Trigonometry Related Part IV

Geometry Level 5

\[\frac{1}{a^2+1}+\frac{4}{b^2+4}+\frac{9}{c^2+9}\]

Given that \(a,b\) and \(c\) are positive real numbers satisfying \(3ab+bc+2ca=6\).

If the maximum value of the expression above can be expressed as \(\frac{A}{B}\) for coprime positive integers \(A\) and \(B\), determine \(A+B\).

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