Given that \[f(x)=x^4+10 x^3+35 x^2+50 x+24\] and that \[S=\prod_{i=1}^{50} \left( \prod_{j=1}^{50} (f(i)-f(j)) \right),\] find the value of \[ \left(\frac{S \cdot (\sum_{k=1}^{f(50)} f(k))}{\sum_{k=1}^{f(49)}f(k)}+f(1) \right) \pmod{1000}\]

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