meaning - "Stop working" vs "stopped to work"

I want to know, is there any difference between "stop + v.(ing)" and "stopped to + v.". These are example sentences.

I stop working for a month. vs I stopped to work for a month.
I stop watching movies. vs I stopped to watch movies.
I stop cooking for a year. vs I stopped to cook.

I read “I like to do (be) something” vs “I like doing (being) something” but I still don't understand what is the difference or when and how to use them.

Answer

Both are correct, but they have very different meanings.

I stopped working means I once worked, and now no longer do. I stopped to work means that I once was doing something (unspecified, based on context), and I ceased from doing it so that I could work. The infinitive (to work) here has the meaning of in order to work and so that I could work.

So I stopped cooking means I once cooked, and no longer do, but I stopped to cook means I stopped [doing something] so that I was then able to cook.