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Geometry Level 4

\[ \begin{cases} a^{2}+b^{2}+c^{2}=50 \\ a^{3}+b^{3}+c^{3}=216 \\ ab+bc+ca=47 \end{cases} \]

For \(\triangle ABC\) with sides \(a,b\) and \(c\), let \(R\) and \(r\) denote the radius of circumcircle and incircle of \(\triangle ABC\). If the values of \(a,b\) and \(c\) satisfy the system of equations above, find the value of \(R \times r\).

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