Don't Exceed but Evaluate

Calculus Level 4

\[ \displaystyle \lim_{n \to \infty} n^{-n^2} \Bigg [ \bigg (n + 1 \bigg ) \left (n+ \frac 1 2 \right ) \left (n + \frac 1 4 \right ) \ldots \left ( n + \frac {1}{2^{n-1}} \right ) \Bigg ]^n \]

If the above limit equals to \( \alpha \), what is the value of \( \lfloor 1000 \alpha \rfloor\)?

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