An algebra problem by Kshetrapal Dashottar

Algebra Level 4

\[ \begin{cases} a^3 = 3 (b^2+c^2)-25 \\ b^3 = 3(c^2+a^2) -25 \\ c^3 = 3 (a^2 + b^2) - 25 \end{cases} \]

Let \(a,b\) and \(c\) be distinct real numbers satisfying the system of equations above. Find the product \(abc \).

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