To evaluate the integral \(\frac{1}{\pi} \left| \iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS \right|\) where \( S \) is given by the surface of the sphere \(x^2 + y^2 + (z - 1)^2 = 9\) for \(1 \leq z \leq 4\), we identify that the surface \( S \) is a sphere centered at \((0,0,1)\) with radius 3. The vector field \(\vec{F} = (y - z)\hat{i} + (z - x)\hat{j} + (x - y)\hat{k}\) needs to be handled appropriately across the surface \( S \).
Using Stokes' theorem, \(\iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS = \iiint_V (\nabla \cdot \vec{F}) \, dV\) where \(V\) is the volume enclosed by \( S \) and within the bounds \(1 \leq z \leq 4\). Upon calculation, \(\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(y-z) + \frac{\partial}{\partial y}(z-x) + \frac{\partial}{\partial z}(x-y) = 0\).
Due to \(\nabla \cdot \vec{F} = 0\), the volume integral evaluates to zero:
\[\iiint_V (\nabla \cdot \vec{F}) \, dV = 0\]
Thus, \(\iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS = 0\).
Consequently,
\(\frac{1}{\pi} \left| \iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS \right| = \frac{1}{\pi} \cdot 0 = 0\).
Hence, the integer value is 0. However, interpreting the range \(18, 18\), if there is a miscalculation in interpreting instructions or additional effects not addressed (potentially relating to other symmetries or field influences), we might need that specific value. But as computed, the logical rigorous process yields 0 barring additional context-driven insights.