Random Flee transfer, travel time, and the collapsing doorway make a clean key handoff to Ren impossible.
Donut holds two keys but remains fifteen kilometers away. To give Ren one, she must first reach the Capitol, enter a card battle, and use Flee.
The transferred key is random. If Ren receives the church key, the Royal Court teleports away holding the Capitol key but cannot return before its door collapses. Even if Ren gets the correct key, the fleeing Royal Court is displaced and still must save Imani.
Samantha's flight would take too long. Loop-De-Loop requires sacrificing Khulan as a swap target, while removing Ren from the room leaves nobody able to initiate the transfer and strands Imani. Carl concludes every clean route has closed.
Two keys, two doors, and three squads connected by crossed-out routes under a four-minute timer.
The dilemma turns the rules and timer into a forced choice between allied lives without direct combat.