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$f(x) = \prod_{i=1}^{\infty} \frac{1+ 2\cos\left( \frac{2x}{3^i} \right)}{3}$

Let $f(x)$ denote an function as described above. The number of points where $|xf(x)| + | |x-2|-1|$ is non-differentiable in $x \in (0,3\pi)$ is $k$, find $k^2$.

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