Firstly, it's pretty rare to use cogitate at all. Using any word related to it, the most common is "cogitation", the action noun for the act of cogitating.
Second, think has two main senses in English. Most of us native speakers don't even necessarily realise it, but if we learn a language that has separate words for the two, like French, it kind of clicks. In French, there is penser, the active sense of think, where we might say think about, and croire, the stative sense of think, which is closely matched in sense to believe (though clearly with some difference).
Cogitate is a close match to penser, not to croire. It means the act of thinking, pondering, and so on. You might, if you wished to be pretentious, say:
Let me cogitate on that a minute.
You can't say what you want to say. It doesn't make sense.