Rat Poison III

Fix a positive integer $k$. You have $r$ very loyal rats that are willing to taste cups of wine for you! Exactly one of the $n$ cups has poison in it at random, which kills rats that drink it one hour after sipping. Otherwise, the rat drinks wine and feels a little tipsy, but it powers through and is fine. The rats are less busy now, so they can drink cups for up to $p$ hours. Find the maximum value of $n$ such that the poisonous cup can be identified with probability at least $1/k$ when $r = 3, k = 4,$ and $p = 3$. Assume the sipping process is instantaneous, and that rats can sip cups exactly at time $p$ hours, if needed.