Want to upgrade my i5-13600K

Hello guys, I have a rx 6950 XT paired with an i5-13600k. In some games that are very cpu intensive they won't run as high as i want. I play 1080 240hz but want to upgrade my monitor to 360hz and yeah i know some of you might say 6950xt is overkill for 1080p but i just want to get steady 360 fps on competitive games, so i think i should upgrade my cpu. Which would be a good one to pair it to? I also want it to be reasonably worth it because i know upgrading only my cpu a tiny bit doesnt make that much sense, thanks for the suggestions.