Let \(n\) be a non-zero real number (not necessarily positive). Find the minimum value of

\[\prod_{i = 1}^{8}(n^{2^{i}} + 1) + \prod_{i = 1}^{8}(\frac{1}{n^{2^{i}}} + 1).\]

According to my thoughts, you need 3 things for solving math problems:

Knowledge

Imagination

Solve this problem to figure that one out.

Hint: The third word starts with \(T\) and ends with \(S\). It may depend on each person whether this hint has actually helped or not.

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