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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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