If any abilities trigger while players are shuffling cards into their library or drawing seven cards, those abilities cease to exist when the turn ends. They won’t be put on the stack.
Day's Undoing
Though other spells and abilities that are exiled won’t get a chance to resolve, they don’t count as being countered.
Any “at the beginning of the next end step” triggered abilities won’t get the chance to trigger that turn because the end step is skipped. Those abilities will trigger at the beginning of the end step of the next turn. The same is true of abilities that trigger at the beginning of other phases or steps.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Jonas De Ro
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