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{T}, Discard your hand, Sacrifice Diamond Lion: Add three mana of any one color. Activate only as an instant.
Even though you can only activate this as an instant, it is still a mana ability. It doesn't use the stack and cannot be responded to.
You can't cast an instant (or activate an ability as an instant) while casting a spell. Therefore, much like Lion's Eye Diamond, you can't activate the ability intending to use the mana to cast a spell from your hand.
