A question on regular local rings (of positive characteristic ) of dimension $2$

Question

Let $R$ be a regular local ring of dimension $2$ and of characteristic $p>0$.

How to show that for every $f_1,f_2,f_3 \in R$, $\exists 0\ne c\in R$ and $n_0\in \mathbb N$ such that $c(f_1f_2f_3)^{2p^n} \in (f_1^{3p^n}, f_2^{3p^n}, f_3^{3p^n}),\forall n >n_0$ ?