Stir sake and 1/4 cup white sugar together in a saucepan over medium heat; cook until sugar dissolves, about 5 minutes. Add soy sauce, 1/3 cup white sugar, orange marmalade, vinegar, garlic powder, sesame oil, ginger, cayenne pepper, and black pepper; stir to combine. Bring to a boil, reduce heat, and simmer until slightly reduced, 5 to 10 minutes.
Whisk cold water and cornstarch together in a small bowl until cornstarch is dissolved. Slowly stir cornstarch mixture into teriyaki sauce, 1 to 2 tablespoons at a time, until sauce is thickened. Remove from heat and cool.
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If you can find Mirin (Japanese sweet rice wine) use 1 cup of it instead of the sake-sugar mixture. Fresh ginger and garlic work if well-minced; add extra simmering time for the flavors to fully incorporate.