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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.

If the second ability affects a creature, it doesn't cause that creature to lose any of its abilities. It just gives that creature flying as well.

The affected creature remains 3/1 and has flying as long as it has a feather counter on it. The effect continues even if Aven Mimeomancer leaves the battlefield.

You may put a feather counter on a creature that already has a feather counter on it. Doing so has no visible effect unless some other effect has changed the creature's power or toughness, or caused it to lose flying, since the last feather counter was put on it.

If all feather counters on a creature are moved to a different creature, the ability doesn't follow them. The first creature stops being affected by the ability because it no longer has a feather counter on it. The second creature is not affected by the ability because Aven Mimeomancer didn't target it.

The effect from the ability overwrites other effects that set power and/or toughness if and only if those effects existed before the ability resolved. It will not overwrite effects that modify power or toughness (whether from a static ability, counters, or a resolved spell or ability), nor will it overwrite effects that set power and toughness which come into existence after the ability resolves. Effects that switch the creature's power and toughness are always applied after any other power or toughness changing effects, including this one, regardless of the order in which they are created.