Sword of Once and Future

When you surveil, you may put all the cards you look at back on top of your library, you may put all of those cards into your graveyard, or you may put some of those cards on top and the rest of them into your graveyard.

You choose whether or not to cast the instant or sorcery card as the triggered ability resolves. If you do, you do so as part of the resolution of that ability. You can't wait to cast it later in the turn. Timing restrictions based on the card's type are ignored.

If you cast a card "without paying its mana cost," you can't pay any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, those must be paid to cast the card.