Trigonometry Related Part III

Geometry Level 5

\[\frac{a}{1+a^2}+\frac{b}{1+b^2}+\frac{3c}{\sqrt{1+c^2}}\]

Given that \(a,b\) and \(c\) are positive real numbers satisfying \(ab+bc+ca=1\).

If the maximum value of the expression above can be expressed as \(\sqrt{M}\) for positive integer \(M\), determine \(M\).

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