Moving along a Circle

Geometry Level 2

For point \((x,y)\) moving along the circle \(x^2+y^2=4\), let \(a\) and \(b\) be the maximum and minimum values, respectively, of \[\frac{1}{1+x^2}+\frac{1}{1+y^2}.\] What is the value of \(ab\)?

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