You probably already know this to some degree, but I wanted to post about it since these few things became quite apparent to me while solving this problem today: https://codeforces.me/contest/1912/problem/K.

(1) Ideas come through a filter-like process in the brain -- they are generated in the subconscious mind, are passed through a filter, and then the best ones come out. The catch is that sometimes you try to concentrate on a problem so much (shifting everything to the weaker conscious mind) that basically nothing comes out of the filter -- it becomes too strict. Sometimes you need to chill and take your time to start making progress.

(2) Write out your thoughts. This is problem-solving 101, and I'm not suggesting anything new here, but it is often the simplest advice in life we ignore. Writing your thoughts does 2 things:

(2.1) cognitive unload. short-term memory can generally hold 3-4 chunks of info at a time before getting overloaded. writing relevant thoughts down frees up the memory for actually putting things together rather than retaining them, which a paper can do.

(2.2) removes errors in your thought process and logic.

Consider this advice (I wrote this as somewhat of a reminder to myself as well).