Add strawberries and 2 cups water to blender, and blend until smooth. Pour the mixture into a pitcher, and set aside. Add limes, remaining sugar, and remaining water to the blender and pulse 6-8 times. Pour through a strainer into the pitcher with the strawberry mixture. Mix well, then serve. (If you're really bugged by strawberry seeds in your drink, you could pour the whole thing through a strainer before serving.)
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