Combinatorial design using finite field

Question

Let $\mathcal{F} = \{A_1, \cdots, A_n\}$ be a family of sets with the following conditions.

1. For all $i$, $|A_i| = p^k+1$ where $p$ is prime and $k$ is an integer
2. If $i \neq j$, $|A_i \cap A_j| = 1$
3. $\bigcap_{i=1}^n A_i = \varnothing$

I should find the largest $n$, with such $\mathcal{F}$ exists, where $p$, $k$ are fixed.

Considering the conditions about $p^k$ and the intersection, I think this is the problem using the finite field. I might construct the family by thinking about "lines" on finite field: namely, let $A_i$'s correspond to the sets of tuples $(x,y) \in GF(p^k)^2$ such that $y = ax+b$.

This gives me a vague idea about the answer: $n = p^{2k} + O(p^k) + \cdots$? However I cannot specify those foggy thoughts to actual proof.

Thanks in advance for any form of help, hint, or solution.