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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.

Paying the activation cost of Elixir of Immortality’s ability doesn’t cause it to leave the battlefield. If you have a way to untap it, you can activate the ability multiple times in response to itself.

If Elixir of Immortality is in your graveyard at the time the ability resolves, you’ll still wind up shuffling it into your library because you shuffle your entire graveyard into your library. If it’s anywhere else by that time, including in another player’s graveyard, it remains where it is and you shuffle just your graveyard into your library.

As the ability resolves, you’ll shuffle Elixir of Immortality into its owner’s library directly from the battlefield, if it’s still there.

If you gain control of another player’s Elixir of Immortality and activate it, the Elixir of Immortality will be shuffled into its owner’s library and the cards in your graveyard will be shuffled into your library. You are considered to have shuffled each affected library (even if, as a shortcut, each player physically shuffles their own library).