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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

You may choose to pay the mana cost or an alternative cost for spells you cast, even if their mana value is less than or equal to the number of lands you control.

If a spell has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If you cast one card twice during a turn (such as by casting it as an Adventure and then casting it from exile as a creature), you've cast two spells.

If you cast a spell that was countered, it still counts as one of your two spells.

Fires of Invention will look at the entire turn to see if you've cast two spells yet that turn, even if Fires of Invention wasn't on the battlefield when those spells were cast. Notably, you can't cast Fires of Invention and then cast two more spells that turn; if Fires of Invention was your second (or third, or fourth) spell that turn, you can't cast another spell at all.

If an effect says that you may cast a card that you couldn't otherwise cast if you pay an alternative cost (such as the flashback keyword), you have to pay that alternative cost if you use that permission to cast the card. You can't choose to cast it but cast it without paying its mana cost.

If you cast a card “without paying its mana cost,” you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs (such as that of Fling), you must pay those to cast the card.

Fires of Invention doesn't impose any restrictions on playing lands or activating abilities.

If an effect allows or instructs you to cast a spell during another player's turn, you can't do so. The restriction of Fires of Invention takes precedence over that permission. Similarly, if an effect allows or instructs you to cast a third spell during your turn, you can't do so.