If the cycled card has multiple card types, you stop when you exile a card that shares any of those types. You don’t exile card to find one that shares all of its types, and you can’t choose to keep going to find a different one of those types.

If a spell has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If you exile your entire library without exiling a card that shares a card type, you put the library back in a random order and continue. You won’t draw a card.

Artifact, creature, enchantment, instant, planeswalker, and sorcery are card types. Supertypes (such as legendary) and subtypes (such as Giant) are not.

If you cast a spell “without paying its mana cost,” you can’t choose to cast it for any alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, such as that of Swallow Whole, those must be paid to cast the card.

If you wish to cast the last exiled card, you do so as part of the resolution of the cycling ability. You can’t wait to cast it later. Timing permissions based on a card’s type are ignored.