Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$. If $\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of $$\{ z: \lvert f(z)\rvert<1\}$$ then determine the infimum of $\Lambda(f)$.

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