Suppose $a$ and $b$ are positive integers with a curious property: $(a^3 - 3ab + \frac{1}{2})^n + (b^3 + \frac{1}{2})^n$ is an integer for at least 3, but at most finitely many different choices of positive integers $n$. What is the least possible value of $a+b$?

Let's think step by step and output the final answer within \boxed{}.
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