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I managed to defeat the mage with a double-shift strategy. I was shocked myself. In the first three rounds, I literally had nothing to do, just sitting there like a useless unit. The hardest part was not to type a sarcastic "thank you" when the mage placed an ice golem, against which I sacrificed my units to execute cockroach combos. And in the end, I defeated her with a fairy and a bird from a spell.