Do the kids need to keep their refillable drink mugs (the $35-for-7-day-cruise-mug) with them to get drinks while they're in their kids programs? Or do they get drinks as part of their club activities?
If your child needs a drink while in the kids clubs, you will be paged and can get it for them. Otherwise, they have water available for them in the clubs.
No, there would be no place to refill them anyway, and the kids cannot leave until you sign them out (unless your child is older and has sign out permission).
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