\( \qquad \quad \displaystyle I_n(x) = \frac{2^{n+1}}{\sqrt{\pi}}e^{x^2}\int\limits_0^{\infty}\frac{t^n \cos(2xt - \frac{n \pi}{2})}{e^{t^2}} dt\)

Then , the following integral :

\(\displaystyle \int\limits_{-\infty}^{\infty}e^{-(1-t)^2}I_2(t) dt = k\)

Calculate \(\lfloor k \rfloor\).

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