Given that $\color{#20A900}{a+b+c =1} \\ \color{#3D99F6}{a^2+b^2+c^2=2} \\ \color{#D61F06}{a^3+b^3+c^3=3}.$ 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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