Step 1: Interpret the integrand correctly
The integrand is of the form
\[
(\nabla \times \vec F)\cdot \hat n,
\]
so the surface integral can be evaluated using Stokes’ Theorem:
\[ \iint_S (\nabla \times \vec F)\cdot \hat n\,dS = \oint_{\partial S} \vec F\cdot d\vec r. \]
Step 2: Find the curl of the vector field
Given
\[
\vec F=(y-z)\hat i+(z-x)\hat j+(x-y)\hat k,
\]
compute its curl:
\[ \nabla\times\vec F= \begin{vmatrix} \hat i & \hat j & \hat k\\ \partial_x & \partial_y & \partial_z\\ y-z & z-x & x-y \end{vmatrix} = (1,1,1). \]
Thus, the curl is a constant vector.
Step 3: Convert the surface integral into a flux integral
Since \(\nabla\times\vec F=(1,1,1)\),
\[ \iint_S (\nabla\times\vec F)\cdot \hat n\,dS = \iint_S (1,1,1)\cdot \hat n\,dS. \]
This represents the flux of a constant vector through the surface \(S\). By the divergence theorem, this flux equals the volume integral of \(\nabla\cdot(1,1,1)=3\) over the solid \(V\) enclosed by \(S\) and its base.
Step 4: Compute the enclosed volume
The surface \(S\) is a spherical cap of the sphere
\[
x^2+y^2+(z-1)^2=9
\]
between \(z=1\) and \(z=4\).
The cap has height \(h=3\) and radius \(R=3\). The volume of a spherical cap is:
\[ V=\frac{\pi h^2}{3}(3R-h). \]
Substitute \(R=3,\ h=3\):
\[ V=\frac{\pi\cdot9}{3}(9-3)=3\pi\cdot6=18\pi. \]
Step 5: Evaluate the surface integral
\[ \iint_S (\nabla\times\vec F)\cdot \hat n\,dS = 3\times V = 3\times18\pi = 54\pi. \]
Step 6: Final computation
The required quantity is:
\[ \frac{1}{\pi}\left|\iint_S (\nabla\times\vec F)\cdot \hat n\,dS\right| = \frac{1}{\pi}(54\pi)=18. \]
Final Answer:
\[ \boxed{18} \]