내다 is producing something and putting it out where it counts — handing in an answer (안 보고 내겠습니다, I'm submitting without looking), putting forward an idea or an opinion (참지 않고 낸 의견), producing a note with effort in singing, and in 사고를 내다, causing an accident. The -고 clause gives what came first: 각 잡고 낸 (put out after getting properly serious about it), 깊게 생각은 안 하고 낸 아이디어.