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If an effect increases or decreases the cost of spells you cast, that cost increase or decrease is applied to the alternative cost you chose to pay. In that case, the cost was still paid for the purposes of the effect, even if you paid more or less for it when it was cast.

If you choose to pay one alternative cost, you can't pay any other alternative costs. For example, if an effect lets you cast a "Mastery" spell "without paying its mana cost," you can't also choose to pay its given alternative cost.

If you copy a "Mastery" spell and the alternative cost was paid, the copy will resolve as though the cost was paid.

In a multiplayer game, you choose which opponent takes the prescribed action as the spell resolves.

The mana value of a spell on the stack is determined by its mana cost, not any alternative costs you used to pay for it.