How to prove this identity involving dot product of solid angle and gradient

Question

How to prove following for $n\geq0$.
$$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$ Where, at any point $\vec{r}$, the $\vec{\Omega}$ can be described by the polar angle $\theta$ measured with respect to the z axis and an azimuthal angle $\phi$ measured with respect to the x axis, i.e.
$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$,
$\cos\theta=\vec{\Omega}\cdot\hat{k}$,
$\cos\phi=sin^{-1}\theta(\vec{\Omega}\cdot\hat{i})$
Here, $\vec{\Omega}$ is a unit vector situated at $\vec{r}$ and pointing in the direction ($\theta,\phi$) which is different than the direction of $\vec{r}$.