Artifact
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If multiple replacement effects would modify the mana an ability you control produces, choose one to apply. After that, determine if any others are applicable. A replacement effect can’t apply to the same event more than once.
Damping Sphere’s second ability counts spells that were cast during a turn even if Damping Sphere wasn’t on the battlefield as they were cast. For example, if Damping Sphere itself is the third spell you cast in a turn, the next spell you cast costs 
more to cast.
To determine the total cost of a spell, start with the mana cost or alternative cost (such as a flashback cost) you’re paying, add any cost increases, then apply any cost reductions. The mana value of the spell is determined only by its mana cost, no matter what the total cost to cast the spell was.
instead of any other type and amount.
more to cast for each other spell that player has cast this turn.