# Double maxed out

Algebra Level 5

For a positive real number $$r$$, let $$M(r)$$ be the largest possible value of $$F(x,y,z) = x^2 +yz^3$$ over all ordered triples of non-negative real numbers $$(x,y,z),$$ such that $$x^2+y^2+z^2=\sqrt{r}.$$

There is a value of $$r,$$ denoted by $$r^*$$, such that $$F(x,y,z)=M(r^*)$$ for two different triples $$(x,y,z)$$ which satisfy $$x^2+y^2+z^2=\sqrt{r^*}.$$ If $$r^* = \frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers, what is $$a + b$$?

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