Ariawan's Ratio

Level pending

Given \[\mathrm{A}=\lim_{N \to \infty}\left|\frac{\sum\limits_{\alpha=0}^N \frac{(1 + i\pi)^\alpha}{\alpha!}}{\sum\limits_{\alpha=0}^N \frac{1}{\alpha!}}\right|,\] then determine the value of \(11^\mathrm{A}\)?

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