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Both of Quest for Pure Flame’s abilities interact with any kind of damage, not just combat damage.

If a spell causes damage to be dealt, that spell will always identify the source of the damage. In most cases, the source is the spell itself. For example, Lightning Bolt says “Lightning Bolt deals 3 damage to any target.”

If a spell or ability divides damage among multiple recipients (such as Arrow Volley Trap does), the damage is divided before Quest for Pure Fire’s effect doubles it. The same is true for combat damage.

If an ability causes damage to be dealt, that ability will always identify the source of the damage. The ability itself is never the source. However, the source of the ability is often the source of the damage. For example, Prodigal Pyromancer’s ability says “Prodigal Pyromancer deals 1 damage to any target.”

If multiple effects modify how damage will be dealt, the player who would be dealt damage or the controller of the creature that would be dealt damage chooses the order to apply the effects. For example, Mending Hands says, “Prevent the next 4 damage that would be dealt to any target.” Suppose a spell controlled by a player who has activated Quest for Pure Fire’s second ability would deal 5 damage to a player who has cast Mending Hands targeting themselves. The player who would be dealt damage can either (a) prevent 4 damage first and then let Quest for Pure Fire’s effect double the remaining 1 damage, taking 2 damage, or (b) double the damage to 10 and then prevent 4 damage, taking 6 damage.

If the source of the damage is a permanent, Quest for Pure Flame checks whether you control that permanent at the time that damage is dealt. If the permanent has left the battlefield by then, its last known information is used. If the source of the damage is a spell, whether you control it is obvious. If the source of the damage is a card in some other zone (such as a cycled Jund Sojourners), Quest for Pure Flame checks whether you’re the card’s owner, rather than whether you’re its controller.

If you activate the second ability of more than one Quest for Pure Flame in the same turn, the effects are cumulative. Two such effects will cause damage from sources you control to be multiplied by four; three such effects will cause damage from sources you control to be multiplied by eight.

The first ability will trigger multiple times if multiple sources you control deal damage to a single opponent at once, or if a single source you control deals damage to multiple opponents at once.

The second ability affects all sources you control that would deal damage to a permanent or player at any point in the rest of the turn.

The source of combat damage is the creature that dealt it.

When the first ability resolves, you may put just one quest counter on Quest for Pure Flame. It doesn’t matter how much damage the source dealt.