Let z1 and z2 be relatively prime positive integers with z2 < z1. Each number in the set S = {1, 2, 3, ... , z1 -1} is assigned to either class A or class B. For each k in S, both k and z1-k are assigned to the same class. For each k in S not equal to z2, both k and |k- z2| are assigned to the same class. If two numbers are randomly chosen from S, what is the probability that they belong to the same class ?

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