The components of the three-dimensional angular momentum operator $\hat{\mathbf{L}}$ are defined as follows:

$$\hat{L}_{x}=-i \hbar\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}\right) \quad \hat{L}_{y}=-i \hbar\left(z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}\right) \quad \hat{L}_{z}=-i \hbar\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right)$$

Given that the wavefunction

$$\psi=(f(x)+i y) z$$

is an eigenfunction of $\hat{L}_{z}$, find all possible values of $f(x)$ and the corresponding eigenvalues of $\psi$. Letting $f(x)=x$, show that $\psi$ is an eigenfunction of $\hat{\mathbf{L}}^{2}$ and calculate the corresponding eigenvalue.