This is a broad question about intuition and chess.
I imagine that there is a limit to how far ahead even the best chess masters can see into the future, and how far they can stretch their memory. When this limit is reached during a chess match, how do expert chess players make their decisions? I guess that the answer must be "chess intuition", and I am hoping that the masters among the community can explain to me what chess intuition looks like (I'm imagining some innate feeling for what the right move to make is based on past experiences/general chess principles) and how it factors into the games of chess masters.
To make a comparison with another field-- intuition plays a huge role in doing mathematics. Having a feel for what the right strategy/approach at any given time and being able to update that feel after making mistakes are two crucial skills to have as a mathematician. The Fields medalist Charles Fefferman once famously compared math to playing chess with the devil: the devil is much better at chess than you, but the catch is that you get infinitely many take-backs. These take-backs are how you develop intuition for a problem beyond what you already know.
In chess, of course, there are no take-backs during a match. So, what does intuition look like during a chess match, and what is its role in chess matches?
EDIT: I am asking about a few things at once: the internal experience, when intuition is needed, how much intuition has to be relied upon. Towards this last question, how often (and when) will chess experts make moves essentially “on faith”, in that they don’t know where they will lead or exactly what good they will provide, but are primarily acting on feeling or gut sense? (thanks to @Michael West for the clarifying question)