cerbumi

This is what -um- is for. Cerbo is the brain. -Um- is the suffix Zamenhof left deliberately empty: it means do something in the general neighbourhood of this root and refuses to say what. So cerbumi is to brain-about. English needs an idiom for it - rack your brains, chew something over - and Esperanto gets there with one suffix and no metaphor at all. It is one of the handful of -um- words that speakers genuinely love, and it turns up in speech far more than its frequency rank suggests.

Why the vagueness is the point. No regular suffix fits here. Cerbi would be to brain, which means nothing. Cerbigi would be to make something into a brain. -Um- is what the language reaches for when a word obviously belongs to a root and no ordinary affix says how, and the price is that you cannot compute the meaning - you learn it. The -um wildcard lays out the rest of the family: plenumi, brakumi, kolumo, butonumi, and mastrumi at rank 2,139, which is this band's other genuine -um-.

Register and use. It is intransitive and takes pri for what you are puzzling over: cerbumi pri la respondo. There is a faint note of unproductive effort in it - a cerbumanto is somebody thinking hard and getting nowhere, which is why ne cerbumu is friendly advice rather than an insult. Pensi is the neutral verb, mediti is calm and deliberate, and cerbumi is the one with the furrowed brow.