Hello CF community!

Recently I was solving some problems (usually on AtCoder) where you are given a starting state and some operations you are allowed to make and the question is in how many different states can you end up. I often have trouble with does kind of questions because I am not always sure whether the order of operations matters and what operations matter. What is your usual approach when you see such kind of problem and how do you find out the properties of the operations and the structure you are doing it on so quickly?