Let \(f(x) = (x-a)(x-b)^3 (x-c)^5 (x-d)^7 \), where \(a,b,c,d\) real numbers with \(a<b<c<d\).

Consider the derivative of \(f(x)\), \(f'(x)\). Find the number of distinct real roots of \(f'(x) = 0 \).

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