Intel CPU for RX 7800XT

Hi!

I have a Gigabyte Z690 UD DDR4 motherboard and a i5-12600kf.

I recently purchased an RX 7800XT and I was thinking maybe I should upgrade my CPU too. If I’m correct my motherboard support 13th and 14th gen although I know these CPUs has problem if I’m correct.

I would buy it used and the i7-12700k is around 200$ and the i9-12900k is 270$. (I am Hungarian the prices are converted)

My question is what CPU would pair best with my GPU?

Does the 13th 14th problem still stands?

Should I first update my RAM from 16GB to 32GB and see the performance and decide after?