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Replacement effects are unaffected by Panharmonicon's ability. For example, a creature that enters the battlefield with one +1/+1 counter on it won't receive an additional +1/+1 counter.
If an artifact or creature entering the battlefield at the same time as Panharmonicon (including Panharmonicon itself) causes a triggered ability of a permanent you control to trigger, that ability triggers an additional time.
In some cases involving linked abilities, an ability requires information about "the exiled card." When this happens, the ability gets multiple answers. If these answers are being used to determine the value of a variable, the sum is used. For example, if Elite Arcanist's enters-the-battlefield ability triggers twice, two cards are exiled. The value of X in the activation cost of Elite Arcanist's other ability is the sum of the two cards' mana values. As the ability resolves, you create copies of both cards and can cast none, one, or both of the copies in any order.
You don't need to control the permanent entering the battlefield, only the permanent that has the triggered ability.
Panharmonicon affects a permanent's own enters-the-battlefield triggered abilities as well as other triggered abilities that trigger when that permanent enters the battlefield. Such triggered abilities start with "when" or "whenever."
If you control two Panharmonicons, an artifact or creature entering the battlefield causes abilities to trigger three times, not four. A third Panharmonicon causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on.
If a triggered ability is linked to a second ability, additional instances of that triggered ability are also linked to that second ability. If the second ability refers to "the exiled card," it refers to all cards exiled by instances of the triggered ability.
Abilities that apply "as [this artifact or creature] enters the battlefield," such as choosing a color with Gauntlet of Power, are also unaffected.
Panharmonicon's effect doesn't copy the triggered ability; it just causes the ability to trigger twice. Any choices made as you put the ability onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to put counters on a permanent, are also made individually.
