Unexpectedly real

Algebra Level 3

Let \(a,b,c\) be real numbers so that \(a+b+c=1\). Find the maximum value of \(\frac { a }{ 1+a^{ 2 } } +\frac { b }{ 1+{ b }^{ 2 } } +\frac { c }{ 1+{ c }^{ 2 } } \).

If the answer can be written as \(\frac{x}{y}\), where \(x,y\) are positive integers so that \(\gcd(x,y)=1\), find \(\left| x-y \right| \).

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