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Put the rice in a blender and process for two to three minutes, until ground to a powder.
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Combine the pulverized rice, ground almonds and cinnamon stick. Cover tightly and leave to sit overnight or for at least six hours.
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Remove the cinnamon stick and place the rice mixture in the blender.
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Blend for four or five minutes until the mixture is smooth, powdery and no longer has a gritty texture. Add a third of the water and blend again for about 15 seconds.
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Strain into a jug through a dampened tea towel or coffee filter. Do not skip this step or the drink will have a chalky taste.
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Add most of the remaining water, little by little, as well as the vanilla and the sugar, stirring until the sugar has dissolved.
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If the mixture is too thick, add some additional water. Horchata should have the consistency of milk.
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Cover and refrigerate. Horchata will keep several days, refrigerated. Serve it chilled in tall glasses over ice.