If a polynomial function \( f \) satisfies the relation \[ \log _{2}(f(x))=\log _{2}\left(2+\f...

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If a polynomial function \( f \) satisfies the relation
\[
\log _{2}(f(x))=\log _{2}\left(2+\frac{2}{3}+\frac{2}{9}+\ldots \infty\right) \cdot \log _{3}\left(1+\frac{f(x)}{f\left(\frac{1}{x}\right)}\right)
\]
and \( f(10)=1001 \), then the value of \( f(20) \) is
(a) 2002
(b) 7999
(c) 8001
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