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To make the watermelon juice, cut the watermelon flesh from the rind. Cut into cubes, removing any seeds. Place in a blender and process until liquefied. Pour through a fine-mesh strainer into a small bowl, pressing the pulp with a flexible spatula to extract as much juice as possible. Discard the solids You should have about 1 cup watermelon juice. Stir in the beet juice, lime juice, and agave sweetener.
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To make the carrot-apple juice, in another small bowl, stir together the carrot juice, apple juice, lime juice, and agave sweetener.
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Using a small measuring cup with a pour spout, fill each mold about one-fourth full with the watermelon juice mixture. Insert sticks and freeze until set, about 30 minutes. Fill each mold another fourth of the way full with the carrot juice mixture, pouring it around the sticks. Return to the freezer and freeze until set, 30 to 45 minutes. Repeat once more with each juice mixture to make 4 layers total. Freeze until firm, at least 6 hours or up to overnight.
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To unmold the pops, run hot water over the outsides of the molds for a few seconds, then gently pull the sticks.