Instant
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The copies of Radstorm created by its storm ability are put directly onto the stack. They aren’t cast and won’t be counted by other spells with storm cast later in the turn.
Spells cast from zones other than a player’s hand and spells that were countered or otherwise failed to resolve are counted by the storm ability.
A copy of a spell can be countered like any other spell, but it must be countered individually. Countering a spell with storm won’t affect the copies.
You can choose any permanent that has a counter, including ones controlled by opponents. You can choose any player who has a counter, including opponents. You can’t choose cards in any zone other than the battlefield, even if they have counters on them.
You don’t have to choose every permanent or player that has a counter—only the ones you want to add counters to. Since “any number” includes zero, you don’t have to choose any permanents at all, and you don’t have to choose any players at all.
If a player or permanent has more than one kind of counter on it, and you choose for it to get additional counters, it must get one of each kind of counter it already has. You can’t have it get just one kind of counter it already has and not the others.
Players can respond to a spell or ability whose effect includes proliferating. Once that spell or ability starts to resolve, however, and its controller chooses which permanents and players will get new counters, it’s too late for anyone to respond.
Some spells and abilities that cause you to proliferate may require targets. If each target chosen is an illegal target as that spell or ability tries to resolve, it won’t resolve. You won’t proliferate.