# Mutant Ellipsoid

Calculus Level pending

The volume of the region bounded by $$16x^{2}+81y^{4}+64z^{6}=1$$ can be expressed, in simplest terms, as $${\Large \frac{\Gamma \left( \frac{A}{B} \right)\Gamma \left( \frac{C}{D} \right)}{E\times \Gamma \left( \frac{F}{G} \right)}\sqrt{\pi }}$$, where $$A, B, C, D, E, F$$ and $$G$$ are positive integers, with $$\gcd(A,B) = \gcd(C,D) = \gcd(F,G) = 1$$.

If we know that $$\dfrac AB, \dfrac CD , \dfrac FG < 1$$, find $$A+B+C+D+E+F+G$$.

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