# Tricky Points of Non-Differentiability

Calculus Level 4

$f(x) = \displaystyle \prod_{k=1}^{\infty} \left( \dfrac{1+ 2\cos\left( \dfrac{2x}{3^k} \right)}{3}\right)$

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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