Best CPU to work with a GTX 1060 6 GB on a Dell Inspiron 3847 computer.

Hello, I am a newbie about computer parts.
I have a Dell Inspiron 3847 and solely looking to try and upgrade this Inspiron 3847 and not build a PC. I was wanting to know what CPU would work well with a ASUS GeForce GTX 1060 6 GB GPU and fit into the Inspiron 3847?

Currently, it has:
88DT1 Socket LGA1150 Motherboard
430W power supply
ASUS GeForce GTX 1060 6 GB GPU
Intel i5-4440 3.1 GHz CPU
2x 4GB 1600 MHz DDR3 SDRAM

What would be some good options for gaming?
I'm also going to upgrade the ram to 16 GB.
Looking to spend $250 but can go to $350.
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More about cpu work gtx 1060 dell inspiron 3847 computer
  1. The cpu should not bottle neck. If you want to upgrade to a i7 4790.
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