Is it true that for $p\neq 2 $, $k\ge 1$ and $n_1,n_2,\dots,n_k\ge 1$, the cyclic group $\mathbb{Z}_p$ has no continuous free action on $ \mathbb{C}P^{n_1} \times \mathbb{C}P^{n_2} \times \cdots \times \mathbb{C}P^{n_k}$?

How to prove it?

Thank you so much in advance.