On ternary Egyptian fractions with prime denominator

Given a positive integer $n$ we let $A_k(n)$ be the number of positive integers $a$ such that $\frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\cdots +\frac{1}{m_k}$ for some $m_1,m_2,\dots ,m_k\in \mathbb{N}$. We show that $x(\log x)^3\ll \sum_{p\le x} A_3(p)\ll x(\log x)^5$as$x\rightarrow\infty$.