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Each nonland card in your hand without foretell has foretell. Its foretell cost is equal to its mana cost reduced by {2}. (During your turn, you may pay {2} and exile it from your hand face down. Cast it on a later turn for its foretell cost.)
Whenever chaos ensues, you may cast a foretold card you own from exile without paying its mana cost this turn.
Because exiling a card with foretell from your hand is a special action, you can do so any time you have priority during your turn, including in response to spells and abilities. Once you announce you're taking the action, no other player can respond by trying to remove the card from your hand.
Cards that are foretold by virtue of Singing Towers of Darillium's ability won't continue to have the foretell ability once they leave your hand, but you can still cast them from exile on a later turn for the foretell costs they were given by Singing Towers of Darillium.
Casting a foretold card from exile follows the timing rules for that card. If you foretell an instant card, you can cast it as soon as the next player's turn. In most cases, if you foretell a card that isn't an instant (or doesn't have flash), you'll have to wait until your next turn to cast it.
If you're casting a foretold card from exile for its foretell cost, you can't choose to cast it for any other alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, those must be paid to cast the spell.
The reduction of the foretell cost applies only to generic mana in the foretell cost. It can't reduce requirements of specific colors of mana. For example, if you foretell a card in your hand with mana cost {1}{G}{G}, its foretell cost will be {G}{G}.