A towering limit

Calculus Level 2

$\Large \lim_{n\to\infty} \left( n \sin \dfrac \pi n\right)^{\left( 1 + \frac \pi n\right)^n}$

The limit above has a closed form. Find the value of this closed form.

If this limit can be approximated as $3.19\times 10^\eta$ , where $\eta$ is an integer, find $\eta$ .

You may use a calculator for the final step of your calculation.