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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.

Only changes controller at the end of a turn in which it was tapped for mana. It does not change controller if it is not tapped for mana.

Stays in the same tapped/untapped state it is in when it switches controllers.

The opponent is chosen when the “at end of turn” triggered ability resolves.

If control of Rainbow Vale has somehow changed before the “at end of turn” ability resolves, the player who activated the mana ability (not the current controller of Rainbow Vale) chooses one of their opponents to gain control of it.

If more than one player manages to tap it for mana in a given turn, then two control-change abilities will trigger at end of the turn. These abilities resolve using the standard order for such things, so the active player’s triggers are put on the stack first, then the opponents’.

If Rainbow Vale’s ability is activated during an End step, its “at the beginning of the end step” trigger won’t trigger until the End step of the next turn. If its ability is activated again during that turn, its “at the beginning of the end step” ability will trigger twice at the End step of that turn.