\[ \large \color{red}{3}x^{5} \color{blue}{=} \color{red}{1}x^{4} + \color{red}{4}x^{3} + \color{red}{1}x^{2} + \color{red}{5}x + \color{red}{9}\]

The above polynomial has exactly one real root \( \alpha\) . Find \( \lfloor 1000 \times \alpha \rfloor \).

**Notation:** \( \lfloor x \rfloor \) denotes the greatest integer smaller than or equal to \(x\). This is known as the greatest integer function.

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