Slice and Dice

When you cycle this card, first the cycling ability goes on the stack, then the triggered ability goes on the stack on top of it. The triggered ability will resolve before you draw a card from the cycling ability.

The cycling ability and the triggered ability are separate. If the triggered ability doesn't resolve (because, for example, it has been countered, or all of its targets have become illegal), the cycling ability will still resolve, and you'll draw a card.