The best/better part of something = nearly all of something. We waited for the best part of an hour. Converse drank the better part of the rum. A good/large part of something = a lot or more than half of something.
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A large part of the budget will be spent on advertising.
So that a large part of every day is not determined by thought at all, it more or less just happens because of habit.
The greater/major part of something = most of something.
But people tend to drink caffeine on a regular basis over long periods of time - often the greater part of a lifetime.