I'm not doing Newton's sum all the way!

Algebra Level 5

\[ \begin{cases} a_1 + a_2 + \cdots +a_9 = 1 \\ a_1^2 + a_2^2 + \cdots +a_9 ^2= 2 \\ a_1^3 + a_2^3 + \cdots +a_9 ^3= 3 \\ \qquad \vdots \\ a_1^9 + a_2^9 + \cdots +a_9 ^9 = 9 \end{cases} \]

Let \(N\) be the unique 9-tuple of complex numbers \((a_1,a_2,\ldots , a_9) \) that satisfies the system of equations above.

What is the smallest positive value of \(k\) such that \( k \times (a_1^{10} + a_2^{10} + \cdots +a_9 ^{10}) \) is an integer?

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