Given that \[\color{Green}{a+b+c =1} \\ \color{Blue}{a^2+b^2+c^2=2} \\ \color{Red}{a^3+b^3+c^3=3}.\] If \(a^4+b^4+c^4= \frac{a_1}{a_2}\) where \(a_1\), \(a_2\) are positive coprime integers, then find \(a_1+a_2\).

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