(Not) just your average problem

Calculus Level 5

\[\large{ \begin{align} x+y+z=0 \\ x^2+y^2+z^2=1\end{align}}\]

Consider the curve \(C\) defined by the two equations above. Find the average value of \(f(x,y,z)=x^2y^2z^2\) on \(C\).

Enter the reciprocal of this average.

Inspiration

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