't need a high end hot air work station to get the job done. iFixit does not have a guide for the 27" but does for the 21.5" and the are going to be almost identical when it comes to the card. Here is that guide, if you decide to do the flow let me know and I'll recommend an economical hot air gun. [guide|5980]
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'''The video card for your machine is Apple Part # iMac & 661-5969-R'''
A white screen is indicative of a bad hard drive. In your case it sounds like you have a fusion drive set up so it's harder to diagnose. Since you have a bootable USB device, I would start with it. Disconnect the hard drive and remove the SSD. Now see if you can boot just from the stick. If so, then install the most likely failure candidate, the hard drive and attempt a repair using Disk Utilities from the USB. Even better would be to put the suspect drive in an external enclosure.
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'''UPDATE 8/7/17'''
+
I would really feel better about this if I knew for a certainty that that USB flash drive would completely boot on another Mac. (Then try it on the iMac with both internal drives out. That said, I've done reflows on the GPUs on this machine and the 21.5" with about a 70% success rate. You don't need a high end hot air work station to get the job done. iFixit does not have a guide for the 27" but does for the 21.5" and the are going to be almost identical when it comes to the card. Here is that guide, if you decide to do the flow let me know and I'll recommend an economical hot air gun. https://www.ifixit.com/Guide/iMac+Intel+21.5-Inch+EMC+2428+GPU+Card+Replacement/5980
A white screen is indicative of a bad hard drive. In your case it sounds like you have a fusion drive set up so it's harder to diagnose. Since you have a bootable USB device, I would start with it. Disconnect the hard drive and remove the SSD. Now see if you can boot just from the stick. If so, then install the most likely failure candidate, the hard drive and attempt a repair using Disk Utilities from the USB. Even better would be to put the suspect drive in an external enclosure.