Associated primes and their heights

Question

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal $p\in\operatorname{Ass}_{S}(R)$, $\operatorname{ht}p=t$. Then, for any prime ideal $p \in \operatorname{Ass}_{S}(R/aR)$, $\operatorname{ht}p=t+1$?