Minima and polynomials?

Algebra Level 5

Let \(a> 0\) be a real number. It is known that the following equation in \(x\) has a real root:

\[x^{6}+3ax^{5}+(3a^{2}+3) x^{4}+6ax^{3}+(3a^{2}+3) x^{2}+3ax+1+(ax)^{3}=0.\]

Find the minimum possible value of \(1000a\).

This problem is part of the set ... and polynomials

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