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If you cast a card "without paying its mana cost," such as with suspend, you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those if you want to cast the card.

Due to a recent rules change to suspend, you are no longer required to cast the suspended card as the second triggered ability of suspend resolves. Instead, as the second triggered ability resolves, you may cast the card. Timing permissions based on the card's type are ignored. If you don't cast the card, it remains exiled with no time counters on it, and it's no longer suspended.

If the first triggered ability of suspend (the one that removes time counters) is countered, no time counter is removed. The ability will trigger again at the beginning of the card's owner's next upkeep.