Show that is $A$ is normal then $A$ commutes with some normal matrix with distinct eigenvalues.

Question

Let's $A$ be square matrix, i need a help to show that

if $A$ is normal then there existe $X$ a normal matrix with distinct eigenvalues such that $AX=XA$

if $A$ is normal then there existe $B$ a self-adjoint matrix with distinct eigenvalues such that $AB=BA$

I think about that, if $A$ is normale then $A^{*}=P(A)$ for some polynomial $P$, then $A$ commute with $P(A)$, so i took $X=P(A)$ but the problem is in eigenvalues, and i took $B=A^{*}A$

Thank you !