To cogitate.

To think out; to find out or discover by thinking; to devise; to contrive.

tr.v. excogitated, excogitating, excogitates To consider or think (something) out carefully and thoroughly.

reflect deeply on a subject; "I mulled over the events of the afternoon"; "philosophers have speculated on the question of God for thousands of years"; "The scientist must stop to observe and start to excogitate"

come up with (an idea, plan, explanation, theory, or priciple) after a mental effort; "excogitate a way to measure the speed of light"