\[\begin{align} P_1 &: y = x^2 - ( 6 + 2 \sqrt{5} ) x + ( 22 + 6 \sqrt{5} ) \\ P_2 &: y = -x^2 + ( 6 + 2 \sqrt{5} ) x - ( 6 + 6 \sqrt{5} ) \end{align} \]

If the equation of the common tangent of the two parabolas \(P_1\) and \(P_2\) above is given by \(L : ax+by+c=0 \), where \(a,b \ge 0\) and \(|b|\) and \(|c|\) are coprime, find \(a+b+c\).

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