Question 13
Let $S$ be the set of real solutions $(x,y,z)$ of $$\frac{4x^2}{1+4x^2}=y,\qquad\frac{4y^2}{1+4y^2}=z,\qquad\frac{4z^2}{1+4z^2}=x.$$ For each $(x,y,z)\in S$, define $m(x,y,z)=2000(|x|+|y|+|z|)$. Determine the maximum value of $m(x,y,z)$ over all $(x,y,z)\in S$.