\[\begin{eqnarray} a^{2}+b^{2}+c^{2}&=&2014 \\ a+2b+3c&=&86 \end{eqnarray} \]

Let \(a,b,c\) be real numbers such that they satisfy the system of equations above.

The maximum value of \(a\) can be expressed as \(\dfrac{m}{n}\), where \(m\) and \(n\) are coprime positive integers. Find the value of \(m+n\).

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