An interpretation that I read and I think makes a whole lot of sense is that Lucy and Jake are dreamed up by the Janitor. Specifically, Jake is an idealized, younger version of the Janitor, who was a smart and diligent kid (winning those trivia and earning that badge) but socially awkward, while Lucy is the idealized version of a woman who the Janitor saw once at a bar and wishes he'd had the courage to talk to (despite that she was at the bar for an anniversary with the girlfriend). Moreover, Jake and Lucy are there as some sort of representation of the Janitor's internal mental struggle whether or not to end himself ("I'm thinking of ending things.")
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Into the woods, but mind the past... Into the woods, but mind the future!