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Gathering Stone's second ability applies only to generic mana in the total cost of spells you cast of the chosen type.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying (such as a flashback cost), add any cost increases (such as kicker costs), then apply any cost reductions (such as that of this ability). The mana value of the spell is determined by only its mana cost, no matter what the total cost to cast that spell was.

You don't have to reveal the card you look at with Gathering Stone's third ability even if it's a card of the chosen type. If you choose not to reveal it this way, you can still put it into your graveyard.