The Uneven Pack Train

A mule and an ass are walking to market, each carrying a whole number of sacks. The mule notices that their loads could be rearranged in two revealing ways. “If I took one sack from you,” the mule tells the ass, “I would then be carrying exactly twice as many sacks as you.” The mule continues, “But if I gave one of my sacks to you instead, our loads would become equal.” Neither animal begins empty-handed, and only the single sack mentioned in each possible exchange is moved. How many sacks is the mule carrying before either exchange takes place?