Bashing unavailable - part 3

Algebra Level 4

Given that

\(\color{Green}{a+b+c =1} \)

\(\color{Blue}{a^2+b^2+c^2=2}\)

\(\color{Red}{a^3+b^3+c^3=3}\)

Then \(a^4+b^4+c^4= \frac{a_1}{a_2}\) .... where \(a_1\) and \(a_2\) are positive coprime integers.

And \(a^5+b^5+c^5=a_3\) ...... where \(a_3\) is an integer.

And \(a^6+b^6+c^6= \frac{a_4}{a_5}\) ....where \(a_4\) and \(a_5\) are positive coprime integers.

Find \(a_1+a_2+a_3+a_4+a_5\).

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