두다 is putting something down and leaving it, and the -고 names what happened first. The recurring shape here is 두고 가다 / 두고 오다, leaving something behind by accident: [숏폼 찍고 두고 왔답니다...], [깜빡하고 두고 간 우정 목도리]. It also covers deliberately leaving something for later (다음에 써야지 하고 두고 이러면) and keeping something somewhere safely (집에서 안심하고 둘 수 있을 거 같다는 생각). One line is a different 두다 entirely — playing a move in a board game against an AI (바둑 AI 하고 두면).