\[\lim_{n\to\infty} n^{-n^{2}} \left((n+1)\left(n+\frac{1}{2}\right)\left(n+\frac{1}{2^{2}}\right)\cdots\left(n+\frac{1}{2^{n-1}}\right)\right)^{n} = e^{k}\]

Find the value of \(k\).

\(\)

**Notation:** \(e \approx 2.71828\) denotes the Euler's number.

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