Let \(f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)\). The intersection of the domain of \(f(x)\) with the interval \([0,1]\) is a union of \(n\) disjoint open intervals. What is \(n\)?

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