Crowdsourced investigations shore up democratic institutions by debunking misinformation and uncovering human rights abuses. However, current crowdsourcing approaches rely on simplistic collaborative or competitive models and lack technological support, limiting their collective impact. Prior research has shown that blending elements of competition and collaboration can lead to greater performance and creativity, but crowdsourced investigations pose unique analytical and ethical challenges. In this paper, we employed a four-month-long Research through Design process to design and evaluate a novel interaction style called collaborative capture the flag competitions (CoCTFs). We instantiated this interaction style through CoSINT, a platform that enables a trained crowd to work with professional investigators to identify and investigate social media misinformation. Our mixed-methods evaluation showed that CoSINT leverages the complementary strengths of competition and collaboration, allowing a crowd to quickly identify and debunk misinformation. We also highlight tensions between competition versus collaboration and discuss implications for the design of crowdsourced investigations.