Minima and polynomials?

Algebra Level 5

Let a>0a> 0 be a real number. It is known that the following equation in xx has a real root:

x6+3ax5+(3a2+3)x4+6ax3+(3a2+3)x2+3ax+1+(ax)3=0.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 1000a1000a.

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

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