Not a Typo 2

Algebra Level 4

Real numbers \(a,b,c\) are such that \(ab+bc+ca=1\) and the maximum value of \(\frac{1}{1+a^2}+\frac{1}{1+b^2}+\frac{6c}{1+c^2}\) is \(m+\frac{p\sqrt{q}}{r}\), where \(m,p,q,r\) are all positive integers with \(\gcd(p,r)=1\) and \(q\) square free. Find the value of \(m+p+q+r\).

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