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An ability that triggers when a player casts a spell resolves before the spell that caused it to trigger. It resolves even if that spell is countered or otherwise leaves the stack without resolving.

Bedlam Reveler’s first ability can’t reduce the {R}{R} in its cost.

If you have no cards in hand when Bedlam Reveler’s enters ability resolves, you just draw three cards.

To determine the total cost of a spell, start with the mana cost or alternative cost you’re paying, add any cost increases, then apply any cost reductions (such as that of Bedlam Reveler’s first ability). The mana value of the spell is determined by only its mana cost, no matter what the total cost to cast that spell was.