Let $p \geq 3$ be a prime, and $k\geq 1$ is an integer. Suppose that $A \subseteq F^n_p$ is a subset of $F^n_p$ that does not contain $k$ different (non-trivial) three-term arithmetic progression with the same middle term (which are disjoint apart from the common middle term). Prove that $|A|\leq 10\sqrt{k} ·(\Gamma _p)^n$, where $$\Gamma_p=\min_{0< t< 1} \frac{1+p+\dots+p^{t-1}}{t^{\frac{(p-1)}{3}}}$$
I can show that if $k=1$, the inequality holds. But, I don't know how to generalize it. Any thoughts?