Bashing unavailable - part 4

Algebra Level 4

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

Let \(a,b\) and \(c\) be complex numbers satisfying the system of equations above.

Given that the following equations are true as well for positive integers \(a_1, a_2, a_3, a_4, a_5, a_6\) and \(a_7\), with \(\gcd(a_1, a_2) = \gcd(a_3, a_4) = 1 = \gcd(a_6, a_7) = 1 \).

Find \(a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7\).

\[ \begin{cases} a^4+b^4+c^4 = \dfrac{a_1}{a_2} \\ a^5+b^5+c^5 = a_3 \\ a^6+b^6+c^6 = \dfrac{a_4}{a_5} \\ a^7+b^7+c^7 = \dfrac{a_6}{a_7} \end{cases} \]

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