\[\large{ \begin{cases} a+b+c=1 \\ a^2+b^2+c^2 = 2 \\ a^3 + b^3+c^3 = 3 \\ \end{cases} } \]

Let \(a,b\) and \(c\) satisfy the system of equations above.

Given that \(a^4 + b^4 + c^4 = \dfrac{a_1}{a_2} \) for coprime positive integers \(a_1 \) and \(a_2 \). And \(a^5 + b^5 + c^5 = a_3 \).

Find the value of \(a_1 + a_2 + a_3\).

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