Let $A$ be an $n\times n$ matrix with complex entries. Prove that if $A^2=A^*$, then  $$\operatorname{rank}(A+A^*)=\operatorname{rank}(A).$$ (Here, $A^*$ is the conjugate transpose of $A$.)

(This is the last problem of this semester. Thank you for participating KAIST Math Problem of the Week.)