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reason (something) out
To come to a logical conclusion for, determination about, or solution to something. Well, let's just reason the problem out—there has to be a way to fix this without costing us a fortune. You're wasting your time trying to reason his motivations out. Dan just does things according to his own odd beliefs.
reason something out
to figure something out; to plan a reasonable course of action. Now let's be calm and try to reason this out. Let us reason out our difficulties.
To determine or conclude something by logical thinking: The detective reasoned out the killer's motive. The mechanics were able to reason out the problem from the little I told them. Rather than worrying about the decision, let's sit down and reason it out.