\[ \large \int \sqrt{\dfrac{1-\cos x}{1 + \cos x}} \left ( \prod_{r=0}^n (1 + \sec (2^r x) ) \right) \, dx \]

Let \(f_{(n)} \) be a function as described above for non-negative integer \(n\). Let \(g(x) = f_{(4)} (x) \). If the solution to \(g(x) =1\) can be written in the form of \( \dfrac1{16} \arccos (e^k) \), find \((|k| - 9)! \).

\(f_{(n)}(0)=0\), and take \( \cos x \ne -1, \cos (nx) \ne 0 \).

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