CPU Compatibility

Hi Guys, Im pretty new in the pc world. I built my first gaming pc 6 months ago and my cpu has quit working. I want to upgrade anyway but I am not sure on cpu compatibility to my motherboard.

My current specs are Intel Pentium G3258, MSI Z97 PC mate motherboard, and corsair CX600 power supply.

I want to upgrade to an i7 5820k but I dont see it listed as compatible with my motherboard and also wasn't sure if I needed a better power supply.

MSI doesn't list the i7 5820k as a compatible cpu for the Z97. The highest they list is i7 4771. If thats the one I have to get, how much better is the 5820k compared to the 4771? Is it worth getting a new motherboard?

Any help/ explanations on how these things work is much appreciated.
 
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