Mix the sugars, juice, and cider thoroughly in the pot and add the spices.
Bring to a boil, stirring to make sure the sugar is dissolved.
Cool until room temperature (because the heat will boil the alcohol away if you add it now), and once ready strain out the solids and add the grain alcohol.
Store in a pitcher or jug in the fridge with one of the cinnamon sticks. It can be enjoyed warm or chilled!