Consider a set of complex numbers \( \omega \) such that \( \omega^6 = 1 \) and \( \omega \neq 1 \). If three complex numbers \( z_1, z_2, z_3 \) are chosen independently and uniformly at random from the set \( \{ \omega, \omega^2, \omega^3, \omega^4, \omega^5 \} \), what is the probability that the product \( z_1 z_2 z_3 = 1 \)? Let's think step by step and output the final answer within \boxed{}.
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